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