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