41, Detailed Solutions to Meta-Knowledge puzzle inspired by the the Puzzle "\
Sums and Products"
in Peter Winkler's Wonderful book "Mathematical Mind-Benders"
By Shalosh B. Ekhad
Here is a detailed solution to the original solution
We will solve the following challenging "meta-knowledge" puzzle from Peter W\
inkler's book Mathematical Mind-Benders (p. 114).
Winkler mentions that the puzzle appeared in Martin Gardner's Sci. Amer. co\
lumn in 1979.
Winkler credits Steve Fenner and Bill Gottesman for this version.
Puzzle: All integers greater than one but less than, 100, are put into a hat \
and two are drawn; Samantha is given their sum and Porfirio
their product. Sanantha says:
" I can see that you don't know the numbers. "
Porfirio replies: NOW I do know them.
Now Samantha says: NOW I do TOO!
What are the numbers?
----------------------------------------------------------------------
Answer: the UNIQE possible pair is, [4, 13]
Detailed Solution
Part I: The Samantha Set
Samantha KNOWS from her number (that happens to be the sum of the unknown nu\
mbers),
that she does not know individualy) that Porfirio does NOT KNOW (for sure) \
the numbers.
We claim that the only possible numbers that Samantha can possibly have, whe\
re the numbers from the hat are distinct and less than, 100, are
{11, 17, 23, 27, 29, 35, 37, 41, 47, 53}
Let's prove it.
------------------------------------------------------
There is no way that Samantha can have the number, 5
If Samantha's number (the sum of the numbers) is, 5,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 3, their product, 6,
would enable Porfirio to deduce that the numbers are, 2, 3
------------------------------------------------------
There is no way that Samantha can have the number, 6
If Samantha's number (the sum of the numbers) is, 6,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 4, their product, 8,
would enable Porfirio to deduce that the numbers are, 2, 4
------------------------------------------------------
There is no way that Samantha can have the number, 7
If Samantha's number (the sum of the numbers) is, 7,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 5, their product, 10,
would enable Porfirio to deduce that the numbers are, 2, 5
------------------------------------------------------
There is no way that Samantha can have the number, 8
If Samantha's number (the sum of the numbers) is, 8,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 5, their product, 15,
would enable Porfirio to deduce that the numbers are, 3, 5
------------------------------------------------------
There is no way that Samantha can have the number, 9
If Samantha's number (the sum of the numbers) is, 9,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 7, their product, 14,
would enable Porfirio to deduce that the numbers are, 2, 7
------------------------------------------------------
There is no way that Samantha can have the number, 10
If Samantha's number (the sum of the numbers) is, 10,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 8, their product, 16,
would enable Porfirio to deduce that the numbers are, 2, 8
------------------------------------------------------
Samantha may have the number, 11
If Samantha's number (the sum of the numbers) is, 11,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 9,
then Porfirio only knows that the numbers belong to the set,
{[2, 9], [3, 6]}, that has more than one element.
If the numbers are, 3, 8,
then Porfirio only knows that the numbers belong to the set,
{[2, 12], [3, 8], [4, 6]}, that has more than one element.
If the numbers are, 4, 7,
then Porfirio only knows that the numbers belong to the set,
{[2, 14], [4, 7]}, that has more than one element.
If the numbers are, 5, 6,
then Porfirio only knows that the numbers belong to the set,
{[2, 15], [3, 10], [5, 6]}, that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 12
If Samantha's number (the sum of the numbers) is, 12,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 9, their product, 27,
would enable Porfirio to deduce that the numbers are, 3, 9
------------------------------------------------------
There is no way that Samantha can have the number, 13
If Samantha's number (the sum of the numbers) is, 13,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 11, their product, 22,
would enable Porfirio to deduce that the numbers are, 2, 11
------------------------------------------------------
There is no way that Samantha can have the number, 14
If Samantha's number (the sum of the numbers) is, 14,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 11, their product, 33,
would enable Porfirio to deduce that the numbers are, 3, 11
------------------------------------------------------
There is no way that Samantha can have the number, 15
If Samantha's number (the sum of the numbers) is, 15,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 13, their product, 26,
would enable Porfirio to deduce that the numbers are, 2, 13
------------------------------------------------------
There is no way that Samantha can have the number, 16
If Samantha's number (the sum of the numbers) is, 16,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 13, their product, 39,
would enable Porfirio to deduce that the numbers are, 3, 13
------------------------------------------------------
Samantha may have the number, 17
If Samantha's number (the sum of the numbers) is, 17,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 15,
then Porfirio only knows that the numbers belong to the set,
{[2, 15], [3, 10], [5, 6]}, that has more than one element.
If the numbers are, 3, 14,
then Porfirio only knows that the numbers belong to the set,
{[2, 21], [3, 14], [6, 7]}, that has more than one element.
If the numbers are, 4, 13,
then Porfirio only knows that the numbers belong to the set,
{[2, 26], [4, 13]}, that has more than one element.
If the numbers are, 5, 12,
then Porfirio only knows that the numbers belong to the set,
{[2, 30], [3, 20], [4, 15], [5, 12], [6, 10]},
that has more than one element.
If the numbers are, 6, 11,
then Porfirio only knows that the numbers belong to the set,
{[2, 33], [3, 22], [6, 11]}, that has more than one element.
If the numbers are, 7, 10,
then Porfirio only knows that the numbers belong to the set,
{[2, 35], [5, 14], [7, 10]}, that has more than one element.
If the numbers are, 8, 9,
then Porfirio only knows that the numbers belong to the set,
{[2, 36], [3, 24], [4, 18], [6, 12], [8, 9]},
that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 18
If Samantha's number (the sum of the numbers) is, 18,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 13, their product, 65,
would enable Porfirio to deduce that the numbers are, 5, 13
------------------------------------------------------
There is no way that Samantha can have the number, 19
If Samantha's number (the sum of the numbers) is, 19,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 17, their product, 34,
would enable Porfirio to deduce that the numbers are, 2, 17
------------------------------------------------------
There is no way that Samantha can have the number, 20
If Samantha's number (the sum of the numbers) is, 20,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 17, their product, 51,
would enable Porfirio to deduce that the numbers are, 3, 17
------------------------------------------------------
There is no way that Samantha can have the number, 21
If Samantha's number (the sum of the numbers) is, 21,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 19, their product, 38,
would enable Porfirio to deduce that the numbers are, 2, 19
------------------------------------------------------
There is no way that Samantha can have the number, 22
If Samantha's number (the sum of the numbers) is, 22,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 19, their product, 57,
would enable Porfirio to deduce that the numbers are, 3, 19
------------------------------------------------------
Samantha may have the number, 23
If Samantha's number (the sum of the numbers) is, 23,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 21,
then Porfirio only knows that the numbers belong to the set,
{[2, 21], [3, 14], [6, 7]}, that has more than one element.
If the numbers are, 3, 20,
then Porfirio only knows that the numbers belong to the set,
{[2, 30], [3, 20], [4, 15], [5, 12], [6, 10]},
that has more than one element.
If the numbers are, 4, 19,
then Porfirio only knows that the numbers belong to the set,
{[2, 38], [4, 19]}, that has more than one element.
If the numbers are, 5, 18,
then Porfirio only knows that the numbers belong to the set,
{[2, 45], [3, 30], [5, 18], [6, 15], [9, 10]},
that has more than one element.
If the numbers are, 6, 17,
then Porfirio only knows that the numbers belong to the set,
{[2, 51], [3, 34], [6, 17]}, that has more than one element.
If the numbers are, 7, 16,
then Porfirio only knows that the numbers belong to the set,
{[2, 56], [4, 28], [7, 16], [8, 14]}, that has more than one element.
If the numbers are, 8, 15,
then Porfirio only knows that the numbers belong to the set,
{[2, 60], [3, 40], [4, 30], [5, 24], [6, 20], [8, 15], [10, 12]},
that has more than one element.
If the numbers are, 9, 14,
then Porfirio only knows that the numbers belong to the set,
{[2, 63], [3, 42], [6, 21], [7, 18], [9, 14]},
that has more than one element.
If the numbers are, 10, 13,
then Porfirio only knows that the numbers belong to the set,
{[2, 65], [5, 26], [10, 13]}, that has more than one element.
If the numbers are, 11, 12,
then Porfirio only knows that the numbers belong to the set,
{[2, 66], [3, 44], [4, 33], [6, 22], [11, 12]},
that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 24
If Samantha's number (the sum of the numbers) is, 24,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 19, their product, 95,
would enable Porfirio to deduce that the numbers are, 5, 19
------------------------------------------------------
There is no way that Samantha can have the number, 25
If Samantha's number (the sum of the numbers) is, 25,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 23, their product, 46,
would enable Porfirio to deduce that the numbers are, 2, 23
------------------------------------------------------
There is no way that Samantha can have the number, 26
If Samantha's number (the sum of the numbers) is, 26,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 23, their product, 69,
would enable Porfirio to deduce that the numbers are, 3, 23
------------------------------------------------------
Samantha may have the number, 27
If Samantha's number (the sum of the numbers) is, 27,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 25,
then Porfirio only knows that the numbers belong to the set,
{[2, 25], [5, 10]}, that has more than one element.
If the numbers are, 3, 24,
then Porfirio only knows that the numbers belong to the set,
{[2, 36], [3, 24], [4, 18], [6, 12], [8, 9]},
that has more than one element.
If the numbers are, 4, 23,
then Porfirio only knows that the numbers belong to the set,
{[2, 46], [4, 23]}, that has more than one element.
If the numbers are, 5, 22,
then Porfirio only knows that the numbers belong to the set,
{[2, 55], [5, 22], [10, 11]}, that has more than one element.
If the numbers are, 6, 21,
then Porfirio only knows that the numbers belong to the set,
{[2, 63], [3, 42], [6, 21], [7, 18], [9, 14]},
that has more than one element.
If the numbers are, 7, 20,
then Porfirio only knows that the numbers belong to the set,
{[2, 70], [4, 35], [5, 28], [7, 20], [10, 14]},
that has more than one element.
If the numbers are, 8, 19,
then Porfirio only knows that the numbers belong to the set,
{[2, 76], [4, 38], [8, 19]}, that has more than one element.
If the numbers are, 9, 18,
then Porfirio only knows that the numbers belong to the set,
{[2, 81], [3, 54], [6, 27], [9, 18]}, that has more than one element.
If the numbers are, 10, 17,
then Porfirio only knows that the numbers belong to the set,
{[2, 85], [5, 34], [10, 17]}, that has more than one element.
If the numbers are, 11, 16,
then Porfirio only knows that the numbers belong to the set,
{[2, 88], [4, 44], [8, 22], [11, 16]}, that has more than one element.
If the numbers are, 12, 15,
then Porfirio only knows that the numbers belong to the set,
{[2, 90], [3, 60], [4, 45], [5, 36], [6, 30], [9, 20], [10, 18], [12, 15]},
that has more than one element.
If the numbers are, 13, 14,
then Porfirio only knows that the numbers belong to the set,
{[2, 91], [7, 26], [13, 14]}, that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 28
If Samantha's number (the sum of the numbers) is, 28,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 23, their product, 115,
would enable Porfirio to deduce that the numbers are, 5, 23
------------------------------------------------------
Samantha may have the number, 29
If Samantha's number (the sum of the numbers) is, 29,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 27,
then Porfirio only knows that the numbers belong to the set,
{[2, 27], [3, 18], [6, 9]}, that has more than one element.
If the numbers are, 3, 26,
then Porfirio only knows that the numbers belong to the set,
{[2, 39], [3, 26], [6, 13]}, that has more than one element.
If the numbers are, 4, 25,
then Porfirio only knows that the numbers belong to the set,
{[2, 50], [4, 25], [5, 20]}, that has more than one element.
If the numbers are, 5, 24,
then Porfirio only knows that the numbers belong to the set,
{[2, 60], [3, 40], [4, 30], [5, 24], [6, 20], [8, 15], [10, 12]},
that has more than one element.
If the numbers are, 6, 23,
then Porfirio only knows that the numbers belong to the set,
{[2, 69], [3, 46], [6, 23]}, that has more than one element.
If the numbers are, 7, 22,
then Porfirio only knows that the numbers belong to the set,
{[2, 77], [7, 22], [11, 14]}, that has more than one element.
If the numbers are, 8, 21,
then Porfirio only knows that the numbers belong to the set,
{[2, 84], [3, 56], [4, 42], [6, 28], [7, 24], [8, 21], [12, 14]},
that has more than one element.
If the numbers are, 9, 20,
then Porfirio only knows that the numbers belong to the set,
{[2, 90], [3, 60], [4, 45], [5, 36], [6, 30], [9, 20], [10, 18], [12, 15]},
that has more than one element.
If the numbers are, 10, 19,
then Porfirio only knows that the numbers belong to the set,
{[2, 95], [5, 38], [10, 19]}, that has more than one element.
If the numbers are, 11, 18,
then Porfirio only knows that the numbers belong to the set,
{[2, 99], [3, 66], [6, 33], [9, 22], [11, 18]},
that has more than one element.
If the numbers are, 12, 17,
then Porfirio only knows that the numbers belong to the set,
{[3, 68], [4, 51], [6, 34], [12, 17]}, that has more than one element.
If the numbers are, 13, 16,
then Porfirio only knows that the numbers belong to the set,
{[4, 52], [8, 26], [13, 16]}, that has more than one element.
If the numbers are, 14, 15,
then Porfirio only knows that the numbers belong to the set,
{[3, 70], [5, 42], [6, 35], [7, 30], [10, 21], [14, 15]},
that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 30
If Samantha's number (the sum of the numbers) is, 30,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 27, their product, 81,
would enable Porfirio to deduce that the numbers are, 3, 27
------------------------------------------------------
There is no way that Samantha can have the number, 31
If Samantha's number (the sum of the numbers) is, 31,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 29, their product, 58,
would enable Porfirio to deduce that the numbers are, 2, 29
------------------------------------------------------
There is no way that Samantha can have the number, 32
If Samantha's number (the sum of the numbers) is, 32,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 29, their product, 87,
would enable Porfirio to deduce that the numbers are, 3, 29
------------------------------------------------------
There is no way that Samantha can have the number, 33
If Samantha's number (the sum of the numbers) is, 33,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 31, their product, 62,
would enable Porfirio to deduce that the numbers are, 2, 31
------------------------------------------------------
There is no way that Samantha can have the number, 34
If Samantha's number (the sum of the numbers) is, 34,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 31, their product, 93,
would enable Porfirio to deduce that the numbers are, 3, 31
------------------------------------------------------
Samantha may have the number, 35
If Samantha's number (the sum of the numbers) is, 35,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 33,
then Porfirio only knows that the numbers belong to the set,
{[2, 33], [3, 22], [6, 11]}, that has more than one element.
