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

Order 4 antimagic squares

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

Order 5 antimagic squares

16 8 20 9 22 19 23 13 10 2 21 6 3 15 25 11 18 7 24 1 12 14 17 4 16

21 18 6 17 4 7 3 13 16 24 5 20 23 11 1 15 8 19 2 25 14 12 9 22 10

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. In the antimagic square of order 5 on the left, the rows, columns and diagonals sum up to numbers between 60 and 71. In the antimagic square on the right, the rows, columns and diagonals add up to numbers between 59-70.

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. 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

External links

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