Three Ships on 10×10

1The board, and why exact methods stop

There are 1,924,020 ways to hide the fleet on 10×10 (straight, inside the board, not overlapping, touching allowed), or 240,669 distinct layouts counting rotations and reflections as one: about 240 times as many as on 5×5, but still few enough to list.

A fixed hunt is a decision tree, so its finishing time on every layout can be computed exactly and stored, one byte per layout. Any mixture of stored hunts can then be scored exactly against any pace on every layout, and its worst layout is known, not searched for: the hider's side is exact.

The hunter's side is not. An exact best reply is a dynamic program over every state of knowledge: 181 million states and about 100 GB of memory already on 5×5, out of reach above it, so new hunts come from a search. With the classic fleet's 15 billion or so layouts, even the hider's side must be searched (the classic page).

2Self-play with a moving pace

With an exact hider and no exact best reply, we used the essay's self-play loop, a form of column generation (each new hunt adds a column to a linear program):

pool ← a few fixed hunts
pace ← a starting pace
repeat:
    # one linear program, pace held fixed:
    # mixture = best win rate on the worst
    #   layout against the pace;
    # q = the hider's lottery over layouts
    #   that hurts mixture most
    mixture, q ← linear program over pool
    # found by search, not exactly
    h ← a hunt that beats mixture against q
    if no such hunt is found: stop
    add h to pool
    pace ← halfway to mixture's finishing times

The linear program chooses the hunts' weights. Each of the 240,669 distinct layouts gives one constraint: the mixture's win rate against the pace on that layout (averaged over rotations and reflections, since each game plays its hunt in a random one) must be at least v, which the program maximizes, to within 0.005 points.

The search. A new hunt is a decision tree built against the hider's lottery (below) and the pace. While many layouts remain it fires at the cell the most lottery-weighted layouts cover. Once at most 5 to 400 remain (usually 100), it plans the rest for the best chance to win the race, exactly except that each step weighs only the two best-covering cells. A second kind looks two moves ahead; together they add about three hunts per round.

The hider's lottery. The linear program's own layout weights sit on few layouts and jump between rounds (on 5×5, aiming at them collapsed the run to 0.40% on its worst layout against a near-perfect player). The search aims instead at regret matching, a standard iterative method for zero-sum games, run for a thousand iterations per round; its answer spreads weight widely and moves smoothly. Below, the mixture's hider is the average of these lotteries over all 2,000 rounds.

The pace starts from the heat map's finishing times and moves halfway toward the mixture's own each round; in practice the program scores against a short forecast of where it is heading, which equals the pace once the loop settles. The goal: a mixture that wins half its races against itself on every layout.

The run started on 25 August 2026 and finished its 2,000th round on 15 September. The pool grew to 5,968 hunts, 964 of them weighted in the final mixture, but the weight is lopsided: the heaviest hunt carries 13.9%, the 10 heaviest about half.

The stop line never fired: the run ended at its 2,000-round budget, with counter-hunts still gaining about half a point (section 3) and both curves in Figure 1 still creeping (the deeper counter-hunt's gain fell from 0.76 to 0.73 points over the last 160 rounds).

3How we know it is close

Flat is not fast: a player who fires at random also wins half its races against itself on every layout, so we checked from outside.

Counter-hunts. A probe is one fixed hunt built by the same search against the mixture's hider and pace. Against the finished mixture the best probe gains 0.52 points with a shallow search (exact planning once 30 layouts remain) and 0.73 with a deeper one (once 100 remain), down from 1.27 and 1.45 after 407 rounds.

A probe changes only the hunt, so these are lower bounds on exploitability. On the hiding side the linear program checks: no layout holds the mixture below about 49.95% (49.948% against its own finishing times), so clever hiding gains only about 0.05 points. A reply that changes both was not searched for; the exploitability is known only from below.

A board where the answer is known. On 5×5 the same loop, never shown a near-perfect player, built a mixture that wins 48.5% head-to-head against one and 48.4% on its worst layout, both exact; the near-perfect player keeps 51.3% on its own worst layout against the mixture's pace. The loop lands within about 1.5 points of near-perfect play (the 5×5 page).

Duels. On random layouts the finished mixture beats 3-parity in 53.6% of races and the checkerboard in 61.1%; head-to-head, 54.0% and 61.4%. On random layouts it needs 39.0 shots on average, 3-parity 41.1 and the checkerboard 42.7. The middle of 3-parity's distribution is close (about 39 shots against 38); it loses in the tail, when the 2 slips between its diagonals.

4What the mixture does

Where it hides. Under random hiding a corner cell holds a ship only 3.8% of the time and a cell two in from the edges up to 9.6% (the center 9.5%), since fewer ship positions fit over an edge: that is the heat map's blind spot. The mixture's hider spreads the fleet almost evenly, 7.7% to 8.4% per cell, edges slightly above the middle (Figure 2).

How it opens. Its first shot is spread over 92 cells, none above 3.5%. Then it commits: each hunt in one of 8 rotations and reflections is a plan, 7,712 in all; after one shot a median of 9 remain consistent, and in 869 of 1,000 games on random layouts one plan is left after a median of 5 shots. The randomness is spent early, which is harmless because the hider has already committed.

Every third diagonal, halves mixed. Before its first hit it puts 88% of its search shots on one of the six sets of every-third diagonals, and it mixes the set's dark-square and light-square halves: 56% of those shots land in the half it fires at more (random order: 59%; a half-first sweep: nearly 100%). Both shares are taken on the set and half that fit the shots best, so even random shooting scores well above a third (47% to 52%) and a half. It opens like the large-board model's plain plan.

