How to Win Battleship — the targeting math
We simulated over a million games of standard 10×10 Battleship to measure exactly how much smarter targeting is worth. Headline: probability-density targeting sinks the whole fleet in about 45.5 shots, versus 95.4 for pure random fire — a 52% reduction — and it beats the classic checkerboard hunt-and-target heuristic (56.3 shots) by another 19%.
Three targeting strategies, compared
Average shots needed to sink the entire 17-cell fleet on a 10×10 board. Fewer is better:
A full probability-density heatmap (recomputing every remaining ship's possible placements after every shot) does even better at 45.5 shots -- 19% fewer than hunt-and-target and 52% fewer than random -- because it plays the CENTER of the board early (the most positions can cover it) rather than a fixed checkerboard pattern.
Full results
| Strategy | Mean shots | Median | P90 | Range |
|---|---|---|---|---|
| Probability-density heatmap | 45.5 | 45 | 58 | 20–74 |
| Hunt + target (checkerboard) | 56.3 | 56 | 66 | 20–100 |
| Pure random fire | 95.4 | 97 | 100 | 48–100 |
P90 = 90% of games finish in this many shots or fewer. Probability-density targeting was run over a smaller sample (20,000 games) than the other two (1,000,000 each) — plenty for a stable mean, and its result is consistent across sub-samples.
Why random fire takes 95.4 shots — exactly
Firing completely at random, sinking the whole fleet takes 95.4 shots on average out of 100 possible cells -- an exact combinatorial fact ((101)(17)/18 = 95.39), not a simulation artifact: it holds no matter how the ships are shaped or placed, because it only depends on the LAST of 17 marked cells showing up in a random shuffle of 100.
The exact formula: (N+1)*K/(K+1) with N = 100 cells and K = 17 ship cells gives 95.39 — matching our 1,000,000-game simulation (95.39) almost exactly, because this particular number doesn't depend on ship shapes or placement at all.
The standard fleet
| Ship | Length |
|---|---|
| Carrier | 5 |
| Battleship | 4 |
| Cruiser | 3 |
| Submarine | 3 |
| Destroyer | 2 |
17 ship cells total, out of 100 cells on the board — 17% of the grid is occupied.
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How we got these numbers
A Monte-Carlo simulation of standard 2-player Battleship (10x10 grid, the classic 5-ship fleet: Carrier-5, Battleship-4, Cruiser-3, Submarine-3, Destroyer-2 = 17 ship cells of 100). For each simulated game we randomly place the fleet (no overlaps, straight horizontal/vertical placement, no adjacency restriction, matching the standard rulebook) and then have one shooter fire shots under a fixed targeting strategy until every ship cell is hit, counting shots. Three strategies are compared: pure random fire; the classic human 'hunt (checkerboard) then target (follow the line)' heuristic; and a probability-density heatmap strategy that recomputes, after every shot, how many ways each remaining ship could still fit the board given all information so far, and fires at the cell covered by the most possible placements. The random strategy's mean is exactly solvable combinatorially (independent of ship shapes): firing in a uniformly random order over 100 cells, the expected position of the last of the 17 special (ship) cells is (101)(17)/18 = 95.39, matching the commonly-cited '~96 shots to sink everything by luck alone' figure -- our simulation validates against that exact number. A guide to how much smarter targeting is worth; real play also depends on the opponent's placement choices, which we model as uniformly random here.
Source code lives in our sims/ folder (battleship_sim.py). See also
the full Battleship entry.
Common questions
What is the best Battleship strategy?
Track probability density, not a fixed pattern. Recompute, after every shot, how many ways each remaining ship could still fit the board given every hit and miss so far, and fire at the cell covered by the most possible placements. In our simulation that sinks the whole fleet in an average of 45.5 shots — 52% fewer than firing at random (95.4) and 19% fewer than the classic checkerboard hunt-and-target heuristic (56.3).
How many shots does it take to sink a Battleship fleet by luck alone?
Exactly 95.39 on average, out of 100 cells — and this isn't a simulation estimate, it's an exact combinatorial fact. With 17 ship cells (K) among 100 total cells (N), firing in a uniformly random order, the expected position of the LAST ship cell to be hit is (N+1)*K/(K+1) = 95.39. It holds no matter how the ships are shaped or placed.
Does checkerboard hunting actually beat random firing in Battleship?
Yes, substantially. Firing only on a checkerboard pattern until you get a hit, then following the line of the ship you found, cuts the average from 95.4 shots to 56.3 — a 41% reduction. It's not the optimal strategy (probability-density targeting beats it by another 19%), but it's a huge, easy-to-remember upgrade over firing at random.