Combinatorial Game Theory
who wins, and by how much
Generator

A green edge is not a number

A green edge is not a number
A green edge is not a number. Green edges may be cut by either player, which makes the position impartial in that part. A single green edge is worth ∗ — a value that is neither positive, negative nor zero, and which no number can equal.

Green edges may be cut by either player, which makes the position impartial in that part. A single green edge is worth ∗ — a value that is neither positive, negative nor zero, and which no number can equal.

3 essays call green-edges. 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

17 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
E ∗ N How many ups · Squash every loop to a point · Hackenbush is a numeral · Three ways to add the same games · Outcomes do not add · The other sum, the one that nests · The sum is the object
EE ∗2 N Squash every loop to a point · Hackenbush is a numeral · The other sum, the one that nests
EEL {0, ∗, ∗2 | 0, ∗} N A green edge on a blue one
EL ↑∗ N A green edge on a blue one · How many ups · Squash every loop to a point · The other sum, the one that nests
ELL {0, ↑∗ | 0} N A green edge on a blue one · How many ups
ELR {0, ∗ | 0, ↑∗} N A green edge on a blue one
ER ↓∗ N A green edge on a blue one
L 1 L A green edge on a blue one · Squash every loop to a point · Hackenbush is a numeral · Nothing worth fighting over · The numbers came out of the game · The other sum, the one that nests · The reading that survives too much · The simplicity rule · The values nobody's game produces · What the colon respects
LE 1∗ L A green edge on a blue one · How many ups · Squash every loop to a point · Hackenbush is a numeral · The numbers came out of the game · The other sum, the one that nests
LEE 1∗2 L A green edge on a blue one
LEEE 1∗3 L A green edge on a blue one
LEL 1↑∗ L A green edge on a blue one
LERL {1, 1↓∗ | 1, 1∗} L A green edge on a blue one
LLE 2∗ L A green edge on a blue one · The numbers came out of the game
LRE 1/2∗ L A green edge on a blue one · The numbers came out of the game
RE −1∗ R A green edge on a blue one
RRE −2∗ R A green edge on a blue one

Where it is called

Changing this generator changes every one of these figures.

How many ups, bracketed. Every position here is all-small, so no number says anything about it and the yardstick has to be ↑ instead. Each bar spans the multiples of ↑ the position lies between: the largest it is at least, and the smallest it is at most. Four of the seven are pinned to a single multiple of ↑; the rest keep a band that comparison cannot narrow, the widest being ∗ at four ups of slack. Sums and comparison

How many ups

When every component of a position is smaller than every positive number, no number can decide it. What decides it is a count of ups — and comparison can pin that count down exactly, except when a star is present, when it cannot.

 a triangle on a stalk, worth ∗2. A Hackenbush position in which every edge is green, so either player may cut any of them and the position is impartial. Its value is a single Nim heap. Two principles find which one: fusion, which collapses every cycle to a point and leaves that many loops behind, and the colon principle, which replaces a branch by a stalk as long as the branch's own value. Particular games

Squash every loop to a point

Colour every Hackenbush edge green and the game becomes impartial, so the whole picture is worth a single Nim heap. Two principles find which one without playing anything — fuse the cycles, then run one pass up the tree — and a nine-vertex lattice that costs 1,283 positions to solve costs twelve steps to read.

 A green edge is not a number. Green edges may be cut by either player, which makes the position impartial in that part. A single green edge is worth ∗ — a value that is neither positive, negative nor zero, and which no number can equal. Particular games

A green edge on a blue one

Blue over green and green over blue are the same two edges in the other order. One is worth 1∗ and the other ↑∗ — a number with a star on it against something smaller than every positive number — so a stalk with all three colours in it stops being a numeral and starts being a position whose value depends on what is underneath.

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