Combinatorial Game Theory
who wins, and by how much
Generator

Tiny and miny: infinitesimals with a scale

Tiny and miny: infinitesimals with a scale
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.

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.

2 essays call tiny-scale. 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

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

PositionWorth OutcomeDrawn in
miny-1 {{1 | 0} | 0} R A pawn ending is a sum · Tiny, miny, and the sizes below every size
miny-2 {{2 | 0} | 0} R A pawn ending is a sum · Tiny, miny, and the sizes below every size
miny-4 {{4 | 0} | 0} R A pawn ending is a sum · Tiny, miny, and the sizes below every size
tiny-1 {0 | {0 | −1}} L A pawn ending is a sum · Tiny, miny, and the sizes below every size
tiny-16 {0 | {0 | −16}} L Tiny, miny, and the sizes below every size
tiny-2 {0 | {0 | −2}} L A pawn ending is a sum · Tiny, miny, and the sizes below every size
tiny-3 {0 | {0 | −3}} L Tiny, miny, and the sizes below every size
tiny-4 {0 | {0 | −4}} L A pawn ending is a sum · Tiny, miny, and the sizes below every size
tiny-8 {0 | {0 | −8}} L Tiny, miny, and the sizes below every size

Where it is called

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

A blocked file, and the tempo it holds. Files of a blocked pawn ending: a White pawn below, a Black pawn above, and a gap between them that either side may close one square at a time. Each file carries the value the game recursion gives it. With only single steps available a file is worth a star or nothing, by the parity of the gap, so the whole position is tempo and no material at all — which is what a chess player means by mutual zugzwang. Out in the world

A pawn ending is a sum

In a blocked pawn ending the material is level, the files never speak to each other, and whoever has to move is the one in trouble. Chess calls that mutual zugzwang; this site calls it a P-position; and the two vocabularies were built four decades and one subject apart to say the same thing.

 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.

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