If the numbers are, 3, 32,
then Porfirio only knows that the numbers belong to the set,
{[2, 48], [3, 32], [4, 24], [6, 16], [8, 12]},
that has more than one element.
If the numbers are, 4, 31,
then Porfirio only knows that the numbers belong to the set,
{[2, 62], [4, 31]}, that has more than one element.
If the numbers are, 5, 30,
then Porfirio only knows that the numbers belong to the set,
{[2, 75], [3, 50], [5, 30], [6, 25], [10, 15]},
that has more than one element.
If the numbers are, 6, 29,
then Porfirio only knows that the numbers belong to the set,
{[2, 87], [3, 58], [6, 29]}, that has more than one element.
If the numbers are, 7, 28,
then Porfirio only knows that the numbers belong to the set,
{[2, 98], [4, 49], [7, 28]}, that has more than one element.
If the numbers are, 8, 27,
then Porfirio only knows that the numbers belong to the set,
{[3, 72], [4, 54], [6, 36], [8, 27], [9, 24], [12, 18]},
that has more than one element.
If the numbers are, 9, 26,
then Porfirio only knows that the numbers belong to the set,
{[3, 78], [6, 39], [9, 26], [13, 18]}, that has more than one element.
If the numbers are, 10, 25,
then Porfirio only knows that the numbers belong to the set,
{[5, 50], [10, 25]}, that has more than one element.
If the numbers are, 11, 24,
then Porfirio only knows that the numbers belong to the set,
{[3, 88], [4, 66], [6, 44], [8, 33], [11, 24], [12, 22]},
that has more than one element.
If the numbers are, 12, 23,
then Porfirio only knows that the numbers belong to the set,
{[3, 92], [4, 69], [6, 46], [12, 23]}, that has more than one element.
If the numbers are, 13, 22,
then Porfirio only knows that the numbers belong to the set,
{[11, 26], [13, 22]}, that has more than one element.
If the numbers are, 14, 21,
then Porfirio only knows that the numbers belong to the set,
{[3, 98], [6, 49], [7, 42], [14, 21]}, that has more than one element.
If the numbers are, 15, 20,
then Porfirio only knows that the numbers belong to the set,
{[4, 75], [5, 60], [6, 50], [10, 30], [12, 25], [15, 20]},
that has more than one element.
If the numbers are, 16, 19,
then Porfirio only knows that the numbers belong to the set,
{[4, 76], [8, 38], [16, 19]}, that has more than one element.
If the numbers are, 17, 18,
then Porfirio only knows that the numbers belong to the set,
{[6, 51], [9, 34], [17, 18]}, that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 36
If Samantha's number (the sum of the numbers) is, 36,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 31, their product, 155,
would enable Porfirio to deduce that the numbers are, 5, 31
------------------------------------------------------
Samantha may have the number, 37
If Samantha's number (the sum of the numbers) is, 37,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 35,
then Porfirio only knows that the numbers belong to the set,
{[2, 35], [5, 14], [7, 10]}, that has more than one element.
If the numbers are, 3, 34,
then Porfirio only knows that the numbers belong to the set,
{[2, 51], [3, 34], [6, 17]}, that has more than one element.
If the numbers are, 4, 33,
then Porfirio only knows that the numbers belong to the set,
{[2, 66], [3, 44], [4, 33], [6, 22], [11, 12]},
that has more than one element.
If the numbers are, 5, 32,
then Porfirio only knows that the numbers belong to the set,
{[2, 80], [4, 40], [5, 32], [8, 20], [10, 16]},
that has more than one element.
If the numbers are, 6, 31,
then Porfirio only knows that the numbers belong to the set,
{[2, 93], [3, 62], [6, 31]}, that has more than one element.
If the numbers are, 7, 30,
then Porfirio only knows that the numbers belong to the set,
{[3, 70], [5, 42], [6, 35], [7, 30], [10, 21], [14, 15]},
that has more than one element.
If the numbers are, 8, 29,
then Porfirio only knows that the numbers belong to the set,
{[4, 58], [8, 29]}, that has more than one element.
If the numbers are, 9, 28,
then Porfirio only knows that the numbers belong to the set,
{[3, 84], [4, 63], [6, 42], [7, 36], [9, 28], [12, 21], [14, 18]},
that has more than one element.
If the numbers are, 10, 27,
then Porfirio only knows that the numbers belong to the set,
{[3, 90], [5, 54], [6, 45], [9, 30], [10, 27], [15, 18]},
that has more than one element.
If the numbers are, 11, 26,
then Porfirio only knows that the numbers belong to the set,
{[11, 26], [13, 22]}, that has more than one element.
If the numbers are, 12, 25,
then Porfirio only knows that the numbers belong to the set,
{[4, 75], [5, 60], [6, 50], [10, 30], [12, 25], [15, 20]},
that has more than one element.
If the numbers are, 13, 24,
then Porfirio only knows that the numbers belong to the set,
{[4, 78], [6, 52], [8, 39], [12, 26], [13, 24]},
that has more than one element.
If the numbers are, 14, 23,
then Porfirio only knows that the numbers belong to the set,
{[7, 46], [14, 23]}, that has more than one element.
If the numbers are, 15, 22,
then Porfirio only knows that the numbers belong to the set,
{[5, 66], [6, 55], [10, 33], [11, 30], [15, 22]},
that has more than one element.
If the numbers are, 16, 21,
then Porfirio only knows that the numbers belong to the set,
{[4, 84], [6, 56], [7, 48], [8, 42], [12, 28], [14, 24], [16, 21]},
that has more than one element.
If the numbers are, 17, 20,
then Porfirio only knows that the numbers belong to the set,
{[4, 85], [5, 68], [10, 34], [17, 20]}, that has more than one element.
If the numbers are, 18, 19,
then Porfirio only knows that the numbers belong to the set,
{[6, 57], [9, 38], [18, 19]}, that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 38
If Samantha's number (the sum of the numbers) is, 38,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 7, 31, their product, 217,
would enable Porfirio to deduce that the numbers are, 7, 31
------------------------------------------------------
There is no way that Samantha can have the number, 39
If Samantha's number (the sum of the numbers) is, 39,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 37, their product, 74,
would enable Porfirio to deduce that the numbers are, 2, 37
------------------------------------------------------
There is no way that Samantha can have the number, 40
If Samantha's number (the sum of the numbers) is, 40,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 37, their product, 111,
would enable Porfirio to deduce that the numbers are, 3, 37
------------------------------------------------------
Samantha may have the number, 41
If Samantha's number (the sum of the numbers) is, 41,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 39,
then Porfirio only knows that the numbers belong to the set,
{[2, 39], [3, 26], [6, 13]}, that has more than one element.
If the numbers are, 3, 38,
then Porfirio only knows that the numbers belong to the set,
{[2, 57], [3, 38], [6, 19]}, that has more than one element.
If the numbers are, 4, 37,
then Porfirio only knows that the numbers belong to the set,
{[2, 74], [4, 37]}, that has more than one element.
If the numbers are, 5, 36,
then Porfirio only knows that the numbers belong to the set,
{[2, 90], [3, 60], [4, 45], [5, 36], [6, 30], [9, 20], [10, 18], [12, 15]},
that has more than one element.
If the numbers are, 6, 35,
then Porfirio only knows that the numbers belong to the set,
{[3, 70], [5, 42], [6, 35], [7, 30], [10, 21], [14, 15]},
that has more than one element.
If the numbers are, 7, 34,
then Porfirio only knows that the numbers belong to the set,
{[7, 34], [14, 17]}, that has more than one element.
If the numbers are, 8, 33,
then Porfirio only knows that the numbers belong to the set,
{[3, 88], [4, 66], [6, 44], [8, 33], [11, 24], [12, 22]},
that has more than one element.
If the numbers are, 9, 32,
then Porfirio only knows that the numbers belong to the set,
{[3, 96], [4, 72], [6, 48], [8, 36], [9, 32], [12, 24], [16, 18]},
that has more than one element.
If the numbers are, 10, 31,
then Porfirio only knows that the numbers belong to the set,
{[5, 62], [10, 31]}, that has more than one element.
If the numbers are, 11, 30,
then Porfirio only knows that the numbers belong to the set,
{[5, 66], [6, 55], [10, 33], [11, 30], [15, 22]},
that has more than one element.
If the numbers are, 12, 29,
then Porfirio only knows that the numbers belong to the set,
{[4, 87], [6, 58], [12, 29]}, that has more than one element.
If the numbers are, 13, 28,
then Porfirio only knows that the numbers belong to the set,
{[4, 91], [7, 52], [13, 28], [14, 26]}, that has more than one element.
If the numbers are, 14, 27,
then Porfirio only knows that the numbers belong to the set,
{[6, 63], [7, 54], [9, 42], [14, 27], [18, 21]},
that has more than one element.
If the numbers are, 15, 26,
then Porfirio only knows that the numbers belong to the set,
{[5, 78], [6, 65], [10, 39], [13, 30], [15, 26]},
that has more than one element.
If the numbers are, 16, 25,
then Porfirio only knows that the numbers belong to the set,
{[5, 80], [8, 50], [10, 40], [16, 25]}, that has more than one element.
If the numbers are, 17, 24,
then Porfirio only knows that the numbers belong to the set,
{[6, 68], [8, 51], [12, 34], [17, 24]}, that has more than one element.
If the numbers are, 18, 23,
then Porfirio only knows that the numbers belong to the set,
{[6, 69], [9, 46], [18, 23]}, that has more than one element.
If the numbers are, 19, 22,
then Porfirio only knows that the numbers belong to the set,
{[11, 38], [19, 22]}, that has more than one element.
If the numbers are, 20, 21,
then Porfirio only knows that the numbers belong to the set, {
[5, 84], [6, 70], [7, 60], [10, 42], [12, 35], [14, 30], [15, 28], [20, 21]
}, that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 42
If Samantha's number (the sum of the numbers) is, 42,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 37, their product, 185,
would enable Porfirio to deduce that the numbers are, 5, 37
------------------------------------------------------
There is no way that Samantha can have the number, 43
If Samantha's number (the sum of the numbers) is, 43,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 41, their product, 82,
would enable Porfirio to deduce that the numbers are, 2, 41
------------------------------------------------------
There is no way that Samantha can have the number, 44
If Samantha's number (the sum of the numbers) is, 44,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 41, their product, 123,
would enable Porfirio to deduce that the numbers are, 3, 41
------------------------------------------------------
There is no way that Samantha can have the number, 45
If Samantha's number (the sum of the numbers) is, 45,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 43, their product, 86,
would enable Porfirio to deduce that the numbers are, 2, 43
------------------------------------------------------
There is no way that Samantha can have the number, 46
If Samantha's number (the sum of the numbers) is, 46,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 43, their product, 129,
would enable Porfirio to deduce that the numbers are, 3, 43
------------------------------------------------------
Samantha may have the number, 47
If Samantha's number (the sum of the numbers) is, 47,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 45,
then Porfirio only knows that the numbers belong to the set,
{[2, 45], [3, 30], [5, 18], [6, 15], [9, 10]},
that has more than one element.
If the numbers are, 3, 44,
then Porfirio only knows that the numbers belong to the set,
{[2, 66], [3, 44], [4, 33], [6, 22], [11, 12]},
that has more than one element.
If the numbers are, 4, 43,
then Porfirio only knows that the numbers belong to the set,
{[2, 86], [4, 43]}, that has more than one element.
If the numbers are, 5, 42,
then Porfirio only knows that the numbers belong to the set,
{[3, 70], [5, 42], [6, 35], [7, 30], [10, 21], [14, 15]},
that has more than one element.
If the numbers are, 6, 41,
then Porfirio only knows that the numbers belong to the set,
{[3, 82], [6, 41]}, that has more than one element.
If the numbers are, 7, 40,
then Porfirio only knows that the numbers belong to the set,
{[4, 70], [5, 56], [7, 40], [8, 35], [10, 28], [14, 20]},
that has more than one element.
If the numbers are, 8, 39,
then Porfirio only knows that the numbers belong to the set,
{[4, 78], [6, 52], [8, 39], [12, 26], [13, 24]},
that has more than one element.
If the numbers are, 9, 38,
then Porfirio only knows that the numbers belong to the set,
{[6, 57], [9, 38], [18, 19]}, that has more than one element.
If the numbers are, 10, 37,
then Porfirio only knows that the numbers belong to the set,
{[5, 74], [10, 37]}, that has more than one element.
If the numbers are, 11, 36,
then Porfirio only knows that the numbers belong to the set,
{[4, 99], [6, 66], [9, 44], [11, 36], [12, 33], [18, 22]},
that has more than one element.
If the numbers are, 12, 35,
then Porfirio only knows that the numbers belong to the set, {
[5, 84], [6, 70], [7, 60], [10, 42], [12, 35], [14, 30], [15, 28], [20, 21]
}, that has more than one element.
If the numbers are, 13, 34,
then Porfirio only knows that the numbers belong to the set,
{[13, 34], [17, 26]}, that has more than one element.
If the numbers are, 14, 33,
then Porfirio only knows that the numbers belong to the set,
{[6, 77], [7, 66], [11, 42], [14, 33], [21, 22]},
that has more than one element.
If the numbers are, 15, 32,
then Porfirio only knows that the numbers belong to the set, {
[5, 96], [6, 80], [8, 60], [10, 48], [12, 40], [15, 32], [16, 30], [20, 24]
}, that has more than one element.
