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
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, . . . , n^2 in an n × n grid so that the cat and the dog obtain the same lists of numbers?
Please provide a mathematically rigorous solution with proofs that could be understood by a person at a high school level.
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