Antimagic square

An antimagic square of order n is an arrangement of the numbers 1 to n² in a square, such that 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.

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

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.

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. 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.

Some 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?

External links

• Mathworld