If the numbers are, 16, 31,
then Porfirio only knows that the numbers belong to the set,
{[8, 62], [16, 31]}, that has more than one element.
If the numbers are, 17, 30,
then Porfirio only knows that the numbers belong to the set,
{[6, 85], [10, 51], [15, 34], [17, 30]}, that has more than one element.
If the numbers are, 18, 29,
then Porfirio only knows that the numbers belong to the set,
{[6, 87], [9, 58], [18, 29]}, that has more than one element.
If the numbers are, 19, 28,
then Porfirio only knows that the numbers belong to the set,
{[7, 76], [14, 38], [19, 28]}, that has more than one element.
If the numbers are, 20, 27,
then Porfirio only knows that the numbers belong to the set,
{[6, 90], [9, 60], [10, 54], [12, 45], [15, 36], [18, 30], [20, 27]},
that has more than one element.
If the numbers are, 21, 26,
then Porfirio only knows that the numbers belong to the set,
{[6, 91], [7, 78], [13, 42], [14, 39], [21, 26]},
that has more than one element.
If the numbers are, 22, 25,
then Porfirio only knows that the numbers belong to the set,
{[10, 55], [11, 50], [22, 25]}, that has more than one element.
If the numbers are, 23, 24,
then Porfirio only knows that the numbers belong to the set,
{[6, 92], [8, 69], [12, 46], [23, 24]}, that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 48
If Samantha's number (the sum of the numbers) is, 48,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 43, their product, 215,
would enable Porfirio to deduce that the numbers are, 5, 43
------------------------------------------------------
There is no way that Samantha can have the number, 49
If Samantha's number (the sum of the numbers) is, 49,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 47, their product, 94,
would enable Porfirio to deduce that the numbers are, 2, 47
------------------------------------------------------
There is no way that Samantha can have the number, 50
If Samantha's number (the sum of the numbers) is, 50,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 47, their product, 141,
would enable Porfirio to deduce that the numbers are, 3, 47
------------------------------------------------------
There is no way that Samantha can have the number, 51
If Samantha's number (the sum of the numbers) is, 51,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 17, 34, their product, 578,
would enable Porfirio to deduce that the numbers are, 17, 34
------------------------------------------------------
There is no way that Samantha can have the number, 52
If Samantha's number (the sum of the numbers) is, 52,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 47, their product, 235,
would enable Porfirio to deduce that the numbers are, 5, 47
------------------------------------------------------
Samantha may have the number, 53
If Samantha's number (the sum of the numbers) is, 53,
then she knows for sure that Porfirio does NOT know the numbers.
Let's prove it.
If the numbers are, 2, 51,
then Porfirio only knows that the numbers belong to the set,
{[2, 51], [3, 34], [6, 17]}, that has more than one element.
If the numbers are, 3, 50,
then Porfirio only knows that the numbers belong to the set,
{[2, 75], [3, 50], [5, 30], [6, 25], [10, 15]},
that has more than one element.
If the numbers are, 4, 49,
then Porfirio only knows that the numbers belong to the set,
{[2, 98], [4, 49], [7, 28]}, that has more than one element.
If the numbers are, 5, 48,
then Porfirio only knows that the numbers belong to the set,
{[3, 80], [4, 60], [5, 48], [6, 40], [8, 30], [10, 24], [12, 20], [15, 16]}
, that has more than one element.
If the numbers are, 6, 47,
then Porfirio only knows that the numbers belong to the set,
{[3, 94], [6, 47]}, that has more than one element.
If the numbers are, 7, 46,
then Porfirio only knows that the numbers belong to the set,
{[7, 46], [14, 23]}, that has more than one element.
If the numbers are, 8, 45,
then Porfirio only knows that the numbers belong to the set, {[4, 90],
[5, 72], [6, 60], [8, 45], [9, 40], [10, 36], [12, 30], [15, 24], [18, 20]}
, that has more than one element.
If the numbers are, 9, 44,
then Porfirio only knows that the numbers belong to the set,
{[4, 99], [6, 66], [9, 44], [11, 36], [12, 33], [18, 22]},
that has more than one element.
If the numbers are, 10, 43,
then Porfirio only knows that the numbers belong to the set,
{[5, 86], [10, 43]}, that has more than one element.
If the numbers are, 11, 42,
then Porfirio only knows that the numbers belong to the set,
{[6, 77], [7, 66], [11, 42], [14, 33], [21, 22]},
that has more than one element.
If the numbers are, 12, 41,
then Porfirio only knows that the numbers belong to the set,
{[6, 82], [12, 41]}, that has more than one element.
If the numbers are, 13, 40,
then Porfirio only knows that the numbers belong to the set,
{[8, 65], [10, 52], [13, 40], [20, 26]}, that has more than one element.
If the numbers are, 14, 39,
then Porfirio only knows that the numbers belong to the set,
{[6, 91], [7, 78], [13, 42], [14, 39], [21, 26]},
that has more than one element.
If the numbers are, 15, 38,
then Porfirio only knows that the numbers belong to the set,
{[6, 95], [10, 57], [15, 38], [19, 30]}, that has more than one element.
If the numbers are, 16, 37,
then Porfirio only knows that the numbers belong to the set,
{[8, 74], [16, 37]}, that has more than one element.
If the numbers are, 17, 36,
then Porfirio only knows that the numbers belong to the set,
{[9, 68], [12, 51], [17, 36], [18, 34]}, that has more than one element.
If the numbers are, 18, 35,
then Porfirio only knows that the numbers belong to the set,
{[7, 90], [9, 70], [10, 63], [14, 45], [15, 42], [18, 35], [21, 30]},
that has more than one element.
If the numbers are, 19, 34,
then Porfirio only knows that the numbers belong to the set,
{[17, 38], [19, 34]}, that has more than one element.
If the numbers are, 20, 33,
then Porfirio only knows that the numbers belong to the set,
{[10, 66], [11, 60], [12, 55], [15, 44], [20, 33], [22, 30]},
that has more than one element.
If the numbers are, 21, 32,
then Porfirio only knows that the numbers belong to the set,
{[7, 96], [8, 84], [12, 56], [14, 48], [16, 42], [21, 32], [24, 28]},
that has more than one element.
If the numbers are, 22, 31,
then Porfirio only knows that the numbers belong to the set,
{[11, 62], [22, 31]}, that has more than one element.
If the numbers are, 23, 30,
then Porfirio only knows that the numbers belong to the set,
{[10, 69], [15, 46], [23, 30]}, that has more than one element.
If the numbers are, 24, 29,
then Porfirio only knows that the numbers belong to the set,
{[8, 87], [12, 58], [24, 29]}, that has more than one element.
If the numbers are, 25, 28,
then Porfirio only knows that the numbers belong to the set,
{[10, 70], [14, 50], [20, 35], [25, 28]}, that has more than one element.
If the numbers are, 26, 27,
then Porfirio only knows that the numbers belong to the set,
{[9, 78], [13, 54], [18, 39], [26, 27]}, that has more than one element.
------------------------------------------------------
There is no way that Samantha can have the number, 54
If Samantha's number (the sum of the numbers) is, 54,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 7, 47, their product, 329,
would enable Porfirio to deduce that the numbers are, 7, 47
------------------------------------------------------
There is no way that Samantha can have the number, 55
If Samantha's number (the sum of the numbers) is, 55,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 53, their product, 106,
would enable Porfirio to deduce that the numbers are, 2, 53
------------------------------------------------------
There is no way that Samantha can have the number, 56
If Samantha's number (the sum of the numbers) is, 56,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 53, their product, 159,
would enable Porfirio to deduce that the numbers are, 3, 53
------------------------------------------------------
There is no way that Samantha can have the number, 57
If Samantha's number (the sum of the numbers) is, 57,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 4, 53, their product, 212,
would enable Porfirio to deduce that the numbers are, 4, 53
------------------------------------------------------
There is no way that Samantha can have the number, 58
If Samantha's number (the sum of the numbers) is, 58,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 53, their product, 265,
would enable Porfirio to deduce that the numbers are, 5, 53
------------------------------------------------------
There is no way that Samantha can have the number, 59
If Samantha's number (the sum of the numbers) is, 59,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 6, 53, their product, 318,
would enable Porfirio to deduce that the numbers are, 6, 53
------------------------------------------------------
There is no way that Samantha can have the number, 60
If Samantha's number (the sum of the numbers) is, 60,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 7, 53, their product, 371,
would enable Porfirio to deduce that the numbers are, 7, 53
------------------------------------------------------
There is no way that Samantha can have the number, 61
If Samantha's number (the sum of the numbers) is, 61,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 59, their product, 118,
would enable Porfirio to deduce that the numbers are, 2, 59
------------------------------------------------------
There is no way that Samantha can have the number, 62
If Samantha's number (the sum of the numbers) is, 62,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 59, their product, 177,
would enable Porfirio to deduce that the numbers are, 3, 59
------------------------------------------------------
There is no way that Samantha can have the number, 63
If Samantha's number (the sum of the numbers) is, 63,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 61, their product, 122,
would enable Porfirio to deduce that the numbers are, 2, 61
------------------------------------------------------
There is no way that Samantha can have the number, 64
If Samantha's number (the sum of the numbers) is, 64,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 61, their product, 183,
would enable Porfirio to deduce that the numbers are, 3, 61
------------------------------------------------------
There is no way that Samantha can have the number, 65
If Samantha's number (the sum of the numbers) is, 65,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 4, 61, their product, 244,
would enable Porfirio to deduce that the numbers are, 4, 61
------------------------------------------------------
There is no way that Samantha can have the number, 66
If Samantha's number (the sum of the numbers) is, 66,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 61, their product, 305,
would enable Porfirio to deduce that the numbers are, 5, 61
------------------------------------------------------
There is no way that Samantha can have the number, 67
If Samantha's number (the sum of the numbers) is, 67,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 6, 61, their product, 366,
would enable Porfirio to deduce that the numbers are, 6, 61
------------------------------------------------------
There is no way that Samantha can have the number, 68
If Samantha's number (the sum of the numbers) is, 68,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 7, 61, their product, 427,
would enable Porfirio to deduce that the numbers are, 7, 61
------------------------------------------------------
There is no way that Samantha can have the number, 69
If Samantha's number (the sum of the numbers) is, 69,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 67, their product, 134,
would enable Porfirio to deduce that the numbers are, 2, 67
------------------------------------------------------
There is no way that Samantha can have the number, 70
If Samantha's number (the sum of the numbers) is, 70,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 67, their product, 201,
would enable Porfirio to deduce that the numbers are, 3, 67
------------------------------------------------------
There is no way that Samantha can have the number, 71
If Samantha's number (the sum of the numbers) is, 71,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 4, 67, their product, 268,
would enable Porfirio to deduce that the numbers are, 4, 67
------------------------------------------------------
There is no way that Samantha can have the number, 72
If Samantha's number (the sum of the numbers) is, 72,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 67, their product, 335,
would enable Porfirio to deduce that the numbers are, 5, 67
------------------------------------------------------
There is no way that Samantha can have the number, 73
If Samantha's number (the sum of the numbers) is, 73,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 71, their product, 142,
would enable Porfirio to deduce that the numbers are, 2, 71
------------------------------------------------------
There is no way that Samantha can have the number, 74
If Samantha's number (the sum of the numbers) is, 74,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 71, their product, 213,
would enable Porfirio to deduce that the numbers are, 3, 71
------------------------------------------------------
There is no way that Samantha can have the number, 75
If Samantha's number (the sum of the numbers) is, 75,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 73, their product, 146,
would enable Porfirio to deduce that the numbers are, 2, 73
------------------------------------------------------
There is no way that Samantha can have the number, 76
If Samantha's number (the sum of the numbers) is, 76,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 73, their product, 219,
would enable Porfirio to deduce that the numbers are, 3, 73
------------------------------------------------------
There is no way that Samantha can have the number, 77
If Samantha's number (the sum of the numbers) is, 77,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 4, 73, their product, 292,
would enable Porfirio to deduce that the numbers are, 4, 73
------------------------------------------------------
There is no way that Samantha can have the number, 78
If Samantha's number (the sum of the numbers) is, 78,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 73, their product, 365,
would enable Porfirio to deduce that the numbers are, 5, 73
------------------------------------------------------
There is no way that Samantha can have the number, 79
If Samantha's number (the sum of the numbers) is, 79,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 6, 73, their product, 438,
would enable Porfirio to deduce that the numbers are, 6, 73
------------------------------------------------------
There is no way that Samantha can have the number, 80
If Samantha's number (the sum of the numbers) is, 80,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 7, 73, their product, 511,
would enable Porfirio to deduce that the numbers are, 7, 73
------------------------------------------------------
There is no way that Samantha can have the number, 81
If Samantha's number (the sum of the numbers) is, 81,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 79, their product, 158,
would enable Porfirio to deduce that the numbers are, 2, 79
------------------------------------------------------
There is no way that Samantha can have the number, 82
If Samantha's number (the sum of the numbers) is, 82,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 79, their product, 237,
would enable Porfirio to deduce that the numbers are, 3, 79
------------------------------------------------------
There is no way that Samantha can have the number, 83
If Samantha's number (the sum of the numbers) is, 83,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 4, 79, their product, 316,
would enable Porfirio to deduce that the numbers are, 4, 79
------------------------------------------------------
There is no way that Samantha can have the number, 84
If Samantha's number (the sum of the numbers) is, 84,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 79, their product, 395,
would enable Porfirio to deduce that the numbers are, 5, 79
------------------------------------------------------
There is no way that Samantha can have the number, 85
If Samantha's number (the sum of the numbers) is, 85,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 83, their product, 166,
would enable Porfirio to deduce that the numbers are, 2, 83
------------------------------------------------------
There is no way that Samantha can have the number, 86
If Samantha's number (the sum of the numbers) is, 86,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 83, their product, 249,
would enable Porfirio to deduce that the numbers are, 3, 83
------------------------------------------------------
There is no way that Samantha can have the number, 87
If Samantha's number (the sum of the numbers) is, 87,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 4, 83, their product, 332,
would enable Porfirio to deduce that the numbers are, 4, 83
------------------------------------------------------
There is no way that Samantha can have the number, 88
If Samantha's number (the sum of the numbers) is, 88,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 83, their product, 415,
would enable Porfirio to deduce that the numbers are, 5, 83
------------------------------------------------------
There is no way that Samantha can have the number, 89
If Samantha's number (the sum of the numbers) is, 89,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 6, 83, their product, 498,
would enable Porfirio to deduce that the numbers are, 6, 83
------------------------------------------------------
There is no way that Samantha can have the number, 90
If Samantha's number (the sum of the numbers) is, 90,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 7, 83, their product, 581,
would enable Porfirio to deduce that the numbers are, 7, 83
------------------------------------------------------
There is no way that Samantha can have the number, 91
If Samantha's number (the sum of the numbers) is, 91,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 89, their product, 178,
