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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 Sundayafternoon 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 fourcolour 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 crossfertilisation 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
Collaborations and top research areas from the last five years
Projects
 6 Finished

Matchings in Combinatorial Structures
Wanless, I., Bryant, D. & Horsley, D.
Australian Research Council (ARC), Monash University, University of Queensland , University of Melbourne
1/01/15 → 10/10/20
Project: Research

Towards the prime power conjecture
Australian Research Council (ARC), Monash University
27/02/12 → 31/12/16
Project: Research

Extremal Problems in Hypergraph Matchings
Wanless, I., Greenhill, C. & Aharoni, R.
Australian Research Council (ARC), University of New South Wales (UNSW)
3/01/12 → 31/12/14
Project: Research

Permanents, permutations and polynomials
Wanless, I. & McKay, B.
Australian Research Council (ARC), Monash University, Australian National University (ANU)
4/01/10 → 31/12/13
Project: Research


On Perfect Sequence Covering Arrays
Gentle, A. R. & Wanless, I. M., Sept 2023, In: Annals of Combinatorics. 27, 3, p. 539564 26 p.Research output: Contribution to journal › Article › Research › peerreview
Open Access1 Citation (Scopus) 
Pairs of MOLS of order ten satisfying nontrivial relations
Gill, M. J. & Wanless, I. M., Apr 2023, In: Designs Codes and Cryptography. 91, 4, p. 12931313 21 p.Research output: Contribution to journal › Article › Research › peerreview

A General Framework for Hypergraph Coloring
Wanless, I. M. & Wood, D. R., 2022, In: SIAM Journal on Discrete Mathematics. 36, 3, p. 16631677 15 p.Research output: Contribution to journal › Article › Research › peerreview
3 Citations (Scopus) 
Enumeration of Latin squares with conjugate symmetry
McKay, B. D. & Wanless, I. M., Feb 2022, In: Journal of Combinatorial Designs. 30, 2, p. 105130 26 p.Research output: Contribution to journal › Article › Research › peerreview
2 Citations (Scopus) 
Degree of orthomorphism polynomials over finite fields
Allsop, J. & Wanless, I. M., Oct 2021, In: Finite Fields and their Applications. 75, 9 p., 101893.Research output: Contribution to journal › Article › Research › peerreview
Open Access