41, Meta-Knowledge puzzle inspired by the the Puzzle "Sums and Products" in \
Peter Winkler's Wonderful book "Mathematical Mind-Benders"
By Shalosh B. Ekhad
On page 114 of Peter Winkler's book,"Mathematical Mind-Benders" (A.K. Peters\
, 2007), the following challenging puzzle appeared
Puzzle Number, 0
All integers greater than one but less than, 100, are put into a hat and two \
are drawn, let's call the smaller number x and the larger number y.
Samantha is given, y + x, and Porfirio is given, y x
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?
----------------------------------------------------------------------
Inspired by this puzzle, I composed, 40,
similar puzzles. The answers are at the bottom
For (very!) detailed solutions see a separate file also available from Doron\
Zeilberger's website.
Puzzle Number, 1
All integers greater than one but less than, 50, are put into a hat and two a\
re 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 2
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 2 y + 3 x, and Porfirio is given, 4 x - 2 x y + 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 3
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 8 y - 6 x, and Porfirio is given, 5 x - 8 x y + 4 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 4
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, y + 3 x, and Porfirio is given, 5 x - 4 x y + 2 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 5
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, y + x, and Porfirio is given, 6 x - 10 x y + 5 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 6
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 3 y + 2 x, and Porfirio is given, 6 x - 4 x y + 2 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 7
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 8 y - 4 x, and Porfirio is given, 6 x - 2 x y + 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 8
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 4 y - 2 x, and Porfirio is given, 7 x - 8 x y + 4 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 9
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 3 y + 3 x, and Porfirio is given, 7 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 10
All integers greater than one but less than, 50, are put into a hat and two a\
re 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, 7 x - 2 x y + 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 11
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 6 y - 5 x, and Porfirio is given, 8 x - 14 x y + 7 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 12
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 2 y - x, and Porfirio is given, 8 x - 10 x y + 5 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 13
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 8 y - 4 x, and Porfirio is given, 8 x - 10 x y + 5 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 14
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 2 y + x, and Porfirio is given, 8 x - 4 x y + 2 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 15
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, y + 5 x, and Porfirio is given, 8 x - 2 x y + 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 16
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, y + x, and Porfirio is given, 9 x - 16 x y + 8 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 17
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 2 y + 2 x, and Porfirio is given, 9 x - 14 x y + 7 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 18
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 6 y - 2 x, and Porfirio is given, 9 x - 12 x y + 6 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 19
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 6 y + 2 x, and Porfirio is given, 9 x - 10 x y + 5 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 20
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 4 y - 2 x, and Porfirio is given, 9 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 21
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 4 y - x, and Porfirio is given, 9 x - 4 x y + 2 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 22
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 2 y + 4 x, and Porfirio is given, 9 x - 2 x y + 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 23
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 8 y - 6 x, and Porfirio is given, 10 x - 16 x y + 8 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 24
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 5 y - 4 x, and Porfirio is given, 10 x - 12 x y + 6 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 25
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 4 y - 2 x, and Porfirio is given, 10 x - 8 x y + 4 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 26
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 4 y + 2 x, and Porfirio is given, 10 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 27
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 5 y - 3 x, and Porfirio is given, 11 x - 16 x y + 8 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 28
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 6 y - x, and Porfirio is given, 11 x - 14 x y + 7 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 29
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 7 y - 4 x, and Porfirio is given, 11 x - 12 x y + 6 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 30
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, y + 4 x, and Porfirio is given, 11 x - 8 x y + 4 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 31
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 8 y - 2 x, and Porfirio is given, 11 x - 8 x y + 4 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 32
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 6 y - 2 x, and Porfirio is given, 12 x - 16 x y + 8 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 33
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 6 y + 2 x, and Porfirio is given, 12 x - 14 x y + 7 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 34
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 8 y - 2 x, and Porfirio is given, 12 x - 10 x y + 5 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 35
All integers greater than one but less than, 50, are put into a hat and two a\
re 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, 13 x - 16 x y + 8 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 36
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 3 y + 5 x, and Porfirio is given, 13 x - 14 x y + 7 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 37
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 7 y + x, and Porfirio is given, 13 x - 12 x y + 6 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 38
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 4 y - 2 x, and Porfirio is given, 14 x - 16 x y + 8 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 39
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, 3 y + 3 x, and Porfirio is given, 14 x - 12 x y + 6 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 numbers?
----------------------------------------------------------------------
Puzzle Number, 40
All integers greater than one but less than, 50, are put into a hat and two a\
re drawn, let's call the smaller number x and the larger number y.
2 2
Samantha is given, y + 4 x, and Porfirio is given, 15 x - 14 x y + 7 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 numbers?
----------------------------------------------------------------------
------------------------------------------------------------------
Answers
Answer to the (original) puzzle 0 is, 4, 13
Answer to Puzzle Number , 1, is , 4, 6
Answer to Puzzle Number , 2, is , 3, 8
Answer to Puzzle Number , 3, is , 18, 36
Answer to Puzzle Number , 4, is , 21, 42
Answer to Puzzle Number , 5, is , 47, 49
Answer to Puzzle Number , 6, is , 5, 7
Answer to Puzzle Number , 7, is , 2, 49
Answer to Puzzle Number , 8, is , 2, 35
Answer to Puzzle Number , 9, is , 33, 35
Answer to Puzzle Number , 10, is , 13, 14
Answer to Puzzle Number , 11, is , 5, 6
Answer to Puzzle Number , 12, is , 3, 4
Answer to Puzzle Number , 13, is , 3, 4
Answer to Puzzle Number , 14, is , 3, 4
Answer to Puzzle Number , 15, is , 2, 8
Answer to Puzzle Number , 16, is , 47, 49
Answer to Puzzle Number , 17, is , 47, 49
Answer to Puzzle Number , 18, is , 24, 42
Answer to Puzzle Number , 19, is , 47, 49
Answer to Puzzle Number , 20, is , 2, 49
Answer to Puzzle Number , 21, is , 2, 49
Answer to Puzzle Number , 22, is , 9, 33
Answer to Puzzle Number , 23, is , 18, 36
Answer to Puzzle Number , 24, is , 4, 5
Answer to Puzzle Number , 25, is , 2, 49
Answer to Puzzle Number , 26, is , 17, 19
Answer to Puzzle Number , 27, is , 6, 8
Answer to Puzzle Number , 28, is , 6, 11
Answer to Puzzle Number , 29, is , 8, 11
Answer to Puzzle Number , 30, is , 2, 7
Answer to Puzzle Number , 31, is , 4, 6
Answer to Puzzle Number , 32, is , 24, 42
Answer to Puzzle Number , 33, is , 2, 6
Answer to Puzzle Number , 34, is , 3, 6
Answer to Puzzle Number , 35, is , 47, 49
Answer to Puzzle Number , 36, is , 44, 49
Answer to Puzzle Number , 37, is , 8, 13
Answer to Puzzle Number , 38, is , 2, 35
Answer to Puzzle Number , 39, is , 33, 35
Answer to Puzzle Number , 40, is , 2, 7