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Harder Sudoku puzzles on the way?
Sudoku games could soon get a whole lot harder, say some mathematicians.

Dr Paul Newton and Stephen DeSalvo of the University of Southern California in Los Angeles report their analysis of Sudoku matrices today in the Proceedings of the Royal Society A.

"I think it will help develop multi-dimensional Sudoku puzzles, and answer questions about how to give the initial [clues] in order to create a hard, but still solvable Sudoku puzzle," says Newton.

A Sudoku puzzle solution consists of a 9 x 9 matrix of numbers from 1 to 9.

Each number can only appear once along any row and once down any column, as well as only once in each of the three 3 x 3 sub-blocks that make up the matrix.

There is believed to be about 1021 different matrices.

Newton and DeSalvo generated a "representative sample" of about 10,000 matricies and compared them to randomly-generated matrices.

They found that Sudoku matrices are more random than randomly-generated arrays.

This is surprising, says Newton, since one would expect the more constraints you have on a matrix, the less random it will be.

But, he says in a randomly generated square, you may end up with a matrix made up entirely of one number, which is something you could never get given Sudoku's rules.

Newton says the findings could help puzzle makers and puzzle solvers alike.

"I think it will give people a lot of insight into how to produce better algorithms for constructing Sudoku matricies and it will enable ultimately the very fast learning algorithms that solve Sudoku matrices," he says.
Harder puzzles?

A Sudoku puzzle builder must provide clues - numbers already in place - to help someone work out the solution.

The more clues there are, the easier it is to solve the puzzle.

But too few clues and there will be more than one solution.

Currently the minimum number of clues required to ensure a unique solution is understood to be 17.

But Newton says it may be possible to use his findings to construct harder puzzles.

"I think it could help push that number down," he says.

Newton says the findings could also help in the development of Sudoku-solving computer algorithms, and 3D Sudoku.

Australian mathematician Dr Marcel Jackson of Latrobe University in Melbourne says while the findings that Sudoku squares are more random than randomly-generated squares initially sound counter-intuitive, he agrees with Newton that it makes sense when you think more deeply.

Jackson says Sudoku are a form of "Latin square", which have a 300-year history in mathematics.

He says understanding these are useful in the coding of information to minimise the effect of errors in transmission.

And he agrees it might help in making harder puzzles.

But mathematician Dr Ian Wanless of Monash University in Melbourne urges a word of caution.

He thinks Newton and DeSalvo's counter-intuitive finding, that Sudoku matricies are more random than randomly-generated ones, is a "red flag".

It suggests that the method they used to generate the Sudoku matricies was wrong, says Wanless.

He says even if the method was right, studying a "representative sample" of Sudoku matricies won't help people make harder puzzles.

The puzzles with the smallest number of clues that still have a unique solution will be "outliers", says Wanless.

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