The Game of Dodgem
by T. W. Marlow
(first published in the Games and Puzzles Journal issue 16)

This game, invented by Colin Vout [Ref.1] is simple in concept but difficult to play well. It is played on a 3×3 board as shown in Figure 1. Each side has two counters which start on the squares shown. The players alternate in making single moves and a move consists of a step by one piece to a laterally adjacent square — there are no diagonal moves. White can move up or left or right but never down. Similarly Black moves up and down or to the right but not to the left. Finally, when a White counter is on the top row then a move can take it off the board. The same applies to Black when a counter is in the right-hand column. The aim is to get both counters off the board and the first to do so is the winner. It is possible for one side to be unable to move at its turn — the situation known as stalemate in chess. The side in this position is then immediately declared the winner, i.e. the giver of stalemate is punished.

 ñðò ñðò ñïð ñïð

A full analysis of the game has been made by Berlekamp, Conway and Guy [Ref 2] which shows that the first to play, customarily White, can always win. However, despite the simple concept, the strategy is tricky and it is easy to go wrong. An example is the position in Figure 2, where White is to play. The obvious move of taking c3 off is fatal as the play runs: 1. c3 off c3 2. a1 c2 etc. and Black gets both his counters off first. Correct is 1. a1 c2 2. a2 c2 off 3. a3 b2 and this wasted move by Black gives the game to White.

Figure 3 shows a curious situation in which White can win whichever turn it is to move. If White started the game then it must be his turn to play and win by 1. c3 a2 2. c3 off c3 3. b3 etc. or 2. ... a3 3. a2 etc. But suppose the players had agreed on Black playing first and so now having the move. Play continues 1. ... c3 2. b3 c3 off 3. c3 etc. or 2. ... a2 3. b3 off c3 off and again White wins. Note that in the last sequence 3. ... b2 would have lost by giving stalemate.

The analysis can be used to set up a computerised version of the game, allowing the computer to know all the strategy. Taking White against the computer, and so playing first, is good practice as any inaccuracy will be punished by the machine. I have such a programme but only for the Acorn platform.

There is an obvious extension of the game to an n×n board with each side having n – 1 counters. It seems that no analysis has yet been made for any n greater than three.

References:

[1]. C. Vout and G. Gray, Challenging Puzzles, Cambridge University Press, Cambridge, 1993.

[2]. E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Vol.2, Academic Press, London, 1982.