Fast enumeration of magic squares (Part I)
Presentation slides
Presentation manuscript
The author's web site:
Sequel video:
日本語版 (Japanese version)
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Erratum (updated on 2025.11.09)
p.5 "Enumeration of magic squares"
wrong: Shroeppel
correct: Schroeppel
p.33 "Arrangement of X is a permutation"
wrong: Let up denote
correct: Let us denote
p.44 "Check function at the deepest level"
The pseudo code in this page is incorrect.
Though the slide is incorrect, the code used in the calculation is correct.
The preceding and subsequent slides are correct as well.
p.58 "To be explained"
wrong: 20,000 threads
craig berube correct: 200,000 threads
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References
General references on related topics:
Walter Trump, Notes on Magic Squares and Cubes
Harry White, Magic Squares
Francis Gaspalou, Structure of Magic and Semi-Magic Squares, Methods and Tools for Enumeration
stephon castle Wikipedia
The Online Encyclopedia of Integer Sequences
Enumeration of semi-magic squares and related works:
A. Ripatti, On the number of semi-magic squares of order 6, arXiv:1807.02983, July 2018
Go Kato and Shin-ichi Minato, Enumeration of Associative Magic Squares of Order 7, June 2020
M-Transformations:
Stochastic estimates for the number of magic squares:
Y. Ohishi, in Japanese, Estimation of number of solutions of 6th order magic square by random sampling, Sugei Puzzle No.177, April 1992
K. Pinn and C. Wieczerkowski, Number of Magic Squares From Parallel Tempering Monte Calrlo, International Journal of Modern Physics C, 9 April 1998.
W. Trump, Estimate of the number of magic squares of order 6.
A. Kitajima and M. Kikuchi, Numerous but Rare: An Exploration of Magic Squares, PROS ONE 10(5) e0125062, 14 May 2015.
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Sequel video in preparation:
・A parallel implementation of magic square enumeration on GPUs.
