Oct 08, 2026 by Offer

IMC Battleships Question 2027: Probability & Strategy

The IMC Trading battleships probability question — commonly reported by candidates as "Battleships on a 4x4 board with 3x1 and 2x1 pieces: how many arrangements, and what is the best first target?" — has exact answers: 264 non-overlapping arrangements, and the best first shot is any of the 4 center squares (each hit with probability ~0.379).

What This IMC Trading Battleships Probability Question Assesses

This tests combinatorial counting under constraints plus probabilistic targeting — the two halves of many trading problems. IMC wants to see you count placements carefully (16 for the 3-ship, 24 for the 2-ship, then enforce non-overlap) and then use the distribution to aim optimally rather than guessing.

How to Answer This IMC Trading Battleships Probability Question

Count first, then target:

Step 1 — Count placements. The 3×1 ship: 8 horizontal + 8 vertical = 16 placements. The 2×1 ship: 12 horizontal + 12 vertical = 24 placements.
Step 2 — Enforce non-overlap. For each of the 16 long-ship placements, count 2-ship placements on the remaining squares and sum — the total is 264 valid arrangements. State the method; the interviewer cares that you handle the constraint, not that you arithmetic it perfectly live.
Step 3 — Target by probability. By symmetry, the 4 center squares are equivalent and each is covered by the most arrangements — hit probability about 0.379, versus 0.227 for corners. So open on a center square.
Step 4 — Update after a hit. A hit on the center constrains the ship's orientation — fire adjacent squares to determine direction, then extend. A miss eliminates all arrangements covering that square; recompute the best next target.

Example line: "264 arrangements by placement counting with the overlap constraint — and the center four squares each cover the most arrangements, so that's my opening shot."

Common Mistakes

Multiplying 16 × 24 = 384. Forgetting the non-overlap constraint is the classic error — overlapping placements aren't valid arrangements.
Targeting corners. Corners feel safe but cover the fewest arrangements; probability-weighted targeting always beats gut feel.
Not updating. Firing a second random shot after a hit wastes the information — every result reshapes the distribution, so re-aim every time.

Candidates commonly report follow-ups: "You missed — now where?" or a different board size. The framework is portable: count valid arrangements, compute coverage probabilities, shoot the max, update. Practice the method, not the numbers.

Keep Reading

FAQ

Do I need the exact count live? The method matters more than 264 — but having the right order of magnitude and the correct constraint handling is what scores. Walk the counting clearly.

Why are center squares best? More ship placements pass through the middle of the board than the edges — longer ships especially favor central squares. Symmetry plus coverage.

What about a checkerboard/parity strategy? Parity helps when hunting a ship of known length after a hit (it constrains orientation), but for the opening shot, raw coverage probability dominates.

How does this relate to trading? It's search under uncertainty with updating beliefs — the same structure as hunting for liquidity or updating on order flow. Count the possibilities, weight by probability, act, update.

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