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?