Poison Square is played on a bar of chocolate, five rows by seven. The top left square is poisoned. On your turn you pick any square and eat it, together with everything below it and to the right of it. Whoever is left with the poisoned square loses.
Mathematicians call the game Chomp. It was described in its chocolate-bar form by David Gale in 1974, and given its name by Martin Gardner, who wrote about puzzles for Scientific American for many years.
It has a property that sounds impossible. We know for certain that the first player can always win. We do not know how.
The argument
It goes like this, and it needs no sums.
The game cannot end in a draw. Somebody eats the poison. And there is no luck in it. In a game like that, one of the two players must have a winning plan: either the first player can force a win, or the second can.
Suppose it were the second player. That would mean that whatever the first player does, the second has a perfect reply.
So let the first player make the smallest possible move: take only the single square in the bottom right corner. By our assumption, the second player has a winning reply to that. They bite somewhere, and it leaves a position from which they will win.
Here is the trick. That bite, whatever it is, would also have removed the bottom right corner. Every bite does. So the first player could have made that very bite on move one and reached exactly the same position, with the roles swapped.
In other words, any winning reply the second player has, the first player could have played first. The second player cannot have a winning plan. Therefore the first player does.
What the argument does not tell you
It does not say what the winning first move is. It only says that one exists. Taking the corner square might be it, or might not.
For small bars, people have worked the answers out by checking every possibility. For a five by seven bar a computer can do it easily. But there is no simple rule that a person can carry around, as there is for the stone game Nim.
This kind of reasoning is called a strategy-stealing argument. It turns up elsewhere in mathematics, and it is always slightly unsettling: a proof that something can be done, with no instructions.
For players, it is good news. It means the game is still a game.
Endgames you can learn
You cannot memorise the whole thing, but several positions near the end are easy.
The L shape
Suppose all that remains is the top row and the left column, meeting at the poison square.
If the two arms are the same length, whoever has to move loses. Whatever they bite off one arm, the other player bites the same off the other. The arms stay equal, shrinking together, until the mover is left with only the poison.
So if it is your turn and the arms are unequal, bite the longer one down to match. If they are already equal, you are in trouble.
The diagonal trap
The square diagonally below and to the right of the poison is special. Taking it leaves exactly an L. If you take it and the arms are unequal, as they are on a five by seven bar, your opponent evens them up and you have lost.
On a perfectly square bar it would be the winning first move, because the arms would be equal. On ours it is a blunder.
Two rows
Suppose only the top two rows are left. If the top row is exactly one square longer than the row beneath, whoever has to move loses.
So if you are faced with two rows, try to bite so that the lower row is one shorter than the upper.
A single row or column
If only the top row is left, take everything except the poison. Your opponent has to eat it.
Putting it to use
In a real game, think of each bite as shaping a staircase. You are trying to hand over one of the losing shapes above, and to avoid being handed one.
A reasonable approach for the first few moves is to take modest bites from the lower right, keep the bar fairly wide, and watch for the moment the position can be turned into an L or a two-row bar that suits you.
When you are close to the end, count. With few squares left, you can often check each possible bite and see whether it leaves a shape you recognise.
Who goes first on our board
The game picks the starting player at random. In theory the first player is winning. In practice, since neither of you knows the perfect line, it makes very little difference.
If you would like to even things up, play two games and swap who starts.
Why it is worth a minute
A match takes about a minute. In that time you are playing a game for which a perfect strategy is known to exist and has never been written down in a form a person can use.
There are not many HTML5 games on TwoSide, or anywhere, of which that can be said.