BULO

Copyright (c) 2001 Piet Hein

This game (aka TacTix) is played on the following 4x4 square board:

 

TURNS - At each turn, each player must remove a connected orthogonal line of stones. GOAL - The player that removes the last stones, loses.

 

An example If the next player removes the marked stones, he wins the game. The adversary cannot avoid removing the last stone(s).

This is a CGT game, sort of 2D NIM. This game can be expanded to any board size (TacTex is one name used for hex boards), and also to allow different shapes to be removed (like rectangles, pentominoes or diagonal crosses).

Eatcake is a misère Bulo: Given a NxM board full of pieces, the first player must remove a connected horizontal line of stones, and second player a vertical line. The last player to move, wins.

Bill Taylor invented a variant, called Bonus NIM played on a 7x7 board (which is expandable to any rectangular board). A player may remove an orthogonal connected line of stones of any size. There is a restriction for the first move, where the move must take less than half the maximum possible. Also, there is a bonus of four stones for making the final capture. Winner is whoever obtains most stones. 

An example First player has 18 stones, second player has 17. Second player's turn. If Second removes the marked stones, First may reply by removing the two below (e5 and f5), creating a null position (i.e., a winning position) which provides him with 4 extra points. Even if Second tries to maximize each move, he will not gather enough stones to win (e.g., if Second replies by capturing three stones like d6,d7,d8, First just replies g2 and still have enough points to win).

This bonus mechanism can be applied to many NIM games, giving a new interest to these mathematical studied games. The bonus size is critical to define the best strategy.

Another NIM-like games: Nineteen, King's Horses.