would enable Porfirio to deduce that the numbers are, 2, 89
------------------------------------------------------
There is no way that Samantha can have the number, 92
If Samantha's number (the sum of the numbers) is, 92,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 89, their product, 267,
would enable Porfirio to deduce that the numbers are, 3, 89
------------------------------------------------------
There is no way that Samantha can have the number, 93
If Samantha's number (the sum of the numbers) is, 93,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 4, 89, their product, 356,
would enable Porfirio to deduce that the numbers are, 4, 89
------------------------------------------------------
There is no way that Samantha can have the number, 94
If Samantha's number (the sum of the numbers) is, 94,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 89, their product, 445,
would enable Porfirio to deduce that the numbers are, 5, 89
------------------------------------------------------
There is no way that Samantha can have the number, 95
If Samantha's number (the sum of the numbers) is, 95,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 6, 89, their product, 534,
would enable Porfirio to deduce that the numbers are, 6, 89
------------------------------------------------------
There is no way that Samantha can have the number, 96
If Samantha's number (the sum of the numbers) is, 96,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 7, 89, their product, 623,
would enable Porfirio to deduce that the numbers are, 7, 89
------------------------------------------------------
There is no way that Samantha can have the number, 97
If Samantha's number (the sum of the numbers) is, 97,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 8, 89, their product, 712,
would enable Porfirio to deduce that the numbers are, 8, 89
------------------------------------------------------
There is no way that Samantha can have the number, 98
If Samantha's number (the sum of the numbers) is, 98,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 9, 89, their product, 801,
would enable Porfirio to deduce that the numbers are, 9, 89
------------------------------------------------------
There is no way that Samantha can have the number, 99
If Samantha's number (the sum of the numbers) is, 99,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 2, 97, their product, 194,
would enable Porfirio to deduce that the numbers are, 2, 97
------------------------------------------------------
There is no way that Samantha can have the number, 100
If Samantha's number (the sum of the numbers) is, 100,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 3, 97, their product, 291,
would enable Porfirio to deduce that the numbers are, 3, 97
------------------------------------------------------
There is no way that Samantha can have the number, 101
If Samantha's number (the sum of the numbers) is, 101,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 4, 97, their product, 388,
would enable Porfirio to deduce that the numbers are, 4, 97
------------------------------------------------------
There is no way that Samantha can have the number, 102
If Samantha's number (the sum of the numbers) is, 102,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 5, 97, their product, 485,
would enable Porfirio to deduce that the numbers are, 5, 97
------------------------------------------------------
There is no way that Samantha can have the number, 103
If Samantha's number (the sum of the numbers) is, 103,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 6, 97, their product, 582,
would enable Porfirio to deduce that the numbers are, 6, 97
------------------------------------------------------
There is no way that Samantha can have the number, 104
If Samantha's number (the sum of the numbers) is, 104,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 7, 97, their product, 679,
would enable Porfirio to deduce that the numbers are, 7, 97
------------------------------------------------------
There is no way that Samantha can have the number, 105
If Samantha's number (the sum of the numbers) is, 105,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 8, 97, their product, 776,
would enable Porfirio to deduce that the numbers are, 8, 97
------------------------------------------------------
There is no way that Samantha can have the number, 106
If Samantha's number (the sum of the numbers) is, 106,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 9, 97, their product, 873,
would enable Porfirio to deduce that the numbers are, 9, 97
------------------------------------------------------
There is no way that Samantha can have the number, 107
If Samantha's number (the sum of the numbers) is, 107,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 10, 97, their product, 970,
would enable Porfirio to deduce that the numbers are, 10, 97
------------------------------------------------------
There is no way that Samantha can have the number, 108
If Samantha's number (the sum of the numbers) is, 108,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 11, 97, their product, 1067,
would enable Porfirio to deduce that the numbers are, 11, 97
------------------------------------------------------
There is no way that Samantha can have the number, 109
If Samantha's number (the sum of the numbers) is, 109,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 12, 97, their product, 1164,
would enable Porfirio to deduce that the numbers are, 12, 97
------------------------------------------------------
There is no way that Samantha can have the number, 110
If Samantha's number (the sum of the numbers) is, 110,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 11, 99, their product, 1089,
would enable Porfirio to deduce that the numbers are, 11, 99
------------------------------------------------------
There is no way that Samantha can have the number, 111
If Samantha's number (the sum of the numbers) is, 111,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 14, 97, their product, 1358,
would enable Porfirio to deduce that the numbers are, 14, 97
------------------------------------------------------
There is no way that Samantha can have the number, 112
If Samantha's number (the sum of the numbers) is, 112,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 15, 97, their product, 1455,
would enable Porfirio to deduce that the numbers are, 15, 97
------------------------------------------------------
There is no way that Samantha can have the number, 113
If Samantha's number (the sum of the numbers) is, 113,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 16, 97, their product, 1552,
would enable Porfirio to deduce that the numbers are, 16, 97
------------------------------------------------------
There is no way that Samantha can have the number, 114
If Samantha's number (the sum of the numbers) is, 114,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 17, 97, their product, 1649,
would enable Porfirio to deduce that the numbers are, 17, 97
------------------------------------------------------
There is no way that Samantha can have the number, 115
If Samantha's number (the sum of the numbers) is, 115,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 18, 97, their product, 1746,
would enable Porfirio to deduce that the numbers are, 18, 97
------------------------------------------------------
There is no way that Samantha can have the number, 116
If Samantha's number (the sum of the numbers) is, 116,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 19, 97, their product, 1843,
would enable Porfirio to deduce that the numbers are, 19, 97
------------------------------------------------------
There is no way that Samantha can have the number, 117
If Samantha's number (the sum of the numbers) is, 117,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 20, 97, their product, 1940,
would enable Porfirio to deduce that the numbers are, 20, 97
------------------------------------------------------
There is no way that Samantha can have the number, 118
If Samantha's number (the sum of the numbers) is, 118,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 21, 97, their product, 2037,
would enable Porfirio to deduce that the numbers are, 21, 97
------------------------------------------------------
There is no way that Samantha can have the number, 119
If Samantha's number (the sum of the numbers) is, 119,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 22, 97, their product, 2134,
would enable Porfirio to deduce that the numbers are, 22, 97
------------------------------------------------------
There is no way that Samantha can have the number, 120
If Samantha's number (the sum of the numbers) is, 120,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 23, 97, their product, 2231,
would enable Porfirio to deduce that the numbers are, 23, 97
------------------------------------------------------
There is no way that Samantha can have the number, 121
If Samantha's number (the sum of the numbers) is, 121,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 24, 97, their product, 2328,
would enable Porfirio to deduce that the numbers are, 24, 97
------------------------------------------------------
There is no way that Samantha can have the number, 122
If Samantha's number (the sum of the numbers) is, 122,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 25, 97, their product, 2425,
would enable Porfirio to deduce that the numbers are, 25, 97
------------------------------------------------------
There is no way that Samantha can have the number, 123
If Samantha's number (the sum of the numbers) is, 123,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 26, 97, their product, 2522,
would enable Porfirio to deduce that the numbers are, 26, 97
------------------------------------------------------
There is no way that Samantha can have the number, 124
If Samantha's number (the sum of the numbers) is, 124,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 27, 97, their product, 2619,
would enable Porfirio to deduce that the numbers are, 27, 97
------------------------------------------------------
There is no way that Samantha can have the number, 125
If Samantha's number (the sum of the numbers) is, 125,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 28, 97, their product, 2716,
would enable Porfirio to deduce that the numbers are, 28, 97
------------------------------------------------------
There is no way that Samantha can have the number, 126
If Samantha's number (the sum of the numbers) is, 126,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 29, 97, their product, 2813,
would enable Porfirio to deduce that the numbers are, 29, 97
------------------------------------------------------
There is no way that Samantha can have the number, 127
If Samantha's number (the sum of the numbers) is, 127,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 30, 97, their product, 2910,
would enable Porfirio to deduce that the numbers are, 30, 97
------------------------------------------------------
There is no way that Samantha can have the number, 128
If Samantha's number (the sum of the numbers) is, 128,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 31, 97, their product, 3007,
would enable Porfirio to deduce that the numbers are, 31, 97
------------------------------------------------------
There is no way that Samantha can have the number, 129
If Samantha's number (the sum of the numbers) is, 129,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 32, 97, their product, 3104,
would enable Porfirio to deduce that the numbers are, 32, 97
------------------------------------------------------
There is no way that Samantha can have the number, 130
If Samantha's number (the sum of the numbers) is, 130,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 33, 97, their product, 3201,
would enable Porfirio to deduce that the numbers are, 33, 97
------------------------------------------------------
There is no way that Samantha can have the number, 131
If Samantha's number (the sum of the numbers) is, 131,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 34, 97, their product, 3298,
would enable Porfirio to deduce that the numbers are, 34, 97
------------------------------------------------------
There is no way that Samantha can have the number, 132
If Samantha's number (the sum of the numbers) is, 132,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 33, 99, their product, 3267,
would enable Porfirio to deduce that the numbers are, 33, 99
------------------------------------------------------
There is no way that Samantha can have the number, 133
If Samantha's number (the sum of the numbers) is, 133,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 36, 97, their product, 3492,
would enable Porfirio to deduce that the numbers are, 36, 97
------------------------------------------------------
There is no way that Samantha can have the number, 134
If Samantha's number (the sum of the numbers) is, 134,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 37, 97, their product, 3589,
would enable Porfirio to deduce that the numbers are, 37, 97
------------------------------------------------------
There is no way that Samantha can have the number, 135
If Samantha's number (the sum of the numbers) is, 135,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 38, 97, their product, 3686,
would enable Porfirio to deduce that the numbers are, 38, 97
------------------------------------------------------
There is no way that Samantha can have the number, 136
If Samantha's number (the sum of the numbers) is, 136,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 37, 99, their product, 3663,
would enable Porfirio to deduce that the numbers are, 37, 99
------------------------------------------------------
There is no way that Samantha can have the number, 137
If Samantha's number (the sum of the numbers) is, 137,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 40, 97, their product, 3880,
would enable Porfirio to deduce that the numbers are, 40, 97
------------------------------------------------------
There is no way that Samantha can have the number, 138
If Samantha's number (the sum of the numbers) is, 138,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 39, 99, their product, 3861,
would enable Porfirio to deduce that the numbers are, 39, 99
------------------------------------------------------
There is no way that Samantha can have the number, 139
If Samantha's number (the sum of the numbers) is, 139,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 42, 97, their product, 4074,
would enable Porfirio to deduce that the numbers are, 42, 97
------------------------------------------------------
There is no way that Samantha can have the number, 140
If Samantha's number (the sum of the numbers) is, 140,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 41, 99, their product, 4059,
would enable Porfirio to deduce that the numbers are, 41, 99
------------------------------------------------------
There is no way that Samantha can have the number, 141
If Samantha's number (the sum of the numbers) is, 141,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 44, 97, their product, 4268,
would enable Porfirio to deduce that the numbers are, 44, 97
------------------------------------------------------
There is no way that Samantha can have the number, 142
If Samantha's number (the sum of the numbers) is, 142,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 43, 99, their product, 4257,
would enable Porfirio to deduce that the numbers are, 43, 99
------------------------------------------------------
There is no way that Samantha can have the number, 143
If Samantha's number (the sum of the numbers) is, 143,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 44, 99, their product, 4356,
would enable Porfirio to deduce that the numbers are, 44, 99
------------------------------------------------------
There is no way that Samantha can have the number, 144
If Samantha's number (the sum of the numbers) is, 144,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 47, 97, their product, 4559,
would enable Porfirio to deduce that the numbers are, 47, 97
------------------------------------------------------
There is no way that Samantha can have the number, 145
If Samantha's number (the sum of the numbers) is, 145,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 48, 97, their product, 4656,
would enable Porfirio to deduce that the numbers are, 48, 97
------------------------------------------------------
There is no way that Samantha can have the number, 146
If Samantha's number (the sum of the numbers) is, 146,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 47, 99, their product, 4653,
would enable Porfirio to deduce that the numbers are, 47, 99
------------------------------------------------------
There is no way that Samantha can have the number, 147
If Samantha's number (the sum of the numbers) is, 147,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 49, 98, their product, 4802,
would enable Porfirio to deduce that the numbers are, 49, 98
------------------------------------------------------
There is no way that Samantha can have the number, 148
If Samantha's number (the sum of the numbers) is, 148,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 50, 98, their product, 4900,
would enable Porfirio to deduce that the numbers are, 50, 98
------------------------------------------------------
There is no way that Samantha can have the number, 149
If Samantha's number (the sum of the numbers) is, 149,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 51, 98, their product, 4998,
would enable Porfirio to deduce that the numbers are, 51, 98
------------------------------------------------------
There is no way that Samantha can have the number, 150
If Samantha's number (the sum of the numbers) is, 150,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 51, 99, their product, 5049,
would enable Porfirio to deduce that the numbers are, 51, 99
------------------------------------------------------
There is no way that Samantha can have the number, 151
