Arts: Variants on chess

When my youngsters were in primary school, they were all in the chess club, and I was roped-in for several years, where I taught the tinies the art of wood-pushing, using a simple variant.

The start of a game of rajah and the sepoys.

There is a simple version of a chess variant called the maharajah and the sepoys, where black has a full set of chess pawns and white has only a king, but the king can move as either a queen or as a knight on each move. It is, in the language of gaming, a solved game, which black can always win, although white can win against an inexperienced opponent.

This simpler version is played with four black pawns and one white pawn, placed as shown. Each piece moves one square diagonally, but the white pawn can go backwards or forwards, while black can only move forwards. The aim is for white to get past the black line, while black wins if the white pawn can be surrounded and stopped from moving.

In my experience, I think black can always win, but I know of no proof of this. The game can go either way when played by young players: black needs to keep the line as level as possible, always advancing when white withdraws, unless this leaves an opening. The game is excellent training for those under the age of nine who are about to begin playing chess. I know I have a book somewhere which documents this game, but it seems to be entirely unknown under this name on the internet.

From the left, a game under way, five moves in; next, a win for black imminent; last, a win for white.

Knight’s Tours

You will need a ruler, pencil, rubber and paper. A knight’s tour is a closed trip around a board with squares like a chess board, visiting each square once and once only. Each “jump” is a “knight’s move” in chess: one square forwards, backwards or sideways, and one square diagonally. A proper knight’s tour visits every square on the board once and once only.

The best way to work out a knight’s tour is to draw up a small grid, and write in the numbers, counting from 1, on the squares that you visit. Here is an example, using a 3 x 3 grid, which cannot be solved completely, as there is no way you can get to (or from) the centre square.

An impossible knight’s tour puzzle.

Problem 1: Find the smallest size grid that can fit in a knight’s tour, even if the tour is not closed (which is where you end up a single knight’s move away from the starting point). It will be no larger than 6 x 4.

Problem 2: Find the smallest grid that can fit in a closed knight’s tour (see task 1 for a definition). (This can be done in 8 x 8, but may be able to be done on a smaller grid.)

Problem 3: Find a solution for a standard chess board (8 x 8)

Problem 4: Can you find a complete 8 x 8 knight’s tour that is also a magic square? Leonhard Euler designed the 8 x 8 “knight’s tour” magic square that appears above. If you are a chess player and understand this, try it for yourself. If you need help, ask for it and then try it. (Hint: do four in each of three quadrants, and eight in the fourth quadrant before doing four in the third, second and first quadrants, and then repeat to complete.)

The main trick is to realise that a chess knight alternates between black squares and white squares. That makes getting in and out of corners a bit of a challenge. There is a fair amount of literature around: look for recreational mathematics or mathematical games. Apart from that, you’re on your own. Good luck!

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