Chomp

Click a square to eat it and everything below and to the right of it. The top-left square is poisoned — whoever is forced to eat it loses. The computer plays perfectly. Everything runs in your browser; nothing is uploaded.

A proof that names no move

The first player wins Chomp on every board bigger than a single square, and the proof is three lines. Suppose every opening lost. Then eating just the bottom-right corner loses, so the second player has a winning reply to it. But eating that corner removed nothing except itself — so the first player could have played that reply as their own opening instead, and won. The assumption contradicts itself.

That argument is complete and it names no move. It never constructs anything; it works purely by ruling a possibility out. So the natural question is what it is declining to tell us — which is answerable by solving the boards outright:

Two things fall out. The first player does win every board bigger than 1 × 1, as promised. And the winning move is unique every single time — across all 37 boards solved here, including long thin ones up to 2 × 12, the number of winning openings is always exactly one. Not one board offers a choice.

So strategy stealing proves the existence of something there is only ever one of, and says nothing whatever about which it is. That is an unusually sharp gap between knowing a thing exists and being able to point at it.

Two families have a rule; the rest wander

Square boards are solved and easy to state: eat the square diagonally next to the poison. That leaves an L — one full row and one full column — and from there you mirror whatever your opponent does on the other arm. Every square board above takes r2c2.

Two-row boards have a rule too: eat the far corner, leaving a row of n above a row of n−1. Every two-row board above does exactly that, out to 2 × 12.

Outside those two families, nothing. Watch the winning move on a three-row board as it widens:

It moves about rather than settling. And it is not always tucked near the poison either — on a 5 × 8 the only winning opening is r5c7, most of the way down and most of the way across. Here are the boards outside the two solved families:

No general formula for Chomp is known, and those are the numbers any formula would have to account for. The whole table above cost 3,202 positions to solve, which is why it is recomputed on every build rather than written down — but the boards grow fast, and that is the reason nobody simply solves their way to a pattern.