Place 16 nonzero base ten digits in the cells of a 4x4 square grid, with each digit occurring in a cell exactly once. The digits can be considered as forming eight 4-digit numbers, read left to right and top to bottom, each of which is a perfect square. Since there are more than nine cells, there will be an obvious repetition of digits.

Since a square number terminates in 1, 4, 5, 6 or 9 and the right side column and lower row are the terminal cells of squares, the only 4 digit squares which contain only those digits are 1156, 1444 and 6561.

Now the problem does not say that all 9 nonzero digits must appear, nor does it say that any square number cannot be repeated.

With that in mind I offer:
  2  1  1  6    46*46
  1  2  2  5    35*35
  1  2  9  6    36*36
  6  5  6  1    81*81

This was derived using a spreadsheet.  I also grouped all of the 4 digit squares according to their terminating digit to make my choices easier.

 When I chose 1296 I then looked to see what squares terminated in "16" and "25".  From then on it became quite clear what my remaining choices were to be.

Posted by brianjn on 2009-10-10 23:22:17