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?

(In reply to heuristic computer exploration (spoiler) by Charlie)

9 and 11 are not valid solutions for the lower left corner as all numbers in the magic square must be distinct.

The algebraic solution for the square in terms of the bottom left corner is:

32-2x        2x-9        46-3x

37-2x        23-x           9

   x           55-4x         14

If x were to exceed 13 the bottom middle would become negative and if x were to fall below 5 the top middle would become negative.

The square containing the largest number (35) is naturally found in the sqaure with maximum row sum (69-3x) which occurs at minimum x (x=5). This sqaure is:

22   1   31
17  18   9
 5   35  14

Posted by Eric on 2006-12-21 15:55:49