Antimagic square

An antimagic square of order n is an arrangement of the numbers 1 to n² in a square, such that the sums of the n rows, the n columns and the two diagonals form a sequence of 2n + 2 consecutive integers. The smallest antimagic squares have order 4.

Examples

2 15 5 13 16 3 7 12 9 8 14 1 6 4 11 10

1 13 3 12 15 9 4 10 7 2 16 8 14 6 11 5

Properties

In each of these two antimagic squares of order 4, the rows, columns and diagonals sum to ten different numbers in the range 29–38.

Antimagic squares form a subset of heterosquares which simply have each row, column and diagonal sum different. They contrast with magic squares where each sum is the same.

Open problems

• How many antimagic squares of a given order exist?
• Do antimagic squares exist for all orders greater than 3?
• Is there a simple proof that no antimagic square of order 3 exists?

Generalizations

A sparse antimagic square (SAM) is a square matrix of size n by n of nonnegative integers whose nonzero entries are the consecutive integers $1,\ldots,m$ for some $m\leq n^2$, and whose row-sums and column-sums constitute a set of consecutive integers.^[1] If the diagonals are included in the set of consecutive integers, the array is known as a sparse totally anti-magic square (STAM). Note that a STAM is not necessarily a SAM, and vice versa.

See also

• Magic square
• Heterosquare
• J. A. Lindon

References

1. Gray, I. D.; MacDougall, J.A. (2006). "Sparse anti-magic squares and vertex-magic labelings of bipartite graphs". Discrete Mathematics 306: 2878–2892. doi:10.1016/j.disc.2006.04.032.

External links

• Weisstein, Eric W., "Antimagic Square", MathWorld.