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.
Contents
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
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