Combinatorial Game Theory
who wins, and by how much
Generator

Which numbers the strings reach

Which numbers the strings reach
Which numbers the strings reach. Every blue-red Hackenbush string of up to four edges, placed at its value. Short strings give integers, longer ones fill in halves and quarters, and the pattern continues — the reachable values are exactly the dyadic rationals, and nothing else.

Every blue-red Hackenbush string of up to four edges, placed at its value. Short strings give integers, longer ones fill in halves and quarters, and the pattern continues — the reachable values are exactly the dyadic rationals, and nothing else.

2 essays call hackenbush-line. 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.

Where it is called

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

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. Particular games

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.

 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. Values

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.

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