Antimagic square

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.[1]

Examples

Order 4 antimagic squares

2 15 5 13
16 3 7 12
9 8 14 1
6 4 11 10
1 13 3 12
15 9 4 10
7 2 16 8
14 6 11 5

Order 5 antimagic squares

5 8 20 9 22
19 23 13 10 2
21 6 3 15 25
11 18 7 24 1
12 14 17 4 16
21 18 6 17 4
7 3 13 16 24
5 20 23 11 1
15 8 19 2 25
14 12 9 22 10

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.[2] In the antimagic square of order 5 on the left, the rows, columns and diagonals sum up to numbers between 60 and 71.[2] In the antimagic square on the right, the rows, columns and diagonals add up to numbers between 59-70.[1]

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.[2]

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.[3] 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

  1. 1 2 Template:Cite 201 web
  2. 1 2 3 "Anti-magic Squares". www.magic-squares.net. Retrieved 2016-12-03.
  3. ↑ Gray, I. D.; MacDougall, J.A. (2006). "Sparse anti-magic squares and vertex-magic labelings of bipartite graphs". Discrete Mathematics. 306: 2878–2892. doi:10.1016/j.disc.2006.04.032.
  • Weisstein, Eric W. "Antimagic Square". MathWorld.
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