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Battleship explained in wiki:

(also Battleships or Sea Battle1) is a guessing game for two players. It is played on ruled grids (paper or board) on which each players fleet of ships (including battleships) are marked. The locations of the fleets are concealed from the other player. Players alternate turns calling "shots" at the other player's ships, and the objective of the game is to destroy the opposing player's fleet.

My question is, is there a "best strategy" in such a game?

For example, the grid is $10\times 10$ size, and each side has 5 ships (1 ship of $1\times 5$, 1 for $1\times 4$, 1 for $1\times 3$ and 2 ships of size $1\times 2$), what is the best strategy to arrange the ships at the beginning and the strategy to target shots?

Of course there are some simple conclusion, e.g. if one is looking for a long ship of size $1\times 5$, areas not big enough shouldn't be targeted.

However, are there more tips?

  • for example, if a shot hits, then the next shot, shall be aimed one grid upper, lower, left or right of the current hit?

  • another example, at the beginning, shall one distribute the fleet among all the $10\times 10$ grids or locate them together?

  • also, should one put the $1\times 2$ small ship near a big ship so it would be neglected by the adversary, or place it away from other ships?

  • considering gaming, if ships are randomly evenly distributed among all possible locations, there are some cells has higher probability of being "part of a ship" than other cells, hence such cells will be on higher priority of attack from the adversary's point of view. Then, shall a player's strategy be simply randomly placing the ships, or, foreseeing the adversary's aiming, redistribute the ships so that the probability of each cell is the same?

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    $\begingroup$ Cheat. Just stack all the ship on top of each other in a U shape bend. $\endgroup$ – Asaf Karagila Feb 6 at 7:44
  • $\begingroup$ Could we just be just a little bit more serious... $\endgroup$ – athos Feb 6 at 7:58
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    $\begingroup$ Sorry. I guess I just don't see where the mathematical part comes in. $\endgroup$ – Asaf Karagila Feb 6 at 8:03
  • $\begingroup$ @AsafKaragila to the simplest case, if there's only one $1\times 5$ ship, and a shot on a grid hit fire, then which grid would you aim for the second shot, the cell above/below/left/right of the one on fire? maybe it doesn't matter? what if the cell hit on fire is located nearer to a corner of the hole grid, does it matter? what if there are other cells already hit and known to be "clear"? does it still make no difference? $\endgroup$ – athos Feb 6 at 8:07
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    $\begingroup$ There are lots of hits on the web if you search for "battleship game theory". That would be a good place to start looking. $\endgroup$ – Rahul Feb 6 at 8:59
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Placing your ships away from the edges of the battlefield and away from one another minimizes the probability of a followup hit. If this strategy is discovered, however, you may be at a disadvantage. Luckily it's not a full information game, so your opponent must include the edges in their uniformly drawn shots. The same applies to your shots. As you shoot, select from a uniform distribution of "one color" on a checkerboard, never aiming for the corners as there are no $1\times 1$ ships.

I am unsure if hitting all enemy ships before sinking them is superior to sinking a newly discovered ship. If the enemy places their ships at random, then there may be a chance their ships are adjacent, and so it is beneficial to attempt to sink a ship immediately, as you may be striking a new ship in the process.

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    $\begingroup$ I think sinking ships on discovery is benefial even if ships are not adjacent. If you do that there is a chance you discover and sink the $2$ small ships before you discover all the others. Then you can thin out your checker board from a $2$-color one to a $3$ color one and still find all the remaining ships. $\endgroup$ – quarague Feb 6 at 12:12

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