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