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.

An antimagic square of order 4. I, the creator of this image, hereby release it into the public domain. This applies worldwide. File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. Click on date to download the file or...

An antimagic square of order 4. I, the creator of this image, hereby release it into the public domain. This applies worldwide. File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. Click on date to download the file or...

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.

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?

See also

• A heterosquare is a square array of consecutive integers whose rowsums, columnsums, and two diagonal sums, are all different. It is named by analogy to magic square. A heterosquare of size is said to be of order . A heterosquare of any even order may be formed by writing the integers... Heterosquare
• In mathematics, magic squares consist of a number of integers arranged in the form of a square in such a way that the sum of the numbers in every row, column and diagonal are the same. A magic square may have odd or even number of rows and columns. Usually... Magic square

External links

• Antimagic Squares (http://www.geocities.com/~harveyh/anti_ms.htm)
• Mathworld  (http://mathworld.wolfram.com/AntimagicSquare.html)