Sudoku has origins in Latin squares described by Euler in the 18th century, in puzzles published in French magazines at the end of the 19th century, and in gerechte designs invented by Walter-Ulrich Behrens in 1956.
Latin Squares
A Latin square is a square of n x n cells filled with the numbers 1 to n so that each number appears only one in each row and each column.
Although it seems that a Latin square was first published by the Korean mathematician Choi Seok-jeong in 17001, the name Latin square appears in a work by the Swiss mathematician Leonhard Euler, entitled Recherches sur une nouvelle espèce de quarrés magiques2 (Investigations into a new type of magic squares), published in 1782 (in French).
It is of course very easy to construct a Latin square of any size n x n but writing the digits 1 to n in the first row and by shifting them by one column to the left (for example) when going to the next row.
Playing with numbers at the end of the 19th century
Around 1890, several French newspapers publish games where 9×9 grids have to be filled with numbers according to some rules. Christian Boyer, a specialist of mathematical games and magic squares, gives an overview of these puzzles in 20063.
The first time a 9×9 grid is cut into 9 squares is perhaps the problem published in the magazine Revue des Jeux on August 21st, 1891 by H. Mary. This is however a magic square, which must be filled with the numbers from 1 to 81.
According to Christian Boyer, the game closest to Sudoku published at that time was published in La France on July 6th, 1895. The puzzle uses only the digits from 1 to 9 and, although the 3×3 blocks are not drawn, the solution is indeed a Sudoku grid with each digit appearing once in the 3×3 blocks.
W.U. Behrens and gerechte designs
In 1956, Walter-Ulrich Behrens (1902 – 24 August 1962), scientific director at the agricultural experimental station of the German agricultural chemicals company Kali Chemie4, publishes a paper entitled Feldversuchsanordnungen mit verbessertem Ausgleich der Bodenunterschiede5 (Field trial arrangements with improved compensation for ground differences).
According to a lecture by Peter J. Cameron6, Behrens describes in this paper a special case of Latin squares where the n x n square is divided in n regions of n squares and is filled with the numbers 1, . . . , n in such a way that each row, column,or region contains each of the numbers just once. Behrens calls this particular case gerechte design (fair design).
The regions of a gerechte design Latin square need not be of a square shape, nor equal to each other. Cameron gives the following example of a 5 x 5 gereche design Latin square:
A Sudoku completed grid is thus a particular case of a gerechte design Latin square where the regions are all 3 x 3 squares.
To be continued in the next article…
Notes and References
- I’ve not been able to find the origianl work for the time being ↩︎
- A pdf version is available here: https://scholarlycommons.pacific.edu/cgi/viewcontent.cgi?article=1529&context=euler-works ↩︎
- An overview of these publications has been given in the French magazine Pour la Science (French version of Scientific American) in its issue 344 (June 2006). A summary of the article can be found here. A digital version of the magazine can be bought here. ↩︎
- https://en.wikipedia.org/wiki/Walter-Ulrich_Behrens ↩︎
- The complete reference, according to https://pi.math.cornell.edu/~connelly/pdf/10.1080_00029890.2008.11920542.pdf, is W. U. Behrens, Feldversuchsanordnungen mit verbessertem Ausgleich der Bodenunterschiede, Zeitschrift fur Landwirtschaftliches Versuchs- und Untersuchungswesen ¨ 2 (1956) 176–193. I’ve not been able yet to find this paper. ↩︎
- https://webspace.maths.qmul.ac.uk/p.j.cameron/talks/sudoku3.pdf ↩︎
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