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Math Help - Battleship Problem

  1. #1
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    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.
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, Kimberly!

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


    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
    joining two adjacent points.
    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}<br />
o & --- & o & --- & o & --- & o \\<br />
| && | && | && | \\<br />
o & --- & o & --- & o & --- & o \\<br />
| && | && | && | \\<br />
o & --- & o & --- & o & --- & o<br />
\end{array}



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

    . . \begin{array}{ccccccc}<br />
o & --- & o & --- & o & --- & \o \\<br />
| && | && | && | \\<br />
o & --- & o & --- & o & --- & \bullet \\<br />
| && | && | && | \\<br />
o & --- & o & --- & o & --- & o<br />
\end{array}



    The ship is placed "along a line".

    . . \begin{array}{ccccccc}<br />
o & --- & o & --- & o & --- & o \\<br />
| && | && | && | \\<br />
o & --- & o & -{\color{red}\nabla}- & o & --- & \bullet \\<br />
| && | && | && | \\<br />
o & --- & o & --- & o & --- & o<br />
\end{array}



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

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  3. #3
    Newbie
    Joined
    Nov 2008
    Posts
    5

    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
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