Antimagic square
Encyclopedia
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.
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 heterosquare
s which simply have each row, column and diagonal sum different. They contrast with magic square
s 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
for some 
, 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.
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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
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of heterosquare
Heterosquare
A heterosquare of order n is an arrangement of the integers 1 to n2 in a square, such that the rows, columns, and diagonals all sum to different values. There are no heterosquares of order 2, but heterosquares exist for any order n ≥ 3....
s which simply have each row, column and diagonal sum different. They contrast with magic square
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
s 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
for some , 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?