Antimagic square

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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] Antimagic squares contrast with magic squares, where each row, column, and diagonal sum must have the same value. [2]

Contents

Examples

Order 4 antimagic squares

215513
163712
98141
641110
113312
159410
72168
146115

In both of these antimagic squares of order 4, the rows, columns and diagonals sum to ten different numbers in the range 29–38. [2]

Order 5 antimagic squares

5820922
192313102
21631525
11187241
121417416
21186174
73131624
52023111
15819225
141292210

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]

Open problems

The following questions about antimagic squares have not been solved.[ citation needed ]

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.

A filling of the n × n square with the numbers 1 to n2 in a square, such that the rows, columns, and diagonals all sum to different values has been called a heterosquare. [4] (Thus, they are the relaxation in which no particular values are required for the row, column, and diagonal sums.) There are no heterosquares of order 2, but heterosquares exist for any order n  3: if n is odd, filling the square in a spiral pattern will produce a heterosquare. [4] And if n is even, a heterosquare results from writing the numbers 1 to n2 in order, then exchanging 1 and 2. It is suspected that there are exactly 3120 essentially different heterosquares of order 3. [5]

See also

Related Research Articles

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References

  1. 1 2 W., Weisstein, Eric. "Antimagic Square". mathworld.wolfram.com. Retrieved 2016-12-03.
  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 (22): 2878–2892. doi: 10.1016/j.disc.2006.04.032 .
  4. 1 2 Weisstein, Eric W. "Heterosquare". MathWorld .
  5. Peter Bartsch's Heterosquares at magic-squares.net