A 3x3 magic square is an array of nine distinct positive integers such that the sum of the numbers in each row, each column, and each of the two diagonals is the same. In the following magic square:

What are the possible values of the lower left corner? And what is the maximum possible value of any number in the array?

First, a proof that the middle square must always be 1/3 the magic sum can be found here: http://perplexus.info/show.php?pid=1769&cid=35554

Then, by varying the center cell (and thus the magic sum) all the other cells can be filled in. (First the top-left and top-right, then the top-center, then the bottom right, and then the center-left and bottom-center) By filling out the grid using c as the center cell, we can find the bounds for c.

2c-14, 37-2c, 3c-23
2c-9, c, 9
23-c, 4c-37, 14

Thus, to keep all of 2c-9, 2c-14, 3c-23, 4c-37, positive, c>=10. But to keep both 23-c, 37-2c positive, c<=18.

Ignoring c=14 (which already appears)
If 14<c<=18, then using the fact that 9<14 and c-14>0 the following inequality can be constructed:
37-2c<23-c<9<14<c<2c-14<2c-9<3c-23<4c-37
If 10<=c<14, similar inequalities can be made, or the numbers can be tested with the relavent c values, except for c=12.

Edited on December 21, 2006, 4:59 pm

Posted by Gamer on 2006-12-21 16:55:44