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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'' + 2 consecutive integers. The smallest antimagic squares have order 4. | |||
== 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 29–38. | |||
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]] | |||
Revision as of 02:50, 21 January 2014
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
|
|
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.
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 for some , and whose row-sums and column-sums constitute a set of consecutive integers.[1] 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
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
External links
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