The Maths of Sudoku and Latin Squares
- Extra Reading
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Millions of us regularly solve Sudoku puzzles.
In this lecture, we discuss the mathematics behind them, and the links to other kinds of number grids, like magic squares and so-called Latin squares, which have been studied for centuries. Latin squares have many applications in areas as diverse as experiment design, algebra and coding theory.
References and Further Reading
Robin Wilson’s How to Solve Sudoku, published by Brilliant Books, ISBN 9781904902621.
You can read about antimagic squares at https://mathworld.wolfram.com/AntimagicSquare.html
There’s a very interesting paper about the ancient order 4 magic square we mentioned
Varāhamihira's pandiagonal magic square of the order four, by Takao Hayashi
See Fermat’s magic cube: https://demonstrations.wolfram.com/FermatsMagicCube/
The Numberphile video on magic hexagons is at https://youtu.be/ZkVSRwFWjy0
You can read the gory details about the minimum number of entries in a valid Sudoku puzzle in the
paper There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem via Hitting Set Enumeration, by Gary McGuire, Bastian Tugemann, Gilles Civario at https://arxiv.org/pdf/1201.0749.pdf
Latest update on magic squares of order 6 is at https://magicsquare6.net/doku.php?id=magicsquare6
I made my magic Sudoku with the help of https://www.sudoku-solutions.com/, which checks any partial grid you enter to see if it is valid and has a unique solution. So you can (try to) make your own, subject to whatever crazy constraints you like.
If you want to get a lot further into more of the mathematics relating to Sudoku, try Taking Sudoku Seriously: The math behind the world's most popular pencil puzzle, by Jason Rosenhouse and Laura Taalman (Oxford University Press, 2011).
A note on images used in the lecture
To the best of my knowledge all the images used are either in the public domain, or may be used for educational purposes under fair use rules, or were created by me. The only image I wasn’t able to source definitively was the photograph of the Latin square laid out in 1929 at Beddgelert Forest in Wales; I believe it’s from a Forestry Commission report of circa 1945.
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