# 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.[1]

## 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

 5 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.[2] In the antimagic square of order 5 on the left, the rows, columns and diagonals sum up to numbers between 60 and 71.[2] In the antimagic square on the right, the rows, columns and diagonals add up to numbers between 59-70.[1]

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.[2]

## 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 ${\displaystyle 1,\ldots ,m}$ for some ${\displaystyle m\leq n^{2}}$, and whose row-sums and column-sums constitute a set of consecutive integers.[3] 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.