Dodgem

For the fairground ride, see bumper cars.

Starting position for 3×3 Dodgem

Dodgem is a simple abstract strategy game invented by Colin Vout and described in the book Winning Ways. It is played on an n×n board with n-1 cars for each player—two cars each on a 3×3 board is enough for an interesting game, but larger sizes are also possible.

Play

The board is initially set up with n-1 blue cars along the left edge and n-1 red cars along the bottom edge, the bottom left square remaining empty. Turns alternate: player 1 ("Left")'s turn is to move any one of the blue cars one space forwards (right) or sideways (up or down). Player 2 ("Right")'s turn is to move any one of the red cars one space forwards (up) or sideways (left or right).

Cars may not move onto occupied spaces. They may leave the board, but only by a forward move. A car which leaves the board is out of the game.

The winner is the player who first has no legal move on their turn because all their cars are either off the board or blocked in by their opponent.

Theory

The 3×3 game can be completely analyzed (strongly solved) and is a win for the first player—a table showing who wins from every possible position is given on p. 686 (1st edition pagination) of Winning Ways, and given this information it is easy to read off a winning strategy.

David desJardins showed in 1996 (thread from rec.games.abstract) that the 4×4 and 5×5 games never end with perfect play—both players get stuck shuffling their cars from side to side to prevent the other from winning. He conjectures that this is true for all larger boards.

See also

• Gardner, Martin (1987) Time Travel and Other Mathematical Bewilderments, W.H. Freeman & Company; chapter 12. ISBN 0-7167-1925-8