Combinatorial Game Theory
who wins, and by how much
Concept

Surreal — where it appears

The number system the same construction reaches when nothing stops it, containing the reals, the infinite ordinals and their reciprocals. This site stops at the finite part of it, because the recursion has to halt for a value to be computed at all.

Named by 7 essays across 4 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 simplest number in between. A game whose options are numbers is worth the simplest number strictly between them — and simplest means born earliest, so integers come before halves and halves before quarters. It is not the midpoint, and the difference is the whole content of the rule.

The simplicity rule

When both players' options are numbers, the position is worth the simplest number strictly between them. Not the midpoint, not the average, and the difference between "simplest" and "middle" is the entire content of the rule.

values · Numbers 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 Poker Nim from 3, 5, 7, with reserves of 4 and 4. Nim with one extra kind of move: a player may put any number of counters back onto a heap from a private reserve. It looks as though a losing player could stall for ever. They cannot, and the winner is decided by exactly the same nim-sum as ordinary Nim — checked here over every position within a stated range rather than argued.

The condition the recursion rests on

Not that the moves run out, and not that the options are few. Poker Nim's heaps can grow without bound and it ends; the game called `on` has one option and never does. What every value on this site needs is that no infinite run of moves exists — and there are three separate ways to fail it.

limits · Termination The days this site can compute, and the ones it cannot. Zero on the first day, ±1 on the second, and thereafter the simplest number in every remaining gap — the construction run by the game recursion, which produces only fractions with a power of two underneath however long it goes on. Below it, three objects the same recursion reaches when the stopping rule is removed, each written with its option set and the exact reason this site's machinery cannot hold it. They are named rather than drawn, which is the honest half of a figure-first collection.

The numbers came out of the game

The construction is always taught numbers first and games second, and the discovery ran the other way. Conway arrived at the number system from positions, which is why the definition quantifies over sets of previously built objects rather than over cuts — and why it produces a genuinely different collection at every finite stage.

history · Numbers The days this site can compute, and the ones it cannot. Zero on the first day, ±1 on the second, and thereafter the simplest number in every remaining gap — the construction run by the game recursion, which produces only fractions with a power of two underneath however long it goes on. Below it, three objects the same recursion reaches when the stopping rule is removed, each written with its option set and the exact reason this site's machinery cannot hold it. They are named rather than drawn, which is the honest half of a figure-first collection.

The recursion this site cannot run

Remove the stopping condition from the construction and it reaches ω, its reciprocal, and one third — none of which this site's evaluator can represent, because it interns a position from a finite list of options. The figure draws what it computes and names what it cannot, which is where the boundary belongs.

values · Numbers Hydras, and how long each takes to kill. Six small hydras with the ordinal the termination proof assigns to each and the exact number of chops it takes to finish it. Two of them are not finished here: the fight is guaranteed to end and the machine runs out of memory long before it does, which is the gap between a termination proof and a bound.

It ends, and nothing says when

The recursion this site runs needs every line of play to reach a position with no moves, and the condition is usually met by an obvious decreasing quantity. The hydra meets it with no such quantity anywhere: the tree grows at nearly every step and the fight ends regardless, because the only thing that decreases is an ordinal. A four-node hydra dies in twenty chops; one level deeper and 279 chops reach forty thousand nodes with no end in sight.

limits · Termination

Named alongside it

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

BirthdayDyadic rationalSimplicity ruleHackenbushRecursionNumbersTerminationBinaryCanonical formConstructionEnding conditionInduction

All concepts