Prove that the central cell (the number in the middle cell) of any 3x3 magic square is always one-third the magic constant (the sum of any side, either 2 major diagonals, or either center row in the magic square).
Show that in any larger square (n x n), the central cell does not need to be 1/n the magic constant.

solve

a b -
c - -
- - -

with magic constant m

__a_________b________m-a-b
__c_____2a+b+c-m__-2a-b-2c+2m
m-a-c__-2a-2b-c+2m

The bottom corner is either -3a-b-c+2m by diagonal or 3a+2b+2b-2m by the side. Solving for m gives m=3/2a+3/4b+3/4c

Subbing this value into the center gives 1/2a+1/4b+1/4c which is indeed one-third of m

Part 2 looks harder. You would have to find a pattern that shows this relation doesn't hold.

-Jer

Posted by Jer on 2004-04-16 14:16:44