Combinatorial Game Theory
who wins, and by how much
Generator

The two rules everybody is taught, measured

The two rules everybody is taught, measured
The two rules everybody is taught, measured. Both pieces of Dots and Boxes folklore against exhaustive solves. Above: what taking every available box costs, on boards solved from empty — where it costs nothing at all — and over every position of a larger board and over endgames, where it changes the outcome of most of them. Below: how often the long-chain rule is right, split by whether the position is made of long chains, some of them, or none.

Both pieces of Dots and Boxes folklore against exhaustive solves. Above: what taking every available box costs, on boards solved from empty — where it costs nothing at all — and over every position of a larger board and over endgames, where it changes the outcome of most of them. Below: how often the long-chain rule is right, split by whether the position is made of long chains, some of them, or none.

2 essays call boxes-rules. 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

Changing this generator changes every one of these figures.

A 2 × 3 board of boxes, 6 still on the table. A Dots and Boxes position drawn as dots and lines, and — where the figure asks for it — the same position as a strings-and-coins graph: one coin per box, one string per line, and the border lines running to the ground. Lines already played are solid, lines still available are dashed, and a box with no strings left has been pocketed. The footer carries the exact net score the solver computes from here and the normal-play verdict on the same position. Out in the world

The game in every exercise book

Dots and Boxes is played by more people than every game in this collection put together, and everybody is taught the same rule — take every box available. The rule is wrong. Establishing that takes a solver rather than an opinion, and the solver says how wrong, on which boards, and by how many boxes.

 The impartial game inside the scoring one. For every position of a Dots and Boxes board, two questions asked separately: who wins the scoring game, and who wins Nimstring — the same position under the normal-play convention, with no score kept. The bars show how often the two answers agree, grouped by how many boxes are still on the table. Agreement is near-total when there is enough left to be worth controlling and falls away when there is not. Out in the world

The chains decide it before the boxes do

Under every game of Dots and Boxes there is an impartial game with no score in it, and it settles the question the scoring game keeps asking — who ends up having to open. The rule players learn as folklore falls out of it, and so do the exceptions nobody mentions.

The whole library · The position index · The figures that play back