The University of Queensland’s supercomputer has helped settle a problem in mathematics that has stood open for more than 20 years.
Dr Richard Bean, a data scientist, energy market analyst and affiliate at the UQ Cyber Research Centre, used UQ high-performance computer (HPC) Bunya to solve a question in combinatorics, a branch of mathematics focused on counting and enumerating objects with defined structure. (Combinatorics was the subject of Dr Bean’s PhD at UQ, which he completed in 2001.)
For two decades, proving the existence of a combinatorial “tight single-change covering design” greater than a block size of five has remained an unsolved mystery and has been a major milestone at combinatorial conferences.
In 2026, Dr Bean finally solved this problem by constructing a valid tight single-change covering design (TSCCD) and found an example with 26 elements and block size 6, made up of 63 blocks.
“With recent advances in solver software and hardware speed, I was able to test new ideas and find this new design,” said Dr Bean.
“Some examples of combinatorial objects are Latin squares (which are like a generalisation of Sudoku), graphs, and more generally the concept of 'designs'. These have applications in the design of experiments in fields like agriculture, psychology, and cryptography.
“Tight single-change covering designs originally arose out of the study of core storage on mainframe computers in the 1960s.”
The proof required advanced constraint computer programming, using satisfiability searches to navigate the astronomical search space, proving these structures are rare and exceptionally difficult to find.
RCC Director Jake Carroll said: "What's compelling is that Richard didn't just plug a maths problem into an LLM [large language model] and press the 'solve' button. He used complex theory and technique combined with a very specific understanding of simulation, optimisation and solver technologies to achieve this.”
Dr Bean published his work in July 2026 as a preprint.
The block-size-5 case was first announced by Associate Professor Nicholas (Nick) Phillips of Southern Illinois University at an American Mathematical Society meeting in 1999, and published in 2001.
Whether an example with any larger block size could exist at all was then posed by University of London statistician and combinatorialist Professor Donald Preece (whom Dr Bean has met twice) as Problem 1 at the 19th British Combinatorial Conference in 2003, and it remained unanswered until now.
Prof. Preece wrote an article in 1995 explaining the problem first arose in 1968 through work with British statistician Prof. John Nelder (who invented the concept of “critical sets in Latin squares,” the subject of Dr Bean’s PhD thesis) around central-access and backing stores in computers.
At Dr Bean’s talk at an international combinatorics conference in October 2003 (see his presentation), he mentioned he thought 0-1 integer linear programming of a special kind might help with the search.
His successful search that found the block size 6 example was run on UQ HPC Bunya. The benefit of using the HPC, Dr Bean said, was being able to run many parallel searches in different parts of the search space at once.
“It [Bunya] let me eliminate enough of the search space to work out where to look,” he said.
“My initial effort went into the smallest candidate size, 21 elements, which is where a solution would most naturally be expected. If I had run this part of the search on a desktop machine it may not have finished in any reasonable time and I may have given up at that point.
“I had been running searches on my personal machine for a long time, but this was insufficient.”
For each job on Bunya, he used about 96 cores with wall times typically around 24 hours.
He used the Gurobi linear programming solver and the OR-Tools library, specifically CP-SAT, to write programs to search for a single example of this TSCCD.
“This is similar to the WSAT(OIP)* idea I suggested in 2003, but now the software and hardware has improved hugely (and has probably got much easier to use),” he said.
Dr Bean undertook RCC’s introductory HPC training in 2021, working with RCC HPC Manager Dr David Green to learn how an HPC’s job system worked, and about the quirks of Gurobi, batching and programming language Python.
He worked closely with Dr Green in online sessions to learn how Gurobi worked for a different project of his, which resulted in Dr Bean winning first prize for forecasting in an international renewable energy competition.
In a sense, Dr Bean has won again, this time for solving a long-standing open research frontier in mathematics.
“Being able to run all the searches on a high-performance computer in parallel made the problem finally tractable and I’m grateful to UQ RCC to have had access to this fantastic resource [Bunya],” Dr Bean said.
* WSAT(OIP)* is a domain-independent local search algorithm for integer optimisation that generalises the Walksat procedure for propositional satisfiability to over-constrained integer programs (OIPs).
