Sea battle strategy: parity, probability and real numbers

1. After a hit: finish the line

Fire at one of the four neighbours of a hit. A second hit gives the direction: now only along that line, at both ends. In Classic never fire diagonally: no ship can be there. Our Sailor does not know this, and that one mistake costs it about 10 shots more than the Captain.

2. Parity: do not fire at every cell

A 4-cell ship must cross a cell where column plus row is divisible by 4, and there are only 25 of those. So a quarter of the board is enough to find the battleship. Once it is found, switch to steps of 3, then 2. This is how the Captain plays.

3. Probability: where a ship fits most often

Before each shot, imagine placing the remaining ships in every legal way and fire where most layouts pass. You cannot compute it exactly in your head, but the logic is simple: a cell with open surroundings is worth more, a narrow gap between shots is worth less. That is exactly how the Admiral calculates.

Method	Classic	International
Random + hunt (Sailor)	71.3	70.2
Parity + target (Captain)	60.5	51.8
Probability density (Admiral)	55.9	45.7

The table shows the average number of shots to sink the fleet (2,000 games).

4. The myth of the centre

We counted 60,000 random fleets. Under International rules a centre cell holds a ship 21.7 % of the time and a corner only 7.4 %, so starting in the middle is right. Classic looks completely different: centre 19.7 %, edge 20.6 %, corner 18.1 %. The no-touch rule and many small boats nearly flatten the board, so “fire at the centre first” does not work in Classic.

5. In Classic the patrol boats decide

In Classic a hit gives another shot, so the winner is whoever misses least. Big ships are found quickly, but four one-cell boats cannot be calculated. Even the Admiral fires into water 36 times on average. So put your boats away from the other ships, in open water: the opponent reaches them last.

6. Deploying your fleet

Change your layout every game: a single pattern is quickly learned.
In Classic it helps to keep big ships close together: water around a found ship is marked anyway, and more open water remains for the boats.
In International try placing ships side by side: the opponent may sink one and miss its neighbour.
In International the corner is the least-fired spot, a good home for the destroyer.

Test it against the Admiral.