After the first sink, 67% of its hunting shots stay on the set; the plain plan would keep all of them there until the set is swept. While hunting, only 14% of its shots hit the heat map's hottest cell (while finishing a ship, 79%): under its own hider the heat map is nearly flat, with 41% of hunting shots on cells within 5% of the hottest cell's value.

Replays. On 2,000 replayed games with fleets from its hider (a sample; the exact average is 39.0 shots), it needs 38.8 shots on average; the middle 80% take 26 to 55. The layout that takes it longest on average needs 41.8 shots (exact): both 3s run down column B, one cell in from the edge, with the 2 beside the lower one (B2–B4, B8–B10, C9–C10).

On that layout most hunts finish in 22 to 37 shots, but a minority need 45 to 93 (in 200 sampled games: median 34, sample average 40.9). Figure 3 shows why: in the slow game the drawn hunt's diagonals miss the 2 entirely, and it sinks only at shot 65.

5Hand-written rules on this board

The essay's chart for this board, Figure 4 here, races hand-written rules against the snapshot after 387 rounds. Across: head-to-head (accurate). Up: the win rate against its pace on the rule's worst layout, from a quick pass with a fixed number of games per layout; a minimum over many noisy estimates reads low, so the heights are approximate.

No rule wins half its races. The fastest fire at the densest cell, 3-parity densest first (48.6%) and the heat map (48.2%), and both collapse on their worst layouts, to 23.8% and 8.9%. The heat map's worst packs all three ships into the top-left corner: D1–F1, A2–C2 and the 2 in the corner beside them.

Randomized 3-parity keeps about 37% on its worst layout (37.2% by hand, 37.8% with heat-map finishing) but pays in speed (46.5% to 47.1% head-to-head). The checkerboard family is lower on both.

The highest worst case on the chart belongs to a rule you can play by hand: smart 3-parity's three-ship core (the essay's section 3), at 38.6% on its worst layout and 47.7% head-to-head. It is the adaptive rule of section 6 with a jump in 35% of games and a zone final for the 2: on the checkerboard color fired at more, where a 2 still fits, first the cells outside that color's half (A or B) that touch an unshot cell of the other half, then the rest of its half, then the rest of the color. The zone final adds half a point head-to-head (47.2% before) and leaves the worst case within noise of its earlier 38.5%; that noise, about 1.3 points on a single layout class, also covers its lead over 3-parity with heat-map finishing (37.8%). Never jumping reaches 37.6%, also within that noise, and always jumping 31.2%; each does worst on a different layout, where the other wins about 47% to 48%.

The trade-off sharpens as boards grow. Against each board's self-play mixture, 3-parity by hand beats the heat map's worst case by 21 points on 6×6, 27 on 8×8 and about 28 on 10×10, while the heat map's head-to-head lead shrinks from 6.6 to 4.2 to 1.7 points. These are estimates (the 6×6 one from a pass that had not settled); the 5×5 page has the same chart, exact.

6Rules from the model, on 10×10

The large-board page builds its plans on three sets of cells: A and B, the halves of one set of every-third diagonals on dark and on light squares, and C, the other dark squares. The adaptive rule plays that page's adaptive plan on the real board: sweep A, then B while a 3 is afloat, then hunt the 2 on A's checkerboard color. Its variants change the order inside each step or jump early to B.

We raced them against the finished mixture in a harsher measure, the mutual race: the fleet the rule must find sits on the rule's worst layout found, the fleet the mixture must find on the mixture's worst; a copy of the mixture would score 50%. Each rule played 100 games on every layout, then up to 250,000 on its slowest. A harder layout may exist, so the slowest layouts found are lower bounds in shots and the mutual-race values upper bounds (rounded up).

Every fix that removes a fixed choice helps. With one orientation and a corner-to-corner sweep the rule wins at most 33.6% of mutual races; a random rotation or reflection lifts it to 40.3%, random checkerboard order to 44.7%, and random order inside each half to 45.6%. Jumping to B after the first 3 sinks, in a secret 35% of games, reaches 47.0%. Randomizing costs about 0.7 shots of average.

A hider can aim at fixed priorities. The fastest rule (checkerboard by fewest shot neighbors, heat-map finishing) needs 39.09 shots and wins 49.3% on random layouts, but at most 32.0% in the mutual race. For a rule that orders its checkerboard by fewest missed neighbors, 99% of its 500 worst layouts hide the 2 in the interior (21% at random). Computer finishing buys speed, not robustness: 0.1 to 0.23 shots, with the mutual race at 46.2% to 46.4%.

Jump in some games, not all. On 10×10 the adaptive rule peaks near the jump rate of the large-board model's best blend (about a third): 47.0% at 35% and 46.6% at 40%, within noise of each other, against 45.6% never and 41.5% always (Figure 5): the same shape, though not the same number. Layout by layout, the 35% rule's finishing times are exactly 65% of the never-jump rule's plus 35% of the always-jump rule's, and a hider can aim at only one of the two weaknesses.

The best simple rule stays at least 3 points short of the mixture. With the zone final for the 2 (section 5), it is the core of smart 3-parity on the classic fleet (the classic page).

On their slowest layouts found, these rules keep the large-board model's order: 3-parity and the adaptive rule level, always jumping slowest, jumping in some games fastest (how the model compares).

7What is open, and how to play

Open questions.

• The finished mixture's exploitability is known only from below.
• Figure 4's rules were never re-scored against the finished mixture.
• Open questions about the large-board limit are on its page.

Replays are exact. Each of 2,200 replayed games ends at the finishing time stored for its hunt and layout (the essay's replay shows four). Replaying the mixture needs its 964 hunts and weights; rebuilding it needs the loop of section 2, the 1,924,020-layout enumeration and a linear-program solver. The last 1,600 or so rounds took about 185 hours on one 8-thread workstation, about 8 minutes per round near the end.