Northcott’s Game

The is a game with simple rules and easy to play, yet it is deeply mathematical. Each player has three rooks placed on the back rank. White starts and taking turns, each player moves one of their rooks any number of squares up or down the file. The rooks cannot move sideways and they cannot capture or jump past its opposite rook. Soon the rooks will run out of moves. The last person to move is the winner.  Commence with the simple version of the game with just three rooks each.

Start Position – 3 rook version










A typical end position is shown below. No more moves are available. Black was the last person to move and so is the winner.

A typical end position










The full version of the game involves eight rooks on both sides. (In practice these can be pawns or counters.)

Full version of the game











Each position in the game is either won or lost, provided you know the correct move. The correct move can be calculated simply, if you know the trick. Would you prefer to be White or Black?