If Samantha's number (the sum of the numbers) is, 151,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 53, 98, their product, 5194,
would enable Porfirio to deduce that the numbers are, 53, 98
------------------------------------------------------
There is no way that Samantha can have the number, 152
If Samantha's number (the sum of the numbers) is, 152,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 53, 99, their product, 5247,
would enable Porfirio to deduce that the numbers are, 53, 99
------------------------------------------------------
There is no way that Samantha can have the number, 153
If Samantha's number (the sum of the numbers) is, 153,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 56, 97, their product, 5432,
would enable Porfirio to deduce that the numbers are, 56, 97
------------------------------------------------------
There is no way that Samantha can have the number, 154
If Samantha's number (the sum of the numbers) is, 154,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 55, 99, their product, 5445,
would enable Porfirio to deduce that the numbers are, 55, 99
------------------------------------------------------
There is no way that Samantha can have the number, 155
If Samantha's number (the sum of the numbers) is, 155,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 57, 98, their product, 5586,
would enable Porfirio to deduce that the numbers are, 57, 98
------------------------------------------------------
There is no way that Samantha can have the number, 156
If Samantha's number (the sum of the numbers) is, 156,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 57, 99, their product, 5643,
would enable Porfirio to deduce that the numbers are, 57, 99
------------------------------------------------------
There is no way that Samantha can have the number, 157
If Samantha's number (the sum of the numbers) is, 157,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 59, 98, their product, 5782,
would enable Porfirio to deduce that the numbers are, 59, 98
------------------------------------------------------
There is no way that Samantha can have the number, 158
If Samantha's number (the sum of the numbers) is, 158,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 59, 99, their product, 5841,
would enable Porfirio to deduce that the numbers are, 59, 99
------------------------------------------------------
There is no way that Samantha can have the number, 159
If Samantha's number (the sum of the numbers) is, 159,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 61, 98, their product, 5978,
would enable Porfirio to deduce that the numbers are, 61, 98
------------------------------------------------------
There is no way that Samantha can have the number, 160
If Samantha's number (the sum of the numbers) is, 160,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 61, 99, their product, 6039,
would enable Porfirio to deduce that the numbers are, 61, 99
------------------------------------------------------
There is no way that Samantha can have the number, 161
If Samantha's number (the sum of the numbers) is, 161,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 63, 98, their product, 6174,
would enable Porfirio to deduce that the numbers are, 63, 98
------------------------------------------------------
There is no way that Samantha can have the number, 162
If Samantha's number (the sum of the numbers) is, 162,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 64, 98, their product, 6272,
would enable Porfirio to deduce that the numbers are, 64, 98
------------------------------------------------------
There is no way that Samantha can have the number, 163
If Samantha's number (the sum of the numbers) is, 163,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 66, 97, their product, 6402,
would enable Porfirio to deduce that the numbers are, 66, 97
------------------------------------------------------
There is no way that Samantha can have the number, 164
If Samantha's number (the sum of the numbers) is, 164,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 65, 99, their product, 6435,
would enable Porfirio to deduce that the numbers are, 65, 99
------------------------------------------------------
There is no way that Samantha can have the number, 165
If Samantha's number (the sum of the numbers) is, 165,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 66, 99, their product, 6534,
would enable Porfirio to deduce that the numbers are, 66, 99
------------------------------------------------------
There is no way that Samantha can have the number, 166
If Samantha's number (the sum of the numbers) is, 166,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 67, 99, their product, 6633,
would enable Porfirio to deduce that the numbers are, 67, 99
------------------------------------------------------
There is no way that Samantha can have the number, 167
If Samantha's number (the sum of the numbers) is, 167,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 68, 99, their product, 6732,
would enable Porfirio to deduce that the numbers are, 68, 99
------------------------------------------------------
There is no way that Samantha can have the number, 168
If Samantha's number (the sum of the numbers) is, 168,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 69, 99, their product, 6831,
would enable Porfirio to deduce that the numbers are, 69, 99
------------------------------------------------------
There is no way that Samantha can have the number, 169
If Samantha's number (the sum of the numbers) is, 169,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 71, 98, their product, 6958,
would enable Porfirio to deduce that the numbers are, 71, 98
------------------------------------------------------
There is no way that Samantha can have the number, 170
If Samantha's number (the sum of the numbers) is, 170,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 71, 99, their product, 7029,
would enable Porfirio to deduce that the numbers are, 71, 99
------------------------------------------------------
There is no way that Samantha can have the number, 171
If Samantha's number (the sum of the numbers) is, 171,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 73, 98, their product, 7154,
would enable Porfirio to deduce that the numbers are, 73, 98
------------------------------------------------------
There is no way that Samantha can have the number, 172
If Samantha's number (the sum of the numbers) is, 172,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 73, 99, their product, 7227,
would enable Porfirio to deduce that the numbers are, 73, 99
------------------------------------------------------
There is no way that Samantha can have the number, 173
If Samantha's number (the sum of the numbers) is, 173,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 74, 99, their product, 7326,
would enable Porfirio to deduce that the numbers are, 74, 99
------------------------------------------------------
There is no way that Samantha can have the number, 174
If Samantha's number (the sum of the numbers) is, 174,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 75, 99, their product, 7425,
would enable Porfirio to deduce that the numbers are, 75, 99
------------------------------------------------------
There is no way that Samantha can have the number, 175
If Samantha's number (the sum of the numbers) is, 175,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 76, 99, their product, 7524,
would enable Porfirio to deduce that the numbers are, 76, 99
------------------------------------------------------
There is no way that Samantha can have the number, 176
If Samantha's number (the sum of the numbers) is, 176,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 77, 99, their product, 7623,
would enable Porfirio to deduce that the numbers are, 77, 99
------------------------------------------------------
There is no way that Samantha can have the number, 177
If Samantha's number (the sum of the numbers) is, 177,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 78, 99, their product, 7722,
would enable Porfirio to deduce that the numbers are, 78, 99
------------------------------------------------------
There is no way that Samantha can have the number, 178
If Samantha's number (the sum of the numbers) is, 178,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 79, 99, their product, 7821,
would enable Porfirio to deduce that the numbers are, 79, 99
------------------------------------------------------
There is no way that Samantha can have the number, 179
If Samantha's number (the sum of the numbers) is, 179,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 81, 98, their product, 7938,
would enable Porfirio to deduce that the numbers are, 81, 98
------------------------------------------------------
There is no way that Samantha can have the number, 180
If Samantha's number (the sum of the numbers) is, 180,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 81, 99, their product, 8019,
would enable Porfirio to deduce that the numbers are, 81, 99
------------------------------------------------------
There is no way that Samantha can have the number, 181
If Samantha's number (the sum of the numbers) is, 181,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 82, 99, their product, 8118,
would enable Porfirio to deduce that the numbers are, 82, 99
------------------------------------------------------
There is no way that Samantha can have the number, 182
If Samantha's number (the sum of the numbers) is, 182,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 83, 99, their product, 8217,
would enable Porfirio to deduce that the numbers are, 83, 99
------------------------------------------------------
There is no way that Samantha can have the number, 183
If Samantha's number (the sum of the numbers) is, 183,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 84, 99, their product, 8316,
would enable Porfirio to deduce that the numbers are, 84, 99
------------------------------------------------------
There is no way that Samantha can have the number, 184
If Samantha's number (the sum of the numbers) is, 184,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 85, 99, their product, 8415,
would enable Porfirio to deduce that the numbers are, 85, 99
------------------------------------------------------
There is no way that Samantha can have the number, 185
If Samantha's number (the sum of the numbers) is, 185,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 86, 99, their product, 8514,
would enable Porfirio to deduce that the numbers are, 86, 99
------------------------------------------------------
There is no way that Samantha can have the number, 186
If Samantha's number (the sum of the numbers) is, 186,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 87, 99, their product, 8613,
would enable Porfirio to deduce that the numbers are, 87, 99
------------------------------------------------------
There is no way that Samantha can have the number, 187
If Samantha's number (the sum of the numbers) is, 187,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 88, 99, their product, 8712,
would enable Porfirio to deduce that the numbers are, 88, 99
------------------------------------------------------
There is no way that Samantha can have the number, 188
If Samantha's number (the sum of the numbers) is, 188,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 89, 99, their product, 8811,
would enable Porfirio to deduce that the numbers are, 89, 99
------------------------------------------------------
There is no way that Samantha can have the number, 189
If Samantha's number (the sum of the numbers) is, 189,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 90, 99, their product, 8910,
would enable Porfirio to deduce that the numbers are, 90, 99
------------------------------------------------------
There is no way that Samantha can have the number, 190
If Samantha's number (the sum of the numbers) is, 190,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 91, 99, their product, 9009,
would enable Porfirio to deduce that the numbers are, 91, 99
------------------------------------------------------
There is no way that Samantha can have the number, 191
If Samantha's number (the sum of the numbers) is, 191,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 92, 99, their product, 9108,
would enable Porfirio to deduce that the numbers are, 92, 99
------------------------------------------------------
There is no way that Samantha can have the number, 192
If Samantha's number (the sum of the numbers) is, 192,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 93, 99, their product, 9207,
would enable Porfirio to deduce that the numbers are, 93, 99
------------------------------------------------------
There is no way that Samantha can have the number, 193
If Samantha's number (the sum of the numbers) is, 193,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 94, 99, their product, 9306,
would enable Porfirio to deduce that the numbers are, 94, 99
------------------------------------------------------
There is no way that Samantha can have the number, 194
If Samantha's number (the sum of the numbers) is, 194,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 95, 99, their product, 9405,
would enable Porfirio to deduce that the numbers are, 95, 99
------------------------------------------------------
There is no way that Samantha can have the number, 195
If Samantha's number (the sum of the numbers) is, 195,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 96, 99, their product, 9504,
would enable Porfirio to deduce that the numbers are, 96, 99
------------------------------------------------------
There is no way that Samantha can have the number, 196
If Samantha's number (the sum of the numbers) is, 196,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 97, 99, their product, 9603,
would enable Porfirio to deduce that the numbers are, 97, 99
------------------------------------------------------
There is no way that Samantha can have the number, 197
If Samantha's number (the sum of the numbers) is, 197,
then she can't say about Porfirio's lack of knowledge.
Let's prove it.
Indeed, if, as is possible, the numbers are, 98, 99, their product, 9702,
would enable Porfirio to deduce that the numbers are, 98, 99
---------------------------------------
Hence the only numbers Santha (the sum of the two unknown numbers picked fro\
m the hat) can have are indeed the members of the set
{11, 17, 23, 27, 29, 35, 37, 41, 47, 53}
---------------------------------------
This ends Part I.
-----------------------------------------------------------------
Part II: the Set of Possible Pairs that Would Enable Porfirio to Deduce the \
Numbers from What Samantha Told Him
The set of pairs [x,y] such that their sum belongs to the Samantha set and k\
nowing their product and the fact that
their sum belongs to the Samantha set enables Porfirio to deduce the numbers\
,
in other words, the set of pairs such that Samantha can deduce that Porfirio\
does NOT know (immediately) the numbers,
but once Porfirio knows that Samantha knows that he does not know, he DOES k\
now, is the set
{[2, 9], [2, 25], [2, 27], [3, 8], [3, 32], [3, 38], [4, 7], [4, 13], [4, 19],
[4, 23], [4, 25], [4, 31], [4, 37], [4, 43], [5, 22], [5, 32], [5, 48],
[6, 23], [6, 29], [6, 31], [6, 41], [6, 47], [7, 16], [7, 20], [7, 22],
[7, 34], [7, 40], [8, 19], [8, 21], [8, 27], [8, 29], [8, 45], [9, 18],
[9, 26], [9, 28], [9, 32], [10, 13], [10, 17], [10, 19], [10, 25], [10, 27],
[10, 31], [10, 37], [10, 43], [11, 16], [11, 18], [12, 17], [12, 23],
[12, 29], [12, 41], [13, 14], [13, 16], [13, 28], [13, 34], [13, 40],
[14, 21], [14, 27], [15, 26], [15, 32], [15, 38], [16, 19], [16, 21],
[16, 25], [16, 31], [16, 37], [17, 18], [17, 20], [17, 24], [17, 30],
[17, 36], [18, 23], [18, 29], [18, 35], [19, 22], [19, 28], [19, 34],
[20, 27], [20, 33], [21, 32], [22, 25], [22, 31], [23, 24], [23, 30],
[24, 29], [25, 28], [26, 27]}
Let's explain why.
Recall that the Samantha set is, {11, 17, 23, 27, 29, 35, 37, 41, 47, 53}
For the member, 11,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 9], [3, 8], [4, 7], [5, 6]}
The pair, [2, 9], is a good pair, since its sum, 11,
belongs to the Samantha set, and knowing their product, 18,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 9], [3, 6]},
it is the only one whose sum belongs to the Samantha set.
The pair, [3, 8], is a good pair, since its sum, 11,
belongs to the Samantha set, and knowing their product, 24,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 12], [3, 8], [4, 6]},
it is the only one whose sum belongs to the Samantha set.
The pair, [4, 7], is a good pair, since its sum, 11,
belongs to the Samantha set, and knowing their product, 28,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 14], [4, 7]},
it is the only one whose sum belongs to the Samantha set.
The pair, [5, 6], is a not a good pair, even though its sum, 11,
belongs to the Samantha set, but just knowing their product, 30
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 15], [3, 10], [5, 6]}
the members of the subset, {[2, 15], [5, 6]}, all belong to the Samantha set.
This ends examining the pairs whose sum is, 11
---------------------------------
For the member, 17,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 15], [3, 14], [4, 13], [5, 12], [6, 11], [7, 10], [8, 9]}
The pair, [2, 15], is a not a good pair, even though its sum, 17,
belongs to the Samantha set, but just knowing their product, 30
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 15], [3, 10], [5, 6]}
the members of the subset, {[2, 15], [5, 6]}, all belong to the Samantha set.
The pair, [3, 14], is a not a good pair, even though its sum, 17,
belongs to the Samantha set, but just knowing their product, 42
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 21], [3, 14], [6, 7]}
the members of the subset, {[2, 21], [3, 14]}, all belong to the Samantha set.
The pair, [4, 13], is a good pair, since its sum, 17,
belongs to the Samantha set, and knowing their product, 52,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 26], [4, 13]},
it is the only one whose sum belongs to the Samantha set.
