# Battleship Problem

• Nov 30th 2008, 08:18 PM
Kimberly Smith
Battleship Problem
Consider the following version of Battleship. The first player hides a mine under one of the points of a 2 X 3 grid. The second player, not able to see the mine, places a ship along one of the 7 horizontal or vertical lines joining two adjacent points. The first player wins $1 from the second if the ship lands on the mine (the first player has the right to make appropriate sound effects), and pays the second$1 otherwise. Find the game value and optimal strategies for both players.
• Nov 30th 2008, 09:19 PM
Soroban
Hello, Kimberly!

I don't understand the rules of this game . . .

Quote:

Consider the following version of Battleship.
The first player hides a mine under one of the points of a 2 × 3 grid.
The second player places a ship along one of the 7 horizontal or vertical lines
The first player wins $1 from the second if the ship lands on the mine. and pays the second$1 otherwise.
Find the game value and optimal strategies for both players.

The grid looks like this:

. . $\begin{array}{ccccccc}
o & --- & o & --- & o & --- & o \\
| && | && | && | \\
o & --- & o & --- & o & --- & o \\
| && | && | && | \\
o & --- & o & --- & o & --- & o
\end{array}$

A mine is placed on one of the points . . .

. . $\begin{array}{ccccccc}
o & --- & o & --- & o & --- & \o \\
| && | && | && | \\
o & --- & o & --- & o & --- & \bullet \\
| && | && | && | \\
o & --- & o & --- & o & --- & o
\end{array}$

The ship is placed "along a line".

. . $\begin{array}{ccccccc}
o & --- & o & --- & o & --- & o \\
| && | && | && | \\
o & --- & o & -{\color{red}\nabla}- & o & --- & \bullet \\
| && | && | && | \\
o & --- & o & --- & o & --- & o
\end{array}$

Then how does the ship ever "land on the mine"?

• Dec 3rd 2008, 11:42 PM
Kimberly Smith
Battleship Problem
Hello, Soroban!

I figured this out but could you look at this for me?

Let x = (2/3)^t/5 and y = (1,2,4)^t/7

(a) Find a 3 X 2 matrix A so that x and y are Bob's and Kim's optimal strategies for the game played on A.
(b) Find such an A so that the resulting game is fair