# The Maths of Sudoku and Latin Squares

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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.

• Robin Wilson’s How to Solve Sudoku, published by Brilliant Books, ISBN 9781904902621.

• 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

https://www.sciencedirect.com/science/article/pii/031508608790019X

• 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.

## Professor Sarah Hart

### Professor of Geometry

Sarah Hart is the first woman Professor of Geometry at Gresham College, and was appointed in 2020. She is Professor of Mathematics and until recently was Head of Mathematics and Statistics at Birkbeck, University of London.

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