The pair, [5, 12], is a not a good pair, even though its sum, 17,
belongs to the Samantha set, but just knowing their product, 60
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 30], [3, 20], [4, 15], [5, 12], [6, 10]}
the members of the subset, {[3, 20], [5, 12]}, all belong to the Samantha set.
The pair, [6, 11], is a not a good pair, even though its sum, 17,
belongs to the Samantha set, but just knowing their product, 66
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 33], [3, 22], [6, 11]}
the members of the subset, {[2, 33], [6, 11]}, all belong to the Samantha set.
The pair, [7, 10], is a not a good pair, even though its sum, 17,
belongs to the Samantha set, but just knowing their product, 70
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 35], [5, 14], [7, 10]}
the members of the subset, {[2, 35], [7, 10]}, all belong to the Samantha set.
The pair, [8, 9], is a not a good pair, even though its sum, 17,
belongs to the Samantha set, but just knowing their product, 72
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 36], [3, 24], [4, 18], [6, 12], [8, 9]}
the members of the subset, {[3, 24], [8, 9]}, all belong to the Samantha set.
This ends examining the pairs whose sum is, 17
---------------------------------
For the member, 23,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 21], [3, 20], [4, 19], [5, 18], [6, 17], [7, 16], [8, 15],
[9, 14], [10, 13], [11, 12]}
The pair, [2, 21], is a not a good pair, even though its sum, 23,
belongs to the Samantha set, but just knowing their product, 42
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 21], [3, 14], [6, 7]}
the members of the subset, {[2, 21], [3, 14]}, all belong to the Samantha set.
The pair, [3, 20], is a not a good pair, even though its sum, 23,
belongs to the Samantha set, but just knowing their product, 60
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 30], [3, 20], [4, 15], [5, 12], [6, 10]}
the members of the subset, {[3, 20], [5, 12]}, all belong to the Samantha set.
The pair, [4, 19], is a good pair, since its sum, 23,
belongs to the Samantha set, and knowing their product, 76,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 38], [4, 19]},
it is the only one whose sum belongs to the Samantha set.
The pair, [5, 18], is a not a good pair, even though its sum, 23,
belongs to the Samantha set, but just knowing their product, 90
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 45], [3, 30], [5, 18], [6, 15], [9, 10]}
the members of the subset, {[2, 45], [5, 18]}, all belong to the Samantha set.
The pair, [6, 17], is a not a good pair, even though its sum, 23,
belongs to the Samantha set, but just knowing their product, 102
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 51], [3, 34], [6, 17]}
the members of the subset, {[2, 51], [3, 34], [6, 17]},
all belong to the Samantha set.
The pair, [7, 16], is a good pair, since its sum, 23,
belongs to the Samantha set, and knowing their product, 112,
enables Perforio
to deduce them, since amongst his possibilities,
{[2, 56], [4, 28], [7, 16], [8, 14]},
it is the only one whose sum belongs to the Samantha set.
The pair, [8, 15], is a not a good pair, even though its sum, 23,
belongs to the Samantha set, but just knowing their product, 120
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities,
{[2, 60], [3, 40], [4, 30], [5, 24], [6, 20], [8, 15], [10, 12]}
the members of the subset, {[5, 24], [8, 15]}, all belong to the Samantha set.
The pair, [9, 14], is a not a good pair, even though its sum, 23,
belongs to the Samantha set, but just knowing their product, 126
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 63], [3, 42], [6, 21], [7, 18], [9, 14]}
the members of the subset, {[6, 21], [9, 14]}, all belong to the Samantha set.
The pair, [10, 13], is a good pair, since its sum, 23,
belongs to the Samantha set, and knowing their product, 130,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 65], [5, 26], [10, 13]},
it is the only one whose sum belongs to the Samantha set.
The pair, [11, 12], is a not a good pair, even though its sum, 23,
belongs to the Samantha set, but just knowing their product, 132
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 66], [3, 44], [4, 33], [6, 22], [11, 12]}
the members of the subset, {[3, 44], [4, 33], [11, 12]},
all belong to the Samantha set.
This ends examining the pairs whose sum is, 23
---------------------------------
For the member, 27,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 25], [3, 24], [4, 23], [5, 22], [6, 21], [7, 20], [8, 19],
[9, 18], [10, 17], [11, 16], [12, 15], [13, 14]}
The pair, [2, 25], is a good pair, since its sum, 27,
belongs to the Samantha set, and knowing their product, 50,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 25], [5, 10]},
it is the only one whose sum belongs to the Samantha set.
The pair, [3, 24], is a not a good pair, even though its sum, 27,
belongs to the Samantha set, but just knowing their product, 72
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 36], [3, 24], [4, 18], [6, 12], [8, 9]}
the members of the subset, {[3, 24], [8, 9]}, all belong to the Samantha set.
The pair, [4, 23], is a good pair, since its sum, 27,
belongs to the Samantha set, and knowing their product, 92,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 46], [4, 23]},
it is the only one whose sum belongs to the Samantha set.
The pair, [5, 22], is a good pair, since its sum, 27,
belongs to the Samantha set, and knowing their product, 110,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 55], [5, 22], [10, 11]},
it is the only one whose sum belongs to the Samantha set.
The pair, [6, 21], is a not a good pair, even though its sum, 27,
belongs to the Samantha set, but just knowing their product, 126
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 63], [3, 42], [6, 21], [7, 18], [9, 14]}
the members of the subset, {[6, 21], [9, 14]}, all belong to the Samantha set.
The pair, [7, 20], is a good pair, since its sum, 27,
belongs to the Samantha set, and knowing their product, 140,
enables Perforio
to deduce them, since amongst his possibilities,
{[2, 70], [4, 35], [5, 28], [7, 20], [10, 14]},
it is the only one whose sum belongs to the Samantha set.
The pair, [8, 19], is a good pair, since its sum, 27,
belongs to the Samantha set, and knowing their product, 152,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 76], [4, 38], [8, 19]},
it is the only one whose sum belongs to the Samantha set.
The pair, [9, 18], is a good pair, since its sum, 27,
belongs to the Samantha set, and knowing their product, 162,
enables Perforio
to deduce them, since amongst his possibilities,
{[2, 81], [3, 54], [6, 27], [9, 18]},
it is the only one whose sum belongs to the Samantha set.
The pair, [10, 17], is a good pair, since its sum, 27,
belongs to the Samantha set, and knowing their product, 170,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 85], [5, 34], [10, 17]},
it is the only one whose sum belongs to the Samantha set.
The pair, [11, 16], is a good pair, since its sum, 27,
belongs to the Samantha set, and knowing their product, 176,
enables Perforio
to deduce them, since amongst his possibilities,
{[2, 88], [4, 44], [8, 22], [11, 16]},
it is the only one whose sum belongs to the Samantha set.
The pair, [12, 15], is a not a good pair, even though its sum, 27,
belongs to the Samantha set, but just knowing their product, 180
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities,
{[2, 90], [3, 60], [4, 45], [5, 36], [6, 30], [9, 20], [10, 18], [12, 15]}
the members of the subset, {[5, 36], [9, 20], [12, 15]},
all belong to the Samantha set.
The pair, [13, 14], is a good pair, since its sum, 27,
belongs to the Samantha set, and knowing their product, 182,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 91], [7, 26], [13, 14]},
it is the only one whose sum belongs to the Samantha set.
This ends examining the pairs whose sum is, 27
---------------------------------
For the member, 29,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 27], [3, 26], [4, 25], [5, 24], [6, 23], [7, 22], [8, 21],
[9, 20], [10, 19], [11, 18], [12, 17], [13, 16], [14, 15]}
The pair, [2, 27], is a good pair, since its sum, 29,
belongs to the Samantha set, and knowing their product, 54,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 27], [3, 18], [6, 9]},
it is the only one whose sum belongs to the Samantha set.
The pair, [3, 26], is a not a good pair, even though its sum, 29,
belongs to the Samantha set, but just knowing their product, 78
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 39], [3, 26], [6, 13]}
the members of the subset, {[2, 39], [3, 26]}, all belong to the Samantha set.
The pair, [4, 25], is a good pair, since its sum, 29,
belongs to the Samantha set, and knowing their product, 100,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 50], [4, 25], [5, 20]},
it is the only one whose sum belongs to the Samantha set.
The pair, [5, 24], is a not a good pair, even though its sum, 29,
belongs to the Samantha set, but just knowing their product, 120
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities,
{[2, 60], [3, 40], [4, 30], [5, 24], [6, 20], [8, 15], [10, 12]}
the members of the subset, {[5, 24], [8, 15]}, all belong to the Samantha set.
The pair, [6, 23], is a good pair, since its sum, 29,
belongs to the Samantha set, and knowing their product, 138,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 69], [3, 46], [6, 23]},
it is the only one whose sum belongs to the Samantha set.
The pair, [7, 22], is a good pair, since its sum, 29,
belongs to the Samantha set, and knowing their product, 154,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 77], [7, 22], [11, 14]},
it is the only one whose sum belongs to the Samantha set.
The pair, [8, 21], is a good pair, since its sum, 29,
belongs to the Samantha set, and knowing their product, 168,
enables Perforio
to deduce them, since amongst his possibilities,
{[2, 84], [3, 56], [4, 42], [6, 28], [7, 24], [8, 21], [12, 14]},
it is the only one whose sum belongs to the Samantha set.
The pair, [9, 20], is a not a good pair, even though its sum, 29,
belongs to the Samantha set, but just knowing their product, 180
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities,
{[2, 90], [3, 60], [4, 45], [5, 36], [6, 30], [9, 20], [10, 18], [12, 15]}
the members of the subset, {[5, 36], [9, 20], [12, 15]},
all belong to the Samantha set.
The pair, [10, 19], is a good pair, since its sum, 29,
belongs to the Samantha set, and knowing their product, 190,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 95], [5, 38], [10, 19]},
it is the only one whose sum belongs to the Samantha set.
The pair, [11, 18], is a good pair, since its sum, 29,
belongs to the Samantha set, and knowing their product, 198,
enables Perforio
to deduce them, since amongst his possibilities,
{[2, 99], [3, 66], [6, 33], [9, 22], [11, 18]},
it is the only one whose sum belongs to the Samantha set.
The pair, [12, 17], is a good pair, since its sum, 29,
belongs to the Samantha set, and knowing their product, 204,
enables Perforio
to deduce them, since amongst his possibilities,
{[3, 68], [4, 51], [6, 34], [12, 17]},
it is the only one whose sum belongs to the Samantha set.
The pair, [13, 16], is a good pair, since its sum, 29,
belongs to the Samantha set, and knowing their product, 208,
enables Perforio
to deduce them, since amongst his possibilities, {[4, 52], [8, 26], [13, 16]},
it is the only one whose sum belongs to the Samantha set.
The pair, [14, 15], is a not a good pair, even though its sum, 29,
belongs to the Samantha set, but just knowing their product, 210
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[3, 70], [5, 42], [6, 35], [7, 30], [10, 21], [14, 15]}
the members of the subset, {[5, 42], [6, 35], [7, 30], [14, 15]},
all belong to the Samantha set.
This ends examining the pairs whose sum is, 29
---------------------------------
For the member, 35,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 33], [3, 32], [4, 31], [5, 30], [6, 29], [7, 28], [8, 27],
[9, 26], [10, 25], [11, 24], [12, 23], [13, 22], [14, 21], [15, 20],
[16, 19], [17, 18]}
The pair, [2, 33], is a not a good pair, even though its sum, 35,
belongs to the Samantha set, but just knowing their product, 66
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 33], [3, 22], [6, 11]}
the members of the subset, {[2, 33], [6, 11]}, all belong to the Samantha set.
The pair, [3, 32], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 96,
enables Perforio
to deduce them, since amongst his possibilities,
{[2, 48], [3, 32], [4, 24], [6, 16], [8, 12]},
it is the only one whose sum belongs to the Samantha set.
The pair, [4, 31], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 124,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 62], [4, 31]},
it is the only one whose sum belongs to the Samantha set.
The pair, [5, 30], is a not a good pair, even though its sum, 35,
belongs to the Samantha set, but just knowing their product, 150
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 75], [3, 50], [5, 30], [6, 25], [10, 15]}
the members of the subset, {[3, 50], [5, 30]}, all belong to the Samantha set.
The pair, [6, 29], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 174,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 87], [3, 58], [6, 29]},
it is the only one whose sum belongs to the Samantha set.
The pair, [7, 28], is a not a good pair, even though its sum, 35,
belongs to the Samantha set, but just knowing their product, 196
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 98], [4, 49], [7, 28]}
the members of the subset, {[4, 49], [7, 28]}, all belong to the Samantha set.
The pair, [8, 27], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 216,
enables Perforio
to deduce them, since amongst his possibilities,
{[3, 72], [4, 54], [6, 36], [8, 27], [9, 24], [12, 18]},
it is the only one whose sum belongs to the Samantha set.
The pair, [9, 26], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 234,
enables Perforio
to deduce them, since amongst his possibilities,
{[3, 78], [6, 39], [9, 26], [13, 18]},
it is the only one whose sum belongs to the Samantha set.
The pair, [10, 25], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 250,
enables Perforio
to deduce them, since amongst his possibilities, {[5, 50], [10, 25]},
it is the only one whose sum belongs to the Samantha set.
The pair, [11, 24], is a not a good pair, even though its sum, 35,
belongs to the Samantha set, but just knowing their product, 264
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[3, 88], [4, 66], [6, 44], [8, 33], [11, 24], [12, 22]}
the members of the subset, {[8, 33], [11, 24]},
all belong to the Samantha set.
The pair, [12, 23], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 276,
enables Perforio
to deduce them, since amongst his possibilities,
{[3, 92], [4, 69], [6, 46], [12, 23]},
it is the only one whose sum belongs to the Samantha set.
The pair, [13, 22], is a not a good pair, even though its sum, 35,
belongs to the Samantha set, but just knowing their product, 286
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[11, 26], [13, 22]}
the members of the subset, {[11, 26], [13, 22]},
all belong to the Samantha set.
The pair, [14, 21], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 294,
enables Perforio
to deduce them, since amongst his possibilities,
{[3, 98], [6, 49], [7, 42], [14, 21]},
it is the only one whose sum belongs to the Samantha set.
