A 9x9 grid is filled with numbers from 1 to 81, each number occurring exactly once. The grid is said to be ultramagic if the product of numbers in row k is equal to the product of numbers in column k for each k. Can you construct an ultramagic grid?

- via AMS puzzle corner

No such grid exists. This can be seen from the location of the primes in the grid. Let a prime be located in position for . Then, the ith row product is divisible by . The grid construction rule then stipulates that the column is also divisible by . This is only possible if a multiple of () is located on the jth column. Particularly, it is not possible for any prime above . So, the primes between 41 and 81 cannot lie on the off-diagonal. But there are 10 of them and only 9 diagonal locations.

For a general , the above argument yields a necessary condition which rules out . Here is the prime counting function. It will be very surprising if any ultramagic squares exist at all.

Great logic. I question the posts' definition of ultramagic square in the puzzle. Is the requirement re. products a general one (where) or is it particular to this site only?