The Maths of Sudoku and Latin Squares - Sarah Hart

The speaker begins by introducing the Gresham College lecture series, emphasizing the value of free access to knowledge from leading academics. They encourage donations and sharing of the lectures. The lecture's focus is on the mathematics of Sudoku, a popular number-placement puzzle. The speaker explains the basic rules of Sudoku, its structure, and the importance of a unique solution without guesswork. They also touch on the symmetry and uniqueness of Sudoku puzzles, highlighting that the filled squares often exhibit rotational symmetry.

This paragraph delves into strategies for solving Sudoku puzzles, such as using the rules to deduce missing numbers and the concept of 'slicing' the grid to eliminate possibilities. The speaker introduces the idea of 'Latin squares' as a mathematical concept related to Sudoku and discusses the history of Sudoku, debunking the myth of its ancient Japanese origins. It was actually invented in the United States in the 1970s under the name 'Number Place' and later popularized in Japan and the UK.

The speaker embarks on a historical journey, starting with the oldest known magic square, the 'Lo Shu' square, believed to be divinely revealed on a turtle's back. Magic squares are defined as grids where the sum of numbers in each row, column, and diagonal is the same, known as the 'magic constant'. The speaker explores various types of magic squares, including classical and semi-magic squares, and their significance in art and culture, such as the 'Mellon Collie and the Infinite Sadness' painting by Albrecht Dürer.

Continuing the discussion on magic squares, the speaker describes their cultural importance, such as in ancient Chinese mythology and their use in Indian texts for perfume recipes. The speaker also explains the mathematical properties of magic squares, including the calculation of the magic constant for classical magic squares of any order and the impossibility of creating a classical magic square of order two.

This paragraph explores the geometric aspects of magic squares, discussing the method to create a magic square of any odd order as described by the mathematician Mohamed ib Mohammed Alani Al Kwi. The speaker also explains how symmetries of a square can generate multiple magic squares from a single starting point, creating an 'orbit' of eight related squares.

The speaker discusses the vast number of magic squares that exist, especially when not limited to classical forms. They mention the potential for infinite variations by altering the numbers within the squares or combining different magic squares. The speaker also introduces the concept of magic cubes and other variations, such as squares where the product of numbers in each row, column, and diagonal is constant, or anti-magic squares where the sums are all different.

Shifting focus to Latin squares, the speaker explains their definition and distinguishes them from Sudoku, noting that every Sudoku is a Latin square but not vice versa. They discuss the practical applications of Latin squares in experiment design, such as arranging tree species on a Welsh hillside to control for environmental variables, ensuring fair and robust experimental outcomes.

The speaker recounts the history of Latin squares, from a card puzzle in Catherine the Great's court to the '36 officers problem', which led to the concept of orthogonal Latin squares. They explain the connection between Latin squares and the arrangement of suits and values in a deck of cards, highlighting the work of mathematician Leonard Euler in exploring the conditions under which orthogonal Latin squares can exist.

This paragraph discusses the unique structural use of order 10 orthogonal Latin squares in the novel 'Life: A User's Manual' by Georges Perec. The speaker describes how the novel's 99 chapters correspond to the rooms in a Parisian apartment block, each with a unique combination of characteristics drawn from lists of 10, reflecting the theme of failure in the book.

The speaker explores the application of orthogonal Latin squares in creating error-correcting codes for communication over noisy channels, such as in space probes. They explain how these codes can detect and correct errors by ensuring that messages are sent with enough redundancy to identify and fix corruption in transmission.

The speaker delves into the technical aspect of how Latin squares are used to construct error-correcting codes that can handle multiple errors. They provide an example using three-order Latin squares to create code words that can be uniquely determined by any two of their letters, thus allowing the correction of single errors.

The speaker presents a method to estimate the number of valid Sudoku grids, starting with the number of block grids and adjusting for those that also satisfy row and column rules. They discuss the concept of minimal grids, which have the fewest numbers filled in while still ensuring a unique solution, and mention the ongoing research into the exact number of minimal grids.

In the final paragraph, the speaker introduces various Sudoku variants, such as bigger doku, cube doku, hyper-cube doku, and ven doku, which involve overlapping grids. They also present a novel variant called 'su don't coup', where each row, column, and block must contain exactly three of the four numbers. The speaker invites the audience to explore these variants and reflects on the enjoyment and intellectual challenge of mathematical puzzles.

The final segment consists of a question and answer session where the speaker discusses the practical applications of the logical thinking involved in solving mathematical puzzles. They highlight the relevance of such skills in careers like programming and mathematics, and address questions about the solvability of puzzles and the progress made in the field of orthogonal Latin squares.