The pair, [15, 20], is a not a good pair, even though its sum, 35,
belongs to the Samantha set, but just knowing their product, 300
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[4, 75], [5, 60], [6, 50], [10, 30], [12, 25], [15, 20]}
the members of the subset, {[12, 25], [15, 20]},
all belong to the Samantha set.
The pair, [16, 19], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 304,
enables Perforio
to deduce them, since amongst his possibilities, {[4, 76], [8, 38], [16, 19]},
it is the only one whose sum belongs to the Samantha set.
The pair, [17, 18], is a good pair, since its sum, 35,
belongs to the Samantha set, and knowing their product, 306,
enables Perforio
to deduce them, since amongst his possibilities, {[6, 51], [9, 34], [17, 18]},
it is the only one whose sum belongs to the Samantha set.
This ends examining the pairs whose sum is, 35
---------------------------------
For the member, 37,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 35], [3, 34], [4, 33], [5, 32], [6, 31], [7, 30], [8, 29],
[9, 28], [10, 27], [11, 26], [12, 25], [13, 24], [14, 23], [15, 22],
[16, 21], [17, 20], [18, 19]}
The pair, [2, 35], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 70
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 35], [5, 14], [7, 10]}
the members of the subset, {[2, 35], [7, 10]}, all belong to the Samantha set.
The pair, [3, 34], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 102
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 51], [3, 34], [6, 17]}
the members of the subset, {[2, 51], [3, 34], [6, 17]},
all belong to the Samantha set.
The pair, [4, 33], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 132
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 66], [3, 44], [4, 33], [6, 22], [11, 12]}
the members of the subset, {[3, 44], [4, 33], [11, 12]},
all belong to the Samantha set.
The pair, [5, 32], is a good pair, since its sum, 37,
belongs to the Samantha set, and knowing their product, 160,
enables Perforio
to deduce them, since amongst his possibilities,
{[2, 80], [4, 40], [5, 32], [8, 20], [10, 16]},
it is the only one whose sum belongs to the Samantha set.
The pair, [6, 31], is a good pair, since its sum, 37,
belongs to the Samantha set, and knowing their product, 186,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 93], [3, 62], [6, 31]},
it is the only one whose sum belongs to the Samantha set.
The pair, [7, 30], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 210
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[3, 70], [5, 42], [6, 35], [7, 30], [10, 21], [14, 15]}
the members of the subset, {[5, 42], [6, 35], [7, 30], [14, 15]},
all belong to the Samantha set.
The pair, [8, 29], is a good pair, since its sum, 37,
belongs to the Samantha set, and knowing their product, 232,
enables Perforio
to deduce them, since amongst his possibilities, {[4, 58], [8, 29]},
it is the only one whose sum belongs to the Samantha set.
The pair, [9, 28], is a good pair, since its sum, 37,
belongs to the Samantha set, and knowing their product, 252,
enables Perforio
to deduce them, since amongst his possibilities,
{[3, 84], [4, 63], [6, 42], [7, 36], [9, 28], [12, 21], [14, 18]},
it is the only one whose sum belongs to the Samantha set.
The pair, [10, 27], is a good pair, since its sum, 37,
belongs to the Samantha set, and knowing their product, 270,
enables Perforio
to deduce them, since amongst his possibilities,
{[3, 90], [5, 54], [6, 45], [9, 30], [10, 27], [15, 18]},
it is the only one whose sum belongs to the Samantha set.
The pair, [11, 26], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 286
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[11, 26], [13, 22]}
the members of the subset, {[11, 26], [13, 22]},
all belong to the Samantha set.
The pair, [12, 25], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 300
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[4, 75], [5, 60], [6, 50], [10, 30], [12, 25], [15, 20]}
the members of the subset, {[12, 25], [15, 20]},
all belong to the Samantha set.
The pair, [13, 24], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 312
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[4, 78], [6, 52], [8, 39], [12, 26], [13, 24]}
the members of the subset, {[8, 39], [13, 24]},
all belong to the Samantha set.
The pair, [14, 23], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 322
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[7, 46], [14, 23]}
the members of the subset, {[7, 46], [14, 23]},
all belong to the Samantha set.
The pair, [15, 22], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 330
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[5, 66], [6, 55], [10, 33], [11, 30], [15, 22]}
the members of the subset, {[11, 30], [15, 22]},
all belong to the Samantha set.
The pair, [16, 21], is a good pair, since its sum, 37,
belongs to the Samantha set, and knowing their product, 336,
enables Perforio
to deduce them, since amongst his possibilities,
{[4, 84], [6, 56], [7, 48], [8, 42], [12, 28], [14, 24], [16, 21]},
it is the only one whose sum belongs to the Samantha set.
The pair, [17, 20], is a good pair, since its sum, 37,
belongs to the Samantha set, and knowing their product, 340,
enables Perforio
to deduce them, since amongst his possibilities,
{[4, 85], [5, 68], [10, 34], [17, 20]},
it is the only one whose sum belongs to the Samantha set.
The pair, [18, 19], is a not a good pair, even though its sum, 37,
belongs to the Samantha set, but just knowing their product, 342
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[6, 57], [9, 38], [18, 19]}
the members of the subset, {[9, 38], [18, 19]},
all belong to the Samantha set.
This ends examining the pairs whose sum is, 37
---------------------------------
For the member, 41,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 39], [3, 38], [4, 37], [5, 36], [6, 35], [7, 34], [8, 33],
[9, 32], [10, 31], [11, 30], [12, 29], [13, 28], [14, 27], [15, 26],
[16, 25], [17, 24], [18, 23], [19, 22], [20, 21]}
The pair, [2, 39], is a not a good pair, even though its sum, 41,
belongs to the Samantha set, but just knowing their product, 78
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 39], [3, 26], [6, 13]}
the members of the subset, {[2, 39], [3, 26]}, all belong to the Samantha set.
The pair, [3, 38], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 114,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 57], [3, 38], [6, 19]},
it is the only one whose sum belongs to the Samantha set.
The pair, [4, 37], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 148,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 74], [4, 37]},
it is the only one whose sum belongs to the Samantha set.
The pair, [5, 36], is a not a good pair, even though its sum, 41,
belongs to the Samantha set, but just knowing their product, 180
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities,
{[2, 90], [3, 60], [4, 45], [5, 36], [6, 30], [9, 20], [10, 18], [12, 15]}
the members of the subset, {[5, 36], [9, 20], [12, 15]},
all belong to the Samantha set.
The pair, [6, 35], is a not a good pair, even though its sum, 41,
belongs to the Samantha set, but just knowing their product, 210
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[3, 70], [5, 42], [6, 35], [7, 30], [10, 21], [14, 15]}
the members of the subset, {[5, 42], [6, 35], [7, 30], [14, 15]},
all belong to the Samantha set.
The pair, [7, 34], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 238,
enables Perforio
to deduce them, since amongst his possibilities, {[7, 34], [14, 17]},
it is the only one whose sum belongs to the Samantha set.
The pair, [8, 33], is a not a good pair, even though its sum, 41,
belongs to the Samantha set, but just knowing their product, 264
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[3, 88], [4, 66], [6, 44], [8, 33], [11, 24], [12, 22]}
the members of the subset, {[8, 33], [11, 24]},
all belong to the Samantha set.
The pair, [9, 32], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 288,
enables Perforio
to deduce them, since amongst his possibilities,
{[3, 96], [4, 72], [6, 48], [8, 36], [9, 32], [12, 24], [16, 18]},
it is the only one whose sum belongs to the Samantha set.
The pair, [10, 31], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 310,
enables Perforio
to deduce them, since amongst his possibilities, {[5, 62], [10, 31]},
it is the only one whose sum belongs to the Samantha set.
The pair, [11, 30], is a not a good pair, even though its sum, 41,
belongs to the Samantha set, but just knowing their product, 330
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[5, 66], [6, 55], [10, 33], [11, 30], [15, 22]}
the members of the subset, {[11, 30], [15, 22]},
all belong to the Samantha set.
The pair, [12, 29], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 348,
enables Perforio
to deduce them, since amongst his possibilities, {[4, 87], [6, 58], [12, 29]},
it is the only one whose sum belongs to the Samantha set.
The pair, [13, 28], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 364,
enables Perforio
to deduce them, since amongst his possibilities,
{[4, 91], [7, 52], [13, 28], [14, 26]},
it is the only one whose sum belongs to the Samantha set.
The pair, [14, 27], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 378,
enables Perforio
to deduce them, since amongst his possibilities,
{[6, 63], [7, 54], [9, 42], [14, 27], [18, 21]},
it is the only one whose sum belongs to the Samantha set.
The pair, [15, 26], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 390,
enables Perforio
to deduce them, since amongst his possibilities,
{[5, 78], [6, 65], [10, 39], [13, 30], [15, 26]},
it is the only one whose sum belongs to the Samantha set.
The pair, [16, 25], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 400,
enables Perforio
to deduce them, since amongst his possibilities,
{[5, 80], [8, 50], [10, 40], [16, 25]},
it is the only one whose sum belongs to the Samantha set.
The pair, [17, 24], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 408,
enables Perforio
to deduce them, since amongst his possibilities,
{[6, 68], [8, 51], [12, 34], [17, 24]},
it is the only one whose sum belongs to the Samantha set.
The pair, [18, 23], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 414,
enables Perforio
to deduce them, since amongst his possibilities, {[6, 69], [9, 46], [18, 23]},
it is the only one whose sum belongs to the Samantha set.
The pair, [19, 22], is a good pair, since its sum, 41,
belongs to the Samantha set, and knowing their product, 418,
enables Perforio
to deduce them, since amongst his possibilities, {[11, 38], [19, 22]},
it is the only one whose sum belongs to the Samantha set.
The pair, [20, 21], is a not a good pair, even though its sum, 41,
belongs to the Samantha set, but just knowing their product, 420
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {
[5, 84], [6, 70], [7, 60], [10, 42], [12, 35], [14, 30], [15, 28], [20, 21]
}
the members of the subset, {[12, 35], [20, 21]},
all belong to the Samantha set.
This ends examining the pairs whose sum is, 41
---------------------------------
For the member, 47,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 45], [3, 44], [4, 43], [5, 42], [6, 41], [7, 40], [8, 39],
[9, 38], [10, 37], [11, 36], [12, 35], [13, 34], [14, 33], [15, 32],
[16, 31], [17, 30], [18, 29], [19, 28], [20, 27], [21, 26], [22, 25],
[23, 24]}
The pair, [2, 45], is a not a good pair, even though its sum, 47,
belongs to the Samantha set, but just knowing their product, 90
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 45], [3, 30], [5, 18], [6, 15], [9, 10]}
the members of the subset, {[2, 45], [5, 18]}, all belong to the Samantha set.
The pair, [3, 44], is a not a good pair, even though its sum, 47,
belongs to the Samantha set, but just knowing their product, 132
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 66], [3, 44], [4, 33], [6, 22], [11, 12]}
the members of the subset, {[3, 44], [4, 33], [11, 12]},
all belong to the Samantha set.
The pair, [4, 43], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 172,
enables Perforio
to deduce them, since amongst his possibilities, {[2, 86], [4, 43]},
it is the only one whose sum belongs to the Samantha set.
The pair, [5, 42], is a not a good pair, even though its sum, 47,
belongs to the Samantha set, but just knowing their product, 210
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[3, 70], [5, 42], [6, 35], [7, 30], [10, 21], [14, 15]}
the members of the subset, {[5, 42], [6, 35], [7, 30], [14, 15]},
all belong to the Samantha set.
The pair, [6, 41], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 246,
enables Perforio
to deduce them, since amongst his possibilities, {[3, 82], [6, 41]},
it is the only one whose sum belongs to the Samantha set.
The pair, [7, 40], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 280,
enables Perforio
to deduce them, since amongst his possibilities,
{[4, 70], [5, 56], [7, 40], [8, 35], [10, 28], [14, 20]},
it is the only one whose sum belongs to the Samantha set.
The pair, [8, 39], is a not a good pair, even though its sum, 47,
belongs to the Samantha set, but just knowing their product, 312
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[4, 78], [6, 52], [8, 39], [12, 26], [13, 24]}
the members of the subset, {[8, 39], [13, 24]},
all belong to the Samantha set.
The pair, [9, 38], is a not a good pair, even though its sum, 47,
belongs to the Samantha set, but just knowing their product, 342
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[6, 57], [9, 38], [18, 19]}
the members of the subset, {[9, 38], [18, 19]},
all belong to the Samantha set.
The pair, [10, 37], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 370,
enables Perforio
to deduce them, since amongst his possibilities, {[5, 74], [10, 37]},
it is the only one whose sum belongs to the Samantha set.
The pair, [11, 36], is a not a good pair, even though its sum, 47,
belongs to the Samantha set, but just knowing their product, 396
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[4, 99], [6, 66], [9, 44], [11, 36], [12, 33], [18, 22]}
the members of the subset, {[9, 44], [11, 36]},
all belong to the Samantha set.
The pair, [12, 35], is a not a good pair, even though its sum, 47,
belongs to the Samantha set, but just knowing their product, 420
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {
[5, 84], [6, 70], [7, 60], [10, 42], [12, 35], [14, 30], [15, 28], [20, 21]
}
the members of the subset, {[12, 35], [20, 21]},
all belong to the Samantha set.
The pair, [13, 34], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 442,
enables Perforio
to deduce them, since amongst his possibilities, {[13, 34], [17, 26]},
it is the only one whose sum belongs to the Samantha set.
The pair, [14, 33], is a not a good pair, even though its sum, 47,
belongs to the Samantha set, but just knowing their product, 462
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[6, 77], [7, 66], [11, 42], [14, 33], [21, 22]}
the members of the subset, {[11, 42], [14, 33]},
all belong to the Samantha set.
The pair, [15, 32], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 480,
enables Perforio
to deduce them, since amongst his possibilities, {
[5, 96], [6, 80], [8, 60], [10, 48], [12, 40], [15, 32], [16, 30], [20, 24]
}, it is the only one whose sum belongs to the Samantha set.
