Chess

We have two problems for you today, with two things in common. The first is that they are both help-mates - which means that both sides co-operate to reach a position in which Black is mated - and the second is something you will not recognise until you have solved them both.

The first position is a helpmate in four: Black moves first and both sides conspire to reach a position in which White's fourth move gives mate. This was first shown to me on a long train journey, and it kept me happily infuriated for about three-quarters of an hour. I spent ages trying to checkmate the black king on e7, but first there's the problem that 1.e6 puts White in check, and then there's the other problem of leaving e8 vacant if the king moves to e7.

My next idea was to get the king to d8 and queen to e8, while White's moves are Bxa7, Bxb8, Kb6 and Bc7. But how do the black king and queen swap places in four moves?

The answer (obvious once you've seen it) is very elegant: 1.c6 Bxa7 2.Qb6+! Kxb6 3.Kd8 Kc5 4.Kc7 Bb6 mate.

The way White's pieces each dance forward one step on the diagonal, then skip back to occupy their original squares, gives considerable charm to the composition.

I do not know who composed the first position, but this one is by Leon Loewenton and is a help-mate in five. If it were White's move, he could play Bc1 and it would be over, but Black has no waiting move that lets him do it. What we need to do is somehow spend four moves coming back to much the same position, but with White to move.

Here's how it is done: 1.Kb2 Bc1+ 2.Kxc3 Bb2+ 3.Kxb4 Bc3+ 4.Ka3 Bd2 and now 5.b4! Bc1 mate! In the first position it was White's king and bishop that retraced their steps, in this case both the black king and white bishop danced round in complete circles. The backtracking is what makes these things so difficult to solve.