Puzzles and logic

My father had a selection of books of mathematical puzzles, and as a youngster I used to enjoy trying to solve them. Later, we subscribed to a Sunday newspaper, the Sunday Times, which had a weekly "Brainteaser". These were problems of logic and mathematics which we enjoyed solving together. Much to my mother's dismay, some Sunday lunchtime meals were disturbed as he and I debated how to solve the problem.

These were puzzles where the first stage was to sort out the logic needed to solve them. Recently, a number of puzzles have become popular, such as Sudoku. These need logic (and minimal mathematical skill) but the logic is more or less the same each time. I am interested in the reasoning behind the setting of such problems; how can you guarantee that the puzzle can be solved and has a a unique solution?

The Independent newspaper has carried Sudoku puzzles for several years; recently, it has carried a range of puzzles. We have been looking at the ones that are called "Maths Puzzle". These are based around the nine digits 1-9, arranged in a square, with two mathematical operators between the three digits in each row, and between the three digits in each column. Then, at the three row ends and three column feet, are the results of the "sum". The challenge is to work out where the nine digits are placed, given the six results and the twelve operators. See "Maths Puzzle" for an example.

So, I wonder how such problems are set. With nine digits, the checking could be done by brute force very quickly, and that is how I suspect it is verified. The newspaper's problems have an easy puzzle, with two digits entered, and a hard one with one digit entered. Tina and I tackle the problems ignoring those digits. Much more need for logic.

And the O.R. link? The solution is a problem of combinatorial optimisation.