A050257 Number of distinct antimagic squares of order n (modulo rotations and reflections).

0, 0, 0, 299710

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• bref
• hard
• nice

Comments

An Anti-Magic Square (AMS) is an arrangement of the numbers 1 to n*n in a square matrix such that the row, column and diagonal sums form a sequence of consecutive integers.

Cormie et al. estimated that the total number of 5 X 5 antimagic squares is on the order of 10^17. However, computational evidence suggests that the number of such squares is on the order of 10^15. Out of 19 billion randomly generated 5 X 5 matrices with distinct entries in {1, 2, ..., 25}, only 6 formed antimagic squares (see Examples below). - John M. Campbell, Nov 27 2022

Examples

			The following are antimagic squares of order 5.
  [22  7 20  1 18]   [ 2 22 11  6 21]
  [ 5  2 25 15 17]   [ 3 24 15  8 20]
  [13 19  6 24  3]   [12 19 14 16  7]
  [11  8 14 21 12]   [18  1  9 23 13]
  [16 23  4  9 10]   [25  5 17 10  4]
.
  [ 3 12 18 21 17]   [13 12 19  4 18]
  [23  5  9  4 25]   [11 22  1 21  8]
  [13 19 22  6  1]   [ 6 20  3 25 16]
  [16 14  8 24  2]   [15  5 23 10  9]
  [10 20 11  7 15]   [24  2 14  7 17]
.
  [ 3 24  4 20 18]   [23 22  3  1 12]
  [22 10 12  5 13]   [21  9 17 13  7]
  [17  1 21 25  6]   [ 4  5 16 24 19]
  [ 8  7 19 14 15]   [ 2 10 25 11 18]
  [ 9 23 11  2 16]   [20 14  8 15  6]
		

References

• J. Cormie, V. Linek, S. Jiang, and R.-C. Chen, Investigating the antimagic square, J. Combin. Math. Combin. Comput., 43 (2002), 175-197.

Links

• J. Cormie, The Anti-Magic Square Project
• J. Cormie, Example: sorting the sums (numbers in black on the border) yields the sequence: 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269 (from web page above).
• Eric Weisstein's World of Mathematics, Antimagic Square.
• Index entries for sequences related to magic squares

Extensions

a(n) not known for n >= 5.