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.

Mathematical language in the news

When I heard one of the reporters on BBC radio 4's early news programme saying "X will be a subset of Y" today, the words grabbed my attention. Hearing the language of mathematics used in such a context is unusual. The story concerned the UK government's plans to be "Greener" and the reporter actually said:
"UK economic policy will be a subset of green planning".

To mathematicians, subsets are well defined, so all of X will be in Y; there may be items of Y that are not in X, but no items of X will be outside Y. The implication is that all UK financial planning will have to be seen as part of the desire to preserve the environment. As an O.R. person, it is interetsing to see that politicians are recognising that the "System" in which they plan has enormous boundaries. That must be forthe good.

But I don't foresee the Government's chief economist being replaced by an environmentalist for a little while yet.