Sea battle strategy: where to fire first, and why probability beats luck

The hidden-ships game feels like luck. You call a square and miss, they call a square and hit, and it seems to come down to who guesses well. But play the same kind of sea thousands of times with different methods and the luck washes out fast: one way of firing sinks the whole fleet in about 14 misses, another needs 48. That gap is the game.

We measured it. We simulated 20,000 games for each method on the setup our version uses: an 8×8 sea, four ships of 4, 3, 3 and 2 squares, laid at random, and never touching each other — not even at a corner. A hit earns another shot, so the number that decides a match is misses. Every miss hands the turn to your opponent. Fewer misses, you win.

The scoreboard

Average misses to sink the whole fleet:

• Random shots — 48.
• Hunt, then target (after a hit, fire next to it) — 28.
• …plus cross off the squares around a sunk ship — 21.
• …plus hunt on a checkerboard — 17.5.
• Count the ways (fire where the most ship positions still fit) — 14.

Each step also wins head to head. Hunt-and-target beats random firing in 97% of games. Counting beats plain hunt-and-target in 92%. Every one of these methods is something a person can do at the table, so here they are in order.

Where to fire first: anywhere but the corners

The usual advice is to start in the middle. On a small sea with a no-touch rule, that isn't quite right. We laid 200,000 random fleets and counted how often each square held part of a ship:

• The four corners: 12%.
• The edge squares next to a corner: about 16%.
• Everything else, centre and edges alike: 18–21%.

The middle isn't hot. The corners are cold. A ship can only cover a corner by starting exactly there and running along one of two edges, so the corner misses out on most possible placements. Your opening shots should go anywhere except those twelve cold squares. Beyond that, a square two in from the edge is as good a first shot as the centre.

These numbers assume the game hides the ships at random, which ours does. When a person hides them, you are playing against their habits instead, and that is psychology, not arithmetic.

Hunt, then target

The biggest single gain is also the simplest. While you have no unsunk hits, you are hunting: spreading shots to find a ship. The moment you hit something, stop hunting and start targeting:

1. Fire at the four squares that share an edge with the hit — up, down, left and right.
2. As soon as you have two hits in a line, you know which way the ship runs. Fire along that line, at either end, until it sinks.
3. Never fire diagonally from a hit. Ships are straight and can't touch, so a square diagonal to a hit is always empty.

That alone takes you from 48 misses to 28. One refinement: when you choose which neighbour to try first, pick the direction with more open water. A hit one square from the edge is more likely to belong to a ship that runs inwards, simply because more ships fit that way.

Use the no-touch rule

When a ship sinks, every square around it, diagonals included, must be empty — no other ship is allowed to touch it. Cross them all off and never fire there.

This is worth more than it sounds. A four-square ship sunk in open water rules out the 14 squares around it, so with the ship itself that's 18 squares settled — more than a quarter of the sea, from one sinking. It took our simulated players from 28 misses to 21, and they beat players who ignored it in 73% of games.

Hunt on a checkerboard

The smallest ship is two squares long. Any two squares side by side on a checkerboard are different colours, so every ship, whatever its size and direction, covers at least one dark square. Hunt only the dark squares and you search 32 squares instead of 64 — and you still can't miss a ship.

That took our players from 21 misses to 17.5, and it beat the same player without it in 64% of games. Once the two-square ship has sunk, the smallest ship left is three long, and you can widen the net to every third square on a diagonal pattern. That helps a little more (17 misses), but it's harder to track and the gain on an 8×8 sea is small.

The best method: count the ways

The method that wins most often is the one that sounds most like maths. For each open square, ask: how many ways could the ships still afloat lie across this square, given every miss and every sunk ship so far? Then fire at the square with the most.

A computer counting exactly averages 14 misses and beats the checkerboard hunter in 72% of games. You can't count exactly at the table, but two habits get you most of the way:

• Fire into the biggest open gap. A long stretch of untouched water fits many ship positions. A narrow strip between misses fits few.
• Write off dead water. If the only ships left are the 4 and a 3, any open patch where no straight line of three squares fits can't hold either of them. It's empty. Skip it.

Dead water is where most players waste their late-game turns: firing into small pockets that, given what's still afloat, can't contain a ship at all.

A checklist that will beat most people

1. Open away from the corners.
2. Hunt on one colour of the checkerboard.
3. After a hit, fire only at the four squares beside it — never diagonally.
4. Two hits in a line: follow the line until the ship sinks.
5. When a ship sinks, cross off every square touching it.
6. Late in the game, fire into the biggest gap the remaining ships can still fit, and ignore dead water.

Luck matters less than it looks

Any single game can go either way — a lucky first shot is a lucky first shot. But a good method narrows the spread a lot. Counting finished in 9 to 19 misses in eight games out of ten; random firing needed 42 to 52. Over a run of games between two people, the better method wins almost every time, which is what makes this a rivalry rather than a coin toss.

It also suits slow play. One shot, one answer, and the board tells you everything you need when you come back to it. All the strategy above lives on the board, not in your memory.