×

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.

MSC:

05B15 Orthogonal arrays, Latin squares, Room squares
20M05 Free semigroups, generators and relations, word problems
11B65 Binomial coefficients; factorials; \(q\)-identities
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML EMIS

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.