# Strategy games

• Jun 19th 2011, 12:28 PM
Difficulties
Strategy games
Here is a game I wish I had the solution to:
A cuboid consists of 4x6x9 cubes. Two people play a game where they reduce the cuboid with flat cut. Every time you make a cut you can have one of the two pieces and then the turn passes to the opponent who shares the cuboid in the same way with flat cuts. The game continues as long as the cuboid can be divided into smaller cuboids. Whoever makes the last division is the winner. How should the first player to make the first division to increase their chances of winning?
• Jun 24th 2011, 12:56 AM
Traveller
Re: Strategy games
Quote:

Originally Posted by Difficulties
Here is a game I wish I had the solution to:
A cuboid consists of 4x6x9 cubes. Two people play a game where they reduce the cuboid with flat cut. Every time you make a cut you can have one of the two pieces and then the turn passes to the opponent who shares the cuboid in the same way with flat cuts. The game continues as long as the cuboid can be divided into smaller cuboids. Whoever makes the last division is the winner. How should the first player to make the first division to increase their chances of winning?

Let us think of a slightly different game. As earlier, you can only make flat cuts and pass on one of the pieces. But in this game, a player always has to keep all the dimensions of the cuboid that is passed on, different from each other, and he cannot bring any dimension down to 1. The last player who can cut the cuboid obeying these rules wins. Now it is easy to see that the person who cuts the cuboid to obtain a 2x3x4 piece wins. Show that the first player will have a winning strategy when they start with a 4x6x9 cuboid.

Next prove that if they start with a cuboid with a dimension of 1, then the first player has a winning strategy if and only if the other two dimensions are not equal to each other. Use this and the first proof to solve the original problem.