Journal of Integer Sequences, Vol. 13 (2010), Article 10.6.2

Six Little Squares and How Their Numbers Grow

Matthias Beck
Department of Mathematics
San Francisco State University
1600 Holloway Avenue
San Francisco, CA 94132

Thomas Zaslavsky
Department of Mathematical Sciences
Binghamton University
Binghamton, NY 13902-6000


We count the 3 × 3 magic, semimagic, and magilatin squares, as functions either of the magic sum or of an upper bound on the entries in the square. Our results on magic and semimagic squares differ from previous ones, in that we require the entries in the square to be distinct from each other and we derive our results not by ad hoc reasoning, but from the general geometric and algebraic method of our paper "An enumerative geometry for magic and magilatin labellings". Here we illustrate that method with a detailed analysis of 3 × 3 squares.

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(Concerned with sequences A108235 A108236 A108576 A108577 A108578 A108579 A173546 A173547 A173548 A173549 A173723 A173724 A173725 A173726 A173727 A173728 A173729 A173730 A174018 A174019 A174020 A174021 A174256 A174257.)

Received March 9 2010; revised version received June 1 2010. Published in Journal of Integer Sequences, June 2 2010. Revised, June 8 2010.

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