.
Antimagic square
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 heterosquares which simply have each row, column and diagonal sum different. They contrast with magic squares 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 1,\ldots,m for some m\leq n^2, and whose rowsums and columnsums 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 antimagic square (STAM). Note that a STAM is not necessarily a SAM, and viceversa.
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?
References
^ Gray, I. D.; MacDougall, J.A. (2006). "Sparse antimagic squares and vertexmagic labelings of bipartite graphs". Discrete Mathematics 306: 2878–2892. doi:10.1016/j.disc.2006.04.032.
External links
Mathworld
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License