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An '''antimagic square''' of order ''n'' is an arrangement of the numbers 1 to ''n''<sup>2</sup> in a square, such that the sums of the ''n'' rows, the ''n'' columns and the two diagonals form a sequence of 2''n''&nbsp;+&nbsp;2 consecutive integers. The smallest antimagic squares have order 4.
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== Examples ==
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== 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&ndash;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.
 
== 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 <math>1,\ldots,m</math> for some <math>m\leq n^2</math>, and whose row-sums and column-sums constitute a set of consecutive integers.<ref name="Alon 1987">{{cite journal| doi=10.1016/j.disc.2006.04.032| last=Gray| first=I. D.|coauthor=MacDougall, J.A.|  title=Sparse anti-magic squares and vertex-magic labelings of bipartite graphs| journal=Discrete Mathematics| year=2006| volume=306| pages=2878–2892}}</ref> 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]]
 
==References==
{{reflist}}
 
== External links ==
* {{MathWorld |urlname=AntimagicSquare |title=Antimagic Square}}
 
{{DEFAULTSORT:Antimagic Square}}
[[Category:Magic squares]]

Latest revision as of 04:55, 13 January 2015

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