SAT High School

A monkey has filled in a 3 × 3 grid with the numbers 1, 2,. . . , 9. A cat writes down the three numbers obtained by multiplying the numbers in each horizontal row. A dog writes down the three numbers obtained by multiplying the numbers in each vertical column. Can the monkey fill in the grid in such a way that the cat and dog obtain the same lists of three numbers? what if the monkey writes the numbers 1, 2,. . . , 25 in a 5 × 5 grid? or 1, 2,. . . , 121 in a 11 × 11 grid? can you find any conditions on n that guarantee that it is possible or any conditions that guarantee that it is impossible for the monkey to write the numbers 1, 2,. . . , n2 in an n × n grid so that the cat and the dog obtain the same lists of numbers?

Answers

Answer 1

Yes, it is possible for the monkey to fill in the 3 x 3 grid in such a way that the cat and dog obtain the same lists of three numbers.

What if the monkey writes the numbers 1, 2,. . . , 25 in a 5 × 5 grid? or 1, 2,. . . , 121 in a 11 × 11 grid?For example, the monkey could fill the 3x3 grid with the numbers 1, 4, 7; 2, 5, 8; and 3, 6, 9.Yes, it is possible for the monkey to fill in the 5 x 5 grid with the numbers 1, 2, ... , 25 in such a way that the cat and dog obtain the same lists of three numbers. For example, the monkey could fill the 5x5 grid with the numbers 1, 8, 15, 22, 25; 2, 7, 16, 21, 24; 3, 6, 17, 20, 23; 4, 5, 18, 19, 12; and 9, 10, 11, 14, 13.Yes, it is possible for the monkey to fill in the 11 x 11 grid with the numbers 1, 2, ... , 121 in such a way that the cat and dog obtain the same lists of three numbers. For example, the monkey could fill the 11x11 grid with the numbers 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 121; 2, 9, 20, 27, 38, 45, 56, 63, 74, 81, 120; 3, 8, 21, 26, 39, 44, 57, 62, 75, 80, 119; 4, 7, 22, 25, 40, 43, 58, 61, 76, 79, 118; 5, 6, 23, 24, 41, 42, 59, 60, 77, 78, 117; 11, 12, 13, 14, 15, 16, 17, 18, 33, 34, 115; 10, 13, 22, 25, 34, 37, 46, 49, 68, 71, 114; 9, 14, 21, 26, 33, 38, 45, 50, 67, 72, 113; 8, 15, 20, 27, 32, 39, 44, 51, 66, 73, 112; and so on.In general, it is possible for the monkey to fill in an n x n grid with the numbers 1, 2, ... , n2 in such a way that the cat and dog obtain the same lists of numbers, provided that n is an even number. For odd numbers (n = 3, 5, 7, etc.), it is impossible for the monkey to fill in the grid in such a way that the cat and dog obtain the same lists of numbers.

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