### Magic Squares---At no time do my fingers....

by

Fred Schaal

The kind of magic squares that I am familiar with are the ones with an odd number of cells along an edge.  These I refer to as "odd squares".  I sometimes think that one could assemble an even number of odd squares so as to produce an even magic square- an even square.  What if I were to use the very fundamental; 3 by 3 square with just the first nine integers and assemble them into a 6 by 6 magic square?
 8 1 6 3 5 7 4 9 2
 8 1 6 3 5 7 4 9 2
 8 1 6 3 5 7 4 9 2
 8 1 6 3 5 7 4 9 2

This in indeed an even "magic" square.  Each row and column and diagonal sum to the same number: the magic number, aka magic constant. But it is a trivial case.  My kind of magic squares do not repeat numbers. This one four- peats each number of the original primitive single digit magic square.  This square is unacceptable.

I shall endeavor to find out how to construct even magic squares! This link contains instructions for constructing a 4 by 4 magic square with the first 16 consecutive counting numbers.  The magic constant is 43.  Here it is....

 6 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

Unfortunately, when I try to expand it to 6 by 6 or 8 by 8 the method fails .  But at last and at least I can do a 4 by 4 by a set of rules, that is, systematically. (At http://www.primepuzzles.net/puzzles/puzz_068.htm I found a 16 by 16 square that uses the prime numbers from 11 to 2633.  Gasp!)

Below please find a 12 by 12 table for a magic square:

This is really an awful magic square.  Again I am at a lost to know from whence it came.  Its members are consecutive prime numbers starting with 1.  Whoever was involved with its creation truly have too much time on their hands.

This page seems to be going nowhere slowly.  By way of a summary thought:  The rules for odd squares do not have a counterpart in the land of even squares.  It seems that each size has its own pattern of algorithm.  Naturally I am crushed by this bolt of reality but I hope to learn to live with it. I cannot believe that I have not tried a website from NCTM, the National Council of Teachers of Mathematics.  Is there a message here?