Combinatorial Game Theory
who wins, and by how much
Generator

Domineering on 2 by 3

Domineering on 2 by 3
Domineering on 2 by 3. Left places vertical dominoes, Right horizontal ones, and a player who cannot place loses. The two players see different games on the same board, which is what partizan means — and the value that results is not a number.

Left places vertical dominoes, Right horizontal ones, and a player who cannot place loses. The two players see different games on the same board, which is what partizan means — and the value that results is not a number.

4 essays call domineering. 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.

The positions it draws

6 distinct positions, harvested by drawing the figure again at the options each essay asked for.

PositionWorth OutcomeDrawn in
1×3 −1 R Domineering · The values of every small board
2×2 1 | −1 N A board that is a sum of its regions · One board, two rules · Cooling by exactly one · Domineering · Worth nothing, and worth fighting for · The values of every small board · Tiny, miny, and the sizes below every size · What can be struck out · What counts as the same position, and what that is worth · Which option the reduction keeps · Which shapes are worth fighting over
2×3 2 | −1/2 N A board that is a sum of its regions · "Left wins" has no short proof · One board, two rules · Cooling by exactly one · Domineering · Turn the board through a right angle · Worth nothing, and worth fighting for · The values of every small board · Tiny, miny, and the sizes below every size · What can be struck out · What counts as the same position, and what that is worth · Where the impartial theory stops · Which option the reduction keeps · Which shapes are worth fighting over
2×4 {{2 | 0} | 0} R Cooling by exactly one · Turn the board through a right angle · The values of every small board · Tiny, miny, and the sizes below every size · Which shapes are worth fighting over
3×3 1 | −1 N A board that is a sum of its regions · "Left wins" has no short proof · One board, two rules · Cooling by exactly one · Cram · Domineering · Worth nothing, and worth fighting for · The values of every small board · Tiny, miny, and the sizes below every size · What counts as the same position, and what that is worth · Which option the reduction keeps · Which shapes are worth fighting over
3×4 −3/2 R A board that is a sum of its regions · "Left wins" has no short proof · Domineering · A position reached eleven ways is one position · The values of every small board · Which shapes are worth fighting over

Where it is called

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

Domineering on 2 by 3. Left places vertical dominoes, Right horizontal ones, and a player who cannot place loses. The two players see different games on the same board, which is what partizan means — and the value that results is not a number. Particular games

Domineering

One player places dominoes vertically, the other horizontally, on a shared grid. The rules take one line, the values are a mess, and that mess is the point — this is what the theory looks like applied to a game nobody designed for it.

 Tiny and miny: infinitesimals with a scale. Positions that are greater than zero and smaller than every positive number, and which are nevertheless strictly ordered among themselves — the larger the subscript, the smaller the value. Being smaller than everything positive is not one size of thing; it is a whole scale, and up sits above all of it. Values

Tiny, miny, and the sizes below every size

An empty two-by-four Domineering board is worth less than nothing and more than every negative number. It is not up, not down and not a fraction — it is a miny, and the minies come in sizes, strictly ordered among themselves below a floor no number reaches.

 Cram on 4 by 4: the pairing strategy. Cram is Domineering with the orientations shared: either player may place a domino either way up, so both players have exactly the same moves and the game is impartial. Every position therefore has a Grundy value, and this board's was computed by the mex rule over its own placements. Impartial games

Cram

Domineering with one word of the rule changed: both players may place a domino either way up. That makes the game impartial, and the entire partizan apparatus collapses into a single Grundy value — on the 4 × 4 board, Domineering's canonical form runs to 114 characters of nested braces and Cram's answer is the one character 0.

 What a Domineering region is worth, by size. Every connected shape of at most six free squares, sorted by whether its value is a number, an infinitesimal distance from a number, or hot. Hot shapes do not appear at all until four squares, and the hottest shape of six is the two-by-three rectangle. Particular games

Which shapes are worth fighting over

Forty-four of the 104 Domineering regions of at most six squares are worth numbers and the rest are not, and the rung below said no visible property of a shape predicts which. Half of that is wrong: a region only one orientation fits in is a whole number, on all eleven of them, for a reason a reader can supply in a sentence. The other half stands, and thirty-three shapes are what makes it stand.

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