1997 …2023

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The puzzling mathematician

They say Sudoku puzzles require no mathematics to solve. Australian Mathematical Society 2009 medal winner Dr Ian Wanless disagrees. He views Sudoku as an example of a branch of mathematics called combinatorics. Its outstanding feature is Sunday-afternoon surface simplicity. But mathematical solutions to these puzzles hold surprising conceptual depth and applications in information and communication technology (ICT).

Dr Wanless has always loved solving puzzles and the field of combinatorics mathematics allows him to do just that. The problems he solves resemble those found in puzzle books, including anagrams and crosswords. They tend to involve permutations, combinations and rearrangement of symbols.

Much of the motivation for studying combinatorics comes from ICT and includes applications such as coding mobile phone signals or writing music on a CD. Ultimately, this is where a lot of his work finds application. Personally, he is not interested in applying his solutions.

'I like the problems for their elegance as an intellectual puzzle,' Ian says. The problems he finds most attractive are those that can be stated so simply that a child can understand them. Yet some deep mathematics are needed to arrive at a solution that works for every permutation.

The classic combinatorics example is the four-colour map problem. The problem seems simple. It asks how many colours are needed to colour regions on a map so that no two adjacent regions have the same colour. It was posed in the 19th century but took more than 100 years to solve.

'As mathematicians we are trying to find universal, absolute solutions, with the added bonus that nobody has done it before. You need an inescapable reason why whatever example of a puzzle you tried, the solution you propose would always work.

'A mathematical argument has to be general and cover every contingency,' he says. 'That is the idea of proof and it is very important. Certain things that appear to be obvious can be extremely difficult to prove.'

His current favourite questions deal with Latin squares. A finished Sudoku puzzle is an example of a Latin square. These are problems in which symbols fit in a matrix and each symbol occurs once in each row and column.

'Latin squares arise in the design of statistical experiments in agriculture and psychology,' Ian says. 'They are used to decide in which order to do experiments and they are useful in designing code for communication. I'm interested in basic properties like how many there are. Again, they are problems you can explain to a child yet some tricky math is needed to solve them.'

Contrary to stereotypes of the lone wolf intellectual, Ian enjoys collaborating. He has a dozen projects in the works, many involving colleagues in the US, Europe and Asia.

'I'm very much of the opinion that mathematics is a team sport,' he says. 'I do write papers on my own. But it is more fun to collaborate. A lot of the best projects come from cross-fertilisation from different branches, as is the case in all science. You get surprising results when working between fields rather than in a field.'

Research area keywords

  • Latin Squares
  • Combinatorial Designs
  • Combinatorial Enumeration
  • Matrix Permanents
  • Graph Theory

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