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Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and hamming codes. (English) Zbl 1149.05010

A gerechte design is a specialization of a Latin square; namely, it is an $n×n$ array containing the symbols $1,2,\cdots ,n$ which has been partitioned into $n$ regions, each region containing $n$ cells of the array, so that each symbol appears once in each row, once in each column, and once in each region. In particular, the popular Sudoku puzzle is a gerechte design with $n=9$ in which the regions are $3×3$ subsquares. We say that a set of gerechte designs with the same partitioned grid are mutually orthogonal if any pair is orthogonal when treated as Latin squares. The fascinating article under review describes numerous connections of gerechte designs, and in particular certain Sudoku solutions, with several other notions in statistics and discrete mathematics. This review highlights only a few of these connections.

Using a lovely connection with reguli and regular spreads of projective 3-space over the finite field $GF\left(3\right)$, it is shown that there are six mutually orthogonal Sudoku solutions, and this is indeed the maximum number of mutually orthogonal Sudoku solutions. A Sudoku solution is called symmetric if, in addition to the usual requirements, each symbol occurs once in each “broken row”, once in each “broken column”, and once in each “location”, where the three concepts in quotes are defined in the article in the most natural way. Working in the associated affine 4-space over GF(3), it is shown that there is a set of four mutually orthogonal symmetric Sudoku solutions. Finally, using the fact that a symmetric Sudoku solution corresponds to a partition of the underlying 4-dimensional vector space over GF(3) into nine perfect codes, it is shown that, up to equivalence (again naturally defined), there are precisely two symmetric Sudoku solutions. It should be noted that the total number of Sudoku solutions, up to equivalence, is 5472730538.

##### MSC:
 05B15 Orthogonal arrays, Latin squares, Room squares 05B25 Finite geometries (combinatorics)
DESIGN; GRAPE