A006052 Number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.

1, 0, 1, 880, 275305224, 17753889197660635632

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice
• more

Comments

a(4) computed by Frenicle de Bessy (1605? - 1675), published in 1693. The article mentions the 880 squares and considers also 5*5, 6*6, 8*8, and other squares. - Paul Curtz, Jul 13 and Aug 12 2011

a(5) computed by Richard C. Schroeppel in 1973.

According to Pinn and Wieczerkowski, a(6) = (0.17745 +- 0.00016) * 10^20. - R. K. Guy, May 01 2004

a(6) computed by Hidetoshi Mino in 2024 - Hidetoshi Mino, May 31 2024

Examples

			An illustration of the unique (up to rotations and reflections) magic square of order 3:
  +---+---+---+
  | 2 | 7 | 6 |
  +---+---+---+
  | 9 | 5 | 1 |
  +---+---+---+
  | 4 | 3 | 8 |
  +---+---+---+
		

References

• E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Vol. II, pp. 778-783 gives the 880 4 X 4 squares.
• M. Gardner, Mathematical Games, Sci. Amer. Vol. 249 (No. 1, 1976), p. 118.
• M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 216.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Ian Cameron, Adam Rogers and Peter Loly, "The Library of Magical Squares" -- a summary of the main results for the Shannon entropy of magic and Latin squares: isentropic clans and indexing, in celebration of George Styan's 75th.
• Bernard Frénicle de Bessy, Des carrez ou tables magiques, Divers ouvrages de mathématique et de physique (1693), pp. 423-483.
• Bernard Frénicle de Bessy, Table générale des carrez de quatre, Divers ouvrages de mathématique et de physique (1693), pp. 484-503.
• Skylar R. Croy, Jeremy A. Hansen, and Daniel J. McQuillan, Calculating the Number of Order-6 Magic Squares with Modular Lifting, Proceedings of the Ninth International Symposium on Combinatorial Search (SoCS 2016).
• Mahadi Hasan and Md. Masbaul Alam Polash, An Efficient Constraint-Based Local Search for Maximizing Water Retention on Magic Squares, Emerging Trends in Electrical, Communications, and Information Technologies, Lecture Notes in Electrical Engineering book series (LNEE 2019) Vol. 569, 71-79.
• Hidetoshi Mino, The number of magic squares of order 6.
• Hidetoshi Mino, Fast enumeration of magic squares, YouTube video, 2025.
• I. Peterson, Magic Tesseracts.
• K. Pinn and C. Wieczerkowski, Number of magic squares from parallel tempering Monte Carlo, arXiv:cond-mat/9804109 [cond-mat.stat-mech], 1998; Internat. J. Modern Phys., 9 (4) (1998) 541-546.
• Tyler Pringle, Magic Squares and Using Magic Series for Theory, The College of William and Mary (2024). See pp. 6, 9.
• Artem Ripatti, On the number of semi-magic squares of order 6, arXiv:1807.02983 [math.CO], 2018. See Table 1 p. 2.
• R. Schroeppel, Emails to N. J. A. Sloane, Jun. 1991.
• N. J. A. Sloane & J. R. Hendricks, Correspondence, 1974.
• Walter Trump, How many magic squares are there? - Results of historical and computer enumeration.
• Eric Weisstein's World of Mathematics, Magic Square.
• Index entries for sequences related to magic squares

Crossrefs

Cf. A270876, A271103, A271104.

Extensions

Definition corrected by Max Alekseyev, Dec 25 2015

a(6) from Hidetoshi Mino, Jul 17 2023

Incorrect a(6) removed by Hidetoshi Mino, Sep 07 2023

a(6) from Hidetoshi Mino, May 31 2024