Combinatorial play — Euler’s eighteenth-century squares met by the contemporary puzzle.
Play It
Characterization
A Latin square is an n×n grid filled so that each symbol appears exactly once in each row and column. The structure was formalised by Leonhard Euler in 1782 in his Recherches sur une nouvelle espèce de quarrés magiques, though it had appeared earlier in Korean and Arabic mathematical and divinatory traditions. The contemporary puzzle Sudoku — a 9×9 Latin square partitioned into nine 3×3 sub-grids, each of which must also contain the digits 1–9 — was invented by Howard Garns in 1979 under the name Number Place and given its present form and global reach by the Japanese publisher Nikoli in 1984. The combinatorial structure of Latin squares underlies the design of statistical experiments (R. A. Fisher, 1935), the construction of error-correcting codes, and a corner of pure mathematics that remains active. Sudoku is the wonder of constraint satisfaction made elegant: a single puzzle, hand-set by a human or algorithmically generated, that exercises the same logical mind a serious combinatorialist trains for years.
Lineage
Euler’s Recherches sur une nouvelle espèce de quarrés magiques (1782). R. A. Fisher, The Design of Experiments (1935). The disproof of Euler’s conjecture on Greco-Latin squares of order 4k + 2 by Bose, Shrikhande, and Parker (1959–60). The contemporary puzzle: Howard Garns, Number Place (Dell, 1979); Nikoli (Tokyo, 1984); global proliferation via Wayne Gould from 2004. Minimum-clues proof for a unique Sudoku (17 clues) by McGuire, Tugemann, and Civario (2012).
Quests
The Magus
Choose one: (a) construct a Sudoku puzzle with a unique solution and demonstrate its uniqueness; or (b) construct a Greco-Latin square of order 10 — an order Euler conjectured impossible and which Bose, Shrikhande, and Parker disproved in 1959. Note the method.
The Adventurer
Solve three Sudoku puzzles at increasing difficulty, including at least one “hard” or “expert” grade. For each, record the techniques you used to break in.
The Sage
Explain the combinatorial structure of Latin squares from Euler to the present. Cite Euler’s 1782 paper, Fisher’s use of Latin squares in experimental design, and the 2012 minimum-clues result for Sudoku.