Combinatorial Game Theory
who wins, and by how much
Concept

Number tree — where it appears

The construction that produces every number born on a given day, branching into the halves between the ones before it. Only the dyadic rationals ever appear on it, and how far down a number sits is how many days it took to arrive.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

Hackenbush is a numeral

Draw a stalk of coloured edges. Read it as a string, blue for one and red for zero, and the string is the binary expansion of what the position is worth. Not approximately — exactly, and the site computes it both ways and refuses to build if they disagree.

positions · Hackenbush The numbers, by the day they are born. Zero on the first day, ±1 on the second, and thereafter the simplest number in each remaining gap. Every number reachable in finitely many days is a fraction with a power of two underneath, and every such fraction appears — which is a strange thing for a construction with no arithmetic in it to produce.

The day a number is born

Start with a position in which neither player can move, apply one rule, and the numbers appear — but only the fractions with a power of two underneath, and only in a particular order. That order is what "simplest" means.

values · Numbers

Named alongside it

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

Dyadic rationalHackenbushSimplicity ruleSurrealBinaryBirthdayBorn on dayCanonical formFinite birthdayInfinitesimalNimNimber

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