The free semigroup of magic squares. (Le semi-groupe libre des carrés magiques.) (French) Zbl 0957.05020

The authors studies a law of composition upon magic (semimagic or panmagic) squares, already introduced in the literature, that provides the set of all magic (semimagic or panmagic) squares with a structure of semigroup. He then proves a conjecture of A. Adler and S.-Y. R. Li [Am. Math. Mon. 84, 618-627 (1977; Zbl 0389.05018)] and A. Adler [Electron. J. Comb. 4, No. 1, Research Paper R15 (1997; Zbl 0885.05007)], that this semigroup is free.


05B15 Orthogonal arrays, Latin squares, Room squares
20M05 Free semigroups, generators and relations, word problems
11B65 Binomial coefficients; factorials; \(q\)-identities
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[1] Adler Allan and Li, Shuo Yen Robert, Magic cubes and Prouhet sequences, Amer. Math. Monthly 848 (1977), 618-627. · Zbl 0389.05018
[2] Chee, P.S., Magic squares, Menemui Mat.53 (1983), 111-121. · Zbl 0484.05028
[3] Ku, Y.H. and Chen, Nan Xian, Some theorems on construction of magic squares, J. Franklin Inst.3225-6 (1986), 253-266. · Zbl 0607.05020
[4] Bouteloup, Jacques, Carrés magiques, carrés latins et eulériens (1992), Editions du choix.
[5] Allan, Adler, Magic N-cubes form a free monoid, à paraître dansEuropean J. Combinatorics. · Zbl 0885.05007
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