The pair, [16, 31], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 496,
enables Perforio
to deduce them, since amongst his possibilities, {[8, 62], [16, 31]},
it is the only one whose sum belongs to the Samantha set.
The pair, [17, 30], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 510,
enables Perforio
to deduce them, since amongst his possibilities,
{[6, 85], [10, 51], [15, 34], [17, 30]},
it is the only one whose sum belongs to the Samantha set.
The pair, [18, 29], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 522,
enables Perforio
to deduce them, since amongst his possibilities, {[6, 87], [9, 58], [18, 29]},
it is the only one whose sum belongs to the Samantha set.
The pair, [19, 28], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 532,
enables Perforio
to deduce them, since amongst his possibilities, {[7, 76], [14, 38], [19, 28]},
it is the only one whose sum belongs to the Samantha set.
The pair, [20, 27], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 540,
enables Perforio
to deduce them, since amongst his possibilities,
{[6, 90], [9, 60], [10, 54], [12, 45], [15, 36], [18, 30], [20, 27]},
it is the only one whose sum belongs to the Samantha set.
The pair, [21, 26], is a not a good pair, even though its sum, 47,
belongs to the Samantha set, but just knowing their product, 546
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[6, 91], [7, 78], [13, 42], [14, 39], [21, 26]}
the members of the subset, {[14, 39], [21, 26]},
all belong to the Samantha set.
The pair, [22, 25], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 550,
enables Perforio
to deduce them, since amongst his possibilities, {[10, 55], [11, 50], [22, 25]},
it is the only one whose sum belongs to the Samantha set.
The pair, [23, 24], is a good pair, since its sum, 47,
belongs to the Samantha set, and knowing their product, 552,
enables Perforio
to deduce them, since amongst his possibilities,
{[6, 92], [8, 69], [12, 46], [23, 24]},
it is the only one whose sum belongs to the Samantha set.
This ends examining the pairs whose sum is, 47
---------------------------------
For the member, 53,
of the Samantha set, let us examine all the pairs that sum-up to it.
namely , {[2, 51], [3, 50], [4, 49], [5, 48], [6, 47], [7, 46], [8, 45],
[9, 44], [10, 43], [11, 42], [12, 41], [13, 40], [14, 39], [15, 38],
[16, 37], [17, 36], [18, 35], [19, 34], [20, 33], [21, 32], [22, 31],
[23, 30], [24, 29], [25, 28], [26, 27]}
The pair, [2, 51], is a not a good pair, even though its sum, 53,
belongs to the Samantha set, but just knowing their product, 102
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 51], [3, 34], [6, 17]}
the members of the subset, {[2, 51], [3, 34], [6, 17]},
all belong to the Samantha set.
The pair, [3, 50], is a not a good pair, even though its sum, 53,
belongs to the Samantha set, but just knowing their product, 150
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 75], [3, 50], [5, 30], [6, 25], [10, 15]}
the members of the subset, {[3, 50], [5, 30]}, all belong to the Samantha set.
The pair, [4, 49], is a not a good pair, even though its sum, 53,
belongs to the Samantha set, but just knowing their product, 196
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[2, 98], [4, 49], [7, 28]}
the members of the subset, {[4, 49], [7, 28]}, all belong to the Samantha set.
The pair, [5, 48], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 240,
enables Perforio
to deduce them, since amongst his possibilities,
{[3, 80], [4, 60], [5, 48], [6, 40], [8, 30], [10, 24], [12, 20], [15, 16]}
, it is the only one whose sum belongs to the Samantha set.
The pair, [6, 47], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 282,
enables Perforio
to deduce them, since amongst his possibilities, {[3, 94], [6, 47]},
it is the only one whose sum belongs to the Samantha set.
The pair, [7, 46], is a not a good pair, even though its sum, 53,
belongs to the Samantha set, but just knowing their product, 322
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[7, 46], [14, 23]}
the members of the subset, {[7, 46], [14, 23]},
all belong to the Samantha set.
The pair, [8, 45], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 360,
enables Perforio
to deduce them, since amongst his possibilities, {[4, 90], [5, 72], [6, 60],
[8, 45], [9, 40], [10, 36], [12, 30], [15, 24], [18, 20]},
it is the only one whose sum belongs to the Samantha set.
The pair, [9, 44], is a not a good pair, even though its sum, 53,
belongs to the Samantha set, but just knowing their product, 396
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[4, 99], [6, 66], [9, 44], [11, 36], [12, 33], [18, 22]}
the members of the subset, {[9, 44], [11, 36]},
all belong to the Samantha set.
The pair, [10, 43], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 430,
enables Perforio
to deduce them, since amongst his possibilities, {[5, 86], [10, 43]},
it is the only one whose sum belongs to the Samantha set.
The pair, [11, 42], is a not a good pair, even though its sum, 53,
belongs to the Samantha set, but just knowing their product, 462
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[6, 77], [7, 66], [11, 42], [14, 33], [21, 22]}
the members of the subset, {[11, 42], [14, 33]},
all belong to the Samantha set.
The pair, [12, 41], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 492,
enables Perforio
to deduce them, since amongst his possibilities, {[6, 82], [12, 41]},
it is the only one whose sum belongs to the Samantha set.
The pair, [13, 40], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 520,
enables Perforio
to deduce them, since amongst his possibilities,
{[8, 65], [10, 52], [13, 40], [20, 26]},
it is the only one whose sum belongs to the Samantha set.
The pair, [14, 39], is a not a good pair, even though its sum, 53,
belongs to the Samantha set, but just knowing their product, 546
does not enable him to deduce them, since amongst the set of his initial pos\
sibilities, {[6, 91], [7, 78], [13, 42], [14, 39], [21, 26]}
the members of the subset, {[14, 39], [21, 26]},
all belong to the Samantha set.
The pair, [15, 38], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 570,
enables Perforio
to deduce them, since amongst his possibilities,
{[6, 95], [10, 57], [15, 38], [19, 30]},
it is the only one whose sum belongs to the Samantha set.
The pair, [16, 37], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 592,
enables Perforio
to deduce them, since amongst his possibilities, {[8, 74], [16, 37]},
it is the only one whose sum belongs to the Samantha set.
The pair, [17, 36], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 612,
enables Perforio
to deduce them, since amongst his possibilities,
{[9, 68], [12, 51], [17, 36], [18, 34]},
it is the only one whose sum belongs to the Samantha set.
The pair, [18, 35], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 630,
enables Perforio
to deduce them, since amongst his possibilities,
{[7, 90], [9, 70], [10, 63], [14, 45], [15, 42], [18, 35], [21, 30]},
it is the only one whose sum belongs to the Samantha set.
The pair, [19, 34], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 646,
enables Perforio
to deduce them, since amongst his possibilities, {[17, 38], [19, 34]},
it is the only one whose sum belongs to the Samantha set.
The pair, [20, 33], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 660,
enables Perforio
to deduce them, since amongst his possibilities,
{[10, 66], [11, 60], [12, 55], [15, 44], [20, 33], [22, 30]},
it is the only one whose sum belongs to the Samantha set.
The pair, [21, 32], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 672,
enables Perforio
to deduce them, since amongst his possibilities,
{[7, 96], [8, 84], [12, 56], [14, 48], [16, 42], [21, 32], [24, 28]},
it is the only one whose sum belongs to the Samantha set.
The pair, [22, 31], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 682,
enables Perforio
to deduce them, since amongst his possibilities, {[11, 62], [22, 31]},
it is the only one whose sum belongs to the Samantha set.
The pair, [23, 30], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 690,
enables Perforio
to deduce them, since amongst his possibilities, {[10, 69], [15, 46], [23, 30]},
it is the only one whose sum belongs to the Samantha set.
The pair, [24, 29], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 696,
enables Perforio
to deduce them, since amongst his possibilities, {[8, 87], [12, 58], [24, 29]},
it is the only one whose sum belongs to the Samantha set.
The pair, [25, 28], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 700,
enables Perforio
to deduce them, since amongst his possibilities,
{[10, 70], [14, 50], [20, 35], [25, 28]},
it is the only one whose sum belongs to the Samantha set.
The pair, [26, 27], is a good pair, since its sum, 53,
belongs to the Samantha set, and knowing their product, 702,
enables Perforio
to deduce them, since amongst his possibilities,
{[9, 78], [13, 54], [18, 39], [26, 27]},
it is the only one whose sum belongs to the Samantha set.
This ends examining the pairs whose sum is, 53
---------------------------------
So indeed the set of pairs compatible with the data and meta-data so far is
{[2, 9], [2, 25], [2, 27], [3, 8], [3, 32], [3, 38], [4, 7], [4, 13], [4, 19],
[4, 23], [4, 25], [4, 31], [4, 37], [4, 43], [5, 22], [5, 32], [5, 48],
[6, 23], [6, 29], [6, 31], [6, 41], [6, 47], [7, 16], [7, 20], [7, 22],
[7, 34], [7, 40], [8, 19], [8, 21], [8, 27], [8, 29], [8, 45], [9, 18],
[9, 26], [9, 28], [9, 32], [10, 13], [10, 17], [10, 19], [10, 25], [10, 27],
[10, 31], [10, 37], [10, 43], [11, 16], [11, 18], [12, 17], [12, 23],
[12, 29], [12, 41], [13, 14], [13, 16], [13, 28], [13, 34], [13, 40],
[14, 21], [14, 27], [15, 26], [15, 32], [15, 38], [16, 19], [16, 21],
[16, 25], [16, 31], [16, 37], [17, 18], [17, 20], [17, 24], [17, 30],
[17, 36], [18, 23], [18, 29], [18, 35], [19, 22], [19, 28], [19, 34],
[20, 27], [20, 33], [21, 32], [22, 25], [22, 31], [23, 24], [23, 30],
[24, 29], [25, 28], [26, 27]}
This ends Part II.
-----------------------------------------------------------------
Part III: The Set of Pairs that would enable Samantha to know the Numbers
Samantha (and we) know that the possible pairs are as above. From this SHE c\
an deduce the numbers.
This means that the pair is such that her number only appears ONCE in the se\
t of sums of the above set of pairs.
The set of possible good pairs whose sum is, 11, is , {[2, 9], [3, 8], [4, 7]}
since there are, 3, such pairs, this is not an option.
The set of possible good pairs whose sum is, 17, is , {[4, 13]}
since there is only one such pair , that's indeed a possibility.
The set of possible good pairs whose sum is, 23, is ,
{[4, 19], [7, 16], [10, 13]}
since there are, 3, such pairs, this is not an option.
The set of possible good pairs whose sum is, 27, is , {[2, 25], [4, 23],
[5, 22], [7, 20], [8, 19], [9, 18], [10, 17], [11, 16], [13, 14]}
since there are, 9, such pairs, this is not an option.
The set of possible good pairs whose sum is, 29, is , {[2, 27], [4, 25],
[6, 23], [7, 22], [8, 21], [10, 19], [11, 18], [12, 17], [13, 16]}
since there are, 9, such pairs, this is not an option.
The set of possible good pairs whose sum is, 35, is , {[3, 32], [4, 31],
[6, 29], [8, 27], [9, 26], [10, 25], [12, 23], [14, 21], [16, 19], [17, 18]
}
since there are, 10, such pairs, this is not an option.
The set of possible good pairs whose sum is, 37, is ,
{[5, 32], [6, 31], [8, 29], [9, 28], [10, 27], [16, 21], [17, 20]}
since there are, 7, such pairs, this is not an option.
The set of possible good pairs whose sum is, 41, is , {[3, 38], [4, 37],
[7, 34], [9, 32], [10, 31], [12, 29], [13, 28], [14, 27], [15, 26],
[16, 25], [17, 24], [18, 23], [19, 22]}
since there are, 13, such pairs, this is not an option.
The set of possible good pairs whose sum is, 47, is , {[4, 43], [6, 41],
[7, 40], [10, 37], [13, 34], [15, 32], [16, 31], [17, 30], [18, 29],
[19, 28], [20, 27], [22, 25], [23, 24]}
since there are, 13, such pairs, this is not an option.
The set of possible good pairs whose sum is, 53, is , {[5, 48], [6, 47],
[8, 45], [10, 43], [12, 41], [13, 40], [15, 38], [16, 37], [17, 36],
[18, 35], [19, 34], [20, 33], [21, 32], [22, 31], [23, 30], [24, 29],
[25, 28], [26, 27]}
since there are, 18, such pairs, this is not an option.
-------------------------------------------------------------
This ends part III.
---------------------------------------------------------
To conclude:
Answer: the UNIQE possible pair is, [4, 13]
-----------------------------
This ends the detailed solution. Thanks for reading this.
Detailed Solution to Puzzle Number, 1
We will solve the following challenging "meta-knowledge" puzzle inspired by \
a problem Peter Winkler's book Mathematical Mind-Benders (p. 114).
Puzzle: All integers greater than one but less than, 50, are put into a hat a\
nd two are drawn, let's call the smaller number x and the larger number \
y.
2 2
Samantha is given, 3 y - x, and Porfirio is given, 4 x - 6 x y + 3 y
Sanantha says:
" I can see that you don't know the numbers. "
Porfirio replies: NOW I do know them.
Now Samantha says: NOW I do TOO!
What are the possible pairs of numbers consistent with the above data and me\
ta-data?
----------------------------------------------------------------------
Answer: the UNIQE possible pair is, [4, 6]
Detailed Solution
Part I: The Samantha Set
Samantha KNOWS, from the value of, 3 y - x, that Porfirio is not able to fig\
2 2
ure out the numbers,x,y, from his value, namely, 4 x - 6 x y + 3 y , .
We claim that the only possible numbers that Samantha can possibly have, whe\
re the numbers from the hat are distinct and less than, 50, are
{14, 28, 49, 56, 84}
Let's prove it.
------------------------------------------------------
There is no way that Samantha can have the number, 7
If Samantha's number, 3 y - x, is , 7, then she can't know that Porfirio does\
2 2
NOT know the numbers, where Porfirio's number is, 4 x - 6 x y + 3 y
Let's prove it.
Indeed, if, as is possible, the numbers x