Combinatorial Game Theory
who wins, and by how much
Concept

Green hackenbush — where it appears

Hackenbush with every edge cuttable by either player, so the game is impartial and the whole picture is worth one nimber. Fusion and the colon principle collapse the whole drawing to one heap without any of it being played out.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

What it costs to decide the winner. Three families of game, each at several sizes, with the number of distinct positions an exhaustive solver must evaluate beside the work its closed form does. The bars are logarithmic. Hardness is not about the size of the board or the length of the rules — Domineering has the shortest rule here — it is about whether anybody has found the shortcut.

The game with the shortest rule is the hard one

Deciding a generalised board game is PSPACE-complete, which is a statement about families and encodings rather than about size. Nim in the same subject is settled by one pass over the input at any size, and green Hackenbush by one pass over the edges — while Domineering, whose rules take a single line, has no shortcut anybody has found.

complexity · Complexity 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.

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.

positions · Hackenbush 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.

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.

positions · Hackenbush The picture is the numeral. Blue-red Hackenbush strings and their values. Left may cut a blue edge, Right a red one, and everything above the cut falls. The value of each string is a number, and reading the string from the ground upward gives the binary expansion of exactly that number.

The other sum, the one that nests

A move in one part wipes the other out entirely. That is the ordinal sum, it is what a Hackenbush stalk actually is — 1 : (−1) is a half, and 1 : (−1) : 1 is three quarters — and it is not an operation on values at all: three positions all worth zero give three different answers under it.

sums · Ordinal sum Swapping a branch for another of the same value. The ordinal sum of a base with a branch, and the same sum with the branch replaced by a heap of a different game carrying the same Grundy value. The two are compared by playing their difference, not by inspection — and they agree every time, which is what the colon principle claims and what the partizan case denies.

When the nested sum only sees the value

The ordinal sum reads the form and not the value: three positions all worth zero, placed under a star, give three different answers. On impartial games it reads the value after all — 72 substitutions of an equal-valued heap from a different game, and every ordinal sum comes back unchanged. That difference is the whole reason a green Hackenbush tree can be collapsed one branch at a time.

sums · Ordinal sum Trees, and what each is worth. A row of blue-red Hackenbush trees with the value the recursion returns under each. Every one is a number, and none of them is the binary reading of anything a reader can see in the picture.

A tree is still a number

A Hackenbush string spells its own value in binary. Put a fork in it and the numeral has nothing to read — there is no leftmost anything. The value is still a number, in all 10,066 forests up to six edges; it is still computable, by the ordinal sum, in all 3,238 single-trunk trees; and the reading is right on 762 of them, of which 126 are the strings it was written for.

positions · Hackenbush What the third colour reaches. Every row of Toppling Dominoes up to 7 long, over two colours and over three, with the number of distinct values each set of rows carries. Each value was computed by the recursion; the last column is the count of values three colours reach that two do not, cumulatively.

How long a row a value needs

Add a third colour that either player may topple and a row of seven dominoes reaches 1,047 distinct values where two colours reach 149. That makes the length of the shortest row worth a value into a measure of the value's complexity — one a reader can hold in their hand — and it is not the birthday: 1↑ is born on day three and needs seven dominoes.

positions · Toppling dominoes Two orders of magnitude. The share of positions of a size whose value is one no smaller position exhibits. Every Hackenbush string is a new value and fewer than one Clobber row in a hundred is.

The rate was the alphabet

The rung below asked for a quantity a size cap cannot censor and proposed the rate: how many new values a ruleset produces per extra square. The rate is honest and it measures the notation — every ruleset grows at close to the number of symbols its positions are written in, and the seven span less than a factor of two. What separates them is the yield, which spans a hundred and nineteen.

values · Realisability

Named alongside it

The objects these essays reach for when they reach for this one.

HackenbushColon principleExhaustive searchDisjunctive sumOrdinal sumBinaryFusionGrundy valueImpartialNimNimberStar (∗)

All concepts