Combinatorial Game Theory
who wins, and by how much
Generator

The one board where blocking is not enough

The one board where blocking is not enough
The one board where blocking is not enough. Fifteen small m by n boards with k in a row, each solved as played and as Maker–Breaker with Maker first and second. On every board but noughts and crosses, Maker's result as first player agrees with the first player's result as played.

Fifteen small m by n boards with k in a row, each solved as played and as Maker–Breaker with Maker first and second. On every board but noughts and crosses, Maker's result as first player agrees with the first player's result as played.

3 essays call mnk-board. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.

What a corner adds. The replies to a centre opening at noughts and crosses. As played, corners draw and edges lose. In the game where the second player can only block, every reply loses. Out in the world

Blocking is not enough

Everybody knows noughts and crosses is a draw, and the usual explanation is that the second player can always block. That explanation is false. Against an opponent who can only block, the first player wins, from the centre or any corner. The game is a draw only because the second player's blocks build threats of their own. On fifteen boards solved outright, it is the only one where those two things come apart: everywhere else a draw is a draw because blocking is enough.

 Three certificates and the truth. The drawn m,n,k boards up to twenty cells, with whether a pairing exists, the Erdős–Selfridge potential, the exhaustive Maker–Breaker result, and whether a greedy rule based on the potential holds against every line of play. Out in the world

Three ways to know a draw

A draw on a board too large to search has to be certified, and there are two standard certificates: a pairing of cells and the Erdős–Selfridge sum over lines. Set against exhaustive search, both miss 4 × 4, and a rule taken from the sum holds there anyway. The sum looks far stronger until pairs are allowed to serve several lines at once. Then across 440 boards up to 12 × 12, the sum certifies no draw that a pairing does not, and a pairing certifies 36 that the sum cannot.

 A price in stones per turn. Small boards with the exact number of stones Breaker needs per turn to stop a Maker placing one, the rate the biased Erdős–Selfridge potential guarantees, and the number Maker needs per turn to beat a Breaker placing one. Out in the world

A win priced in stones

Change the rate of play and a yes-or-no question becomes a number. Against a Maker placing one stone a turn, Breaker needs two stones to stop three in a row on every board up to 4 × 4 and six by three, and three on 5 × 4. Against a Breaker placing one, a Maker needs two stones on most boards drawn at one, and three to make four in a row on 4 × 3. The extra stone is enough to erase noughts and crosses altogether, and the biased sum is exact there and high everywhere else.

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