SAT
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?