A271104 -id:A271104 - cp's OEIS frontend

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

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

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

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).

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

Cf. A270876, A271103, A271104.

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

A271103 Number of magic and semi-magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.

Original entry on oeis.org

1, 0, 9, 68688, 579043051200, 94590660245399996601600

Offset: 1

William Walkington, Mar 30 2016

A semi-magic square differs from a magic square in that at least one of its main diagonals does not sum to the magic constant. [Walter Trump]
The number of order 4 magic and semi-magic squares was computed by Mutsumi Suzuki, and could be found on his former web site. Mutsumi Suzuki's pages are now in the Internet Archive.
The number of order 5 magic and semi-magic squares was computed by Walter Trump in March 2000.
The number of order 6 magic and semi-magic squares was calculated by Artem Ripatti in April 2018, and published in his paper dated July 10, 2018. - William Walkington, Jul 17 2018

Artem Ripatti, On the number of semi-magic squares of order 6, arXiv:1807.02983 [math.CO], 2018. See Table 1 p. 2.
Mutsumi Suzuki, Semi-Magic Squares of 4 x 4, first "captured" by the Internet Archive on the 5th October 1999.
Walter Trump, How many magic squares are there? - Results of historical and computer enumeration.
Index entries for sequences related to magic squares

Cf. A006052, A270876, A271104.

a(n) = A271104(n)* n^2.

a(6) added by William Walkington, Jul 17 2018

A270876 Number of magic tori of order n composed of the numbers from 1 to n^2.

Original entry on oeis.org

1, 0, 1, 255, 251449712

Offset: 1

William Walkington, Mar 24 2016

Initially based on empirical observations by William Walkington, the results for the orders 1 to 4 have since been computed and confirmed by Walter Trump. The results for the order 5 have been computed by Walter Trump.

Dwane Campbell, Analysis of order-4 magic squares, (2013).
Dwane Campbell, Order-4 squares grouped by base square quartets, (2013).
Dwane Campbell, Features in order-4 magic squares, (2013).
William Walkington, 255 tores magiques d'ordre 4, et 1 tore magique d'ordre 3, (2011).
William Walkington, Passage du carré au tore magique, (2011).
William Walkington, 255 fourth-order magic tori, and 1 third-order magic torus, (2012).
William Walkington, From the magic square to the magic torus, (2012).
William Walkington, (using findings computed by _Walter Trump_), 251 449 712 fifth-order magic tori, (2012).
William Walkington, A new census of fourth-order magic squares, (2012).
William Walkington, Table of fourth-order magic tori, (2012).
William Walkington, 440 torus-opposite pairs of the 880 Frénicle magic squares of order-4, (2023).

Cf. A006052, A271103, A271104.

cp's OEIS Frontend

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

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A271103 Number of magic and semi-magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.

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A270876 Number of magic tori of order n composed of the numbers from 1 to n^2.

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