MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

June 3, 2007

Sink or Be Sunk

It's sink or be sunk in the game of Battleship, the cat-and-mouse game where players try to figure out where their opponent's fleet is deployed. You probably know the paper-and-pencil, pegboard, or electronic versions. And if you don't, you can play a version of the game online at javascript.internet.com/games/battleship.html.

But Mogens Esrom Larsen of the University of Copenhagen has invented a much trickier version of Battleship, where the ships aren't straight and the goal is to hit the open water rather then them.

In Larsen's game the game pieces are special shapes called pentominoes. Dominoes consist of two squares stuck together; a pentomino is made up of five squares. Each player has 11 pieces: the 12 possible pentominoes, named for the letter of the alphabet they most resemble.

The game board is a standard checkerboard with 64 squares. The 12 pentominoes take up a total of 60 squares. Each player fits together the 12 pentominoes to cover the board, leaving just four empty squares. This can be done in many different ways. Here's one example:

The goal of the game is to find the four empty squares on your opponent's grid. Each player, in turn, fires a volley of four shots. Your opponent then tells you how many shots hit which pentomino.

One good offensive strategy is to hit the corners. In the example below, a player has hit a1, b1, a8, and h8. His opponent must tell him that he hit the N pentomino twice, the Z pentomino once, and the P pentomino once. The player then knows the Z, P, and N pentominoes are in the corners. But he doesn't know how any of these pieces are oriented. The P, for example, could be upside down or flopped.

What about a defensive strategy? You want to come up with a starting arrangement of pentominoes that makes the four empty squares hard to find. One way to make the problem manageable is to build your board around the F pentomino, which is one of the few that are not symmetric. If you put the F on a three-by-three square by itself, you end up with four empty squares. And there are eight different ways to place the F on this grid.

If you placed the F grid away from the corners of the larger board, your opponent wouldn't be able to locate it by hitting the corners. And even if he did hit the F, he wouldn't know which of the eight possible orientations it had. It would probably take him several more turns to find out, enough time for you to find his open squares.

The new game combines the challenge of fitting together jigsaw pieces with the fun of sinking the fleet.

Muse, May/June 2004, p. 28-29.