# The Anti-Magic Square Project: Enumeration

## AMS(1)

Do not exist. The only possible arrangement of 1 is: 1 which yields the four sums 1, 1, 1, and 1, which are not consecutive integers. :)

## AMS(2)

For the 2x2 case, we require 6 consecutive sums. But, the largest sum we could hope to get is 3+4=7, and the smallest is 1+2=3. This means there are at most five possible sums, while we require six. (This argument taken from Madachy's book.)

## AMS(3)

There are no AMS(3)'s, but there doesn't seem to be a quick and easy proof of this. A short computer program can rule them out, or a two page argument of case analysis can exclude them (see our preprint). If anyone knows of a short argument please let us know!

## AMS(4)

299,710 Different AMS(4) * 32 symmetry operations = 9,590,720 Total AMS(4) You can download a listing of them (in gzipped .sqr format -- see file area) but it's fairly large.

## AMS(5)

It seems as if there are just way too many 5x5 Anti-Magic squares to count. Compared to the 4x4 problem, the number of squares to consider gets much bigger while the number of group symmetries stays the same.

Of course anything bigger than 5x5 is even more hopeless.

## Comparison to Magic Squares

From the table we can see that Anti-Magic squares far outnumber the Magic squares in all but the trivial cases.
Order n # MS(n) # AMS(n)
1 1 0
2 0 0
3 1 0
4 880 299710
5 ?? ??

Composed by John Cormie July 1999