Combinatorial Game Theory
who wins, and by how much
Concept

Board — where it appears

The arrangement of pieces a position actually is, as against the value the theory replaces it by. Two boards with one value are interchangeable in every sum and are not interchangeable for a question about what moves exist.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

Every Domineering shape up to four squares. The pieces a partly played board falls into, each with the value the recursion gives it. Left plays vertically and Right horizontally, so a tall shape is worth something positive and a wide one something negative, and the quarter turn is not a symmetry of the game.

A board that is a sum of its regions

A table of rectangles is a table about the openings. A partly played Domineering board is not a rectangle, and evaluating one means splitting it into pieces no domino can straddle, looking each piece up and adding. The catalogue of 104 shapes does it correctly on every one of the 3,227 positions of a 3×4 board it covers — and among the shapes are two worth an up and a down, which no rectangle ever is.

positions · Domineering 7 symmetries, and the one that is a strategy. 4 games and 7 candidate symmetries, each tested by playing the strategy out against every opponent line rather than by argument. A pairing strategy needs a map that fixes the start, is an involution, and carries one player's moves to the other's — and the last condition is where most of these fail.

The strategy that is a symmetry

A pairing strategy is a symmetry of the board that turns one player's moves into the other's, and it wins without computing anything. Tested by playing it out rather than argued, it wins one of seven candidate symmetries across four games — exactly the Cram boards with both sides even, which is exactly where no domino is its own image.

impartial · Pairing When counting the free squares gets Push right. Every Push strip of at most seven squares, split by whether any line of play can bring two coins of opposite colour together. Where none can, the count of free squares in front of each coin is the value, without exception; where one can, the count is right more often than not.

The reading that survives too much

Counting the empty squares in front of each coin gets a Push position right half the time, and the rung below said the failures were exactly the positions with two coins of opposite colour side by side. Sixty-six of the 1,072 failures have no such pair, the smallest is five squares long, and the condition that does decide it is not about the board at all — it is about every position the board can reach.

positions · Push A thousand shapes, and twelve pairings. Cram on every connected shape of at most eight squares, with the search for a symmetry that answers each of the opponent’s moves. Every pairing found is a second-player win, most shapes have no involution at all, and the strategy accounts for a sixth of the second-player wins there are.

Looking for the symmetry

Answering every move with its mirror image wins Cram on a board with both sides even, which is the argument everybody meets. Asked of every connected shape of at most eight squares instead of of thirteen rectangles, it wins twelve — and accounts for a sixth of the second-player wins there are, because 852 of the 1,042 shapes have no symmetry to answer with in the first place.

impartial · Pairing Thickness is not the variable; the wall's own groups are. The same strips split by whether the wall is all one colour. A wall of one colour is a single group with liberties on both sides and it never separates them, at any thickness. A wall of two colours is two groups breathing in opposite directions and it nearly always does.

How thick a wall has to be

A single stone between two empty stretches of a NoGo board couples them, and the obvious repair is a thicker wall. Over 590 walled strips a thicker wall does help — and splitting the same 590 by the colour of the stones shows that thickness was never the variable. A wall of four one colour couples the sides exactly as one stone does.

positions · Nogo Where in a game a board falls apart. Every position reachable from an empty Domineering board, grouped by how many dominoes have been placed, with the share that have fallen into two or more live pieces. The share is nought at both ends of the game and around three fifths in the middle.

How often a board falls apart

A decomposition turns a product into a sum, so a solver wants to know how often one arrives. Over every position of a 4 × 4 Domineering board the answer is 47 per cent — nought for the first two moves, three fifths in the middle, and nought again at the end. What one decomposition is worth is the other half of the answer and it is a factor of 1.8.

complexity · Decomposition The same distance, different room. Two shared Amazons regions with the amazons the same distance apart, differing in how many squares both of them can still reach. The one sharing more is the colder.

Room pulls two ways

The rung below found the distance between two amazons setting a shared region's temperature and asked for something finer — the squares each can reach, or the squares both can. Neither beats the distance on its own. Together they beat it by half as much again, and the reason is that they pull opposite ways: further apart is hotter, and sharing more reachable squares is colder.

positions · Amazons The five hottest regions. Every eight-square Domineering region at the ceiling temperature, with which of the boards swept ever produces it.

Eight squares, and no hotter

The rung below found no Domineering position hotter than three halves on four boards and asked for the position that attains it. It is a region of eight squares, there are five of them up to symmetry, three are the hot core of an attaining board on every size swept — and the ceiling holds at nine and ten squares too, where the obvious extrapolation predicted seven quarters.

temperature · Cold Two symmetries, the same two clauses. The half-turn pairing and the reflection pairing written side by side, with the fixed squares and self-paired dominoes each has to exclude.

A pairing, and the pairing

The rung below repaired the half-turn check and asked whether a reflection would fire where it does not. It does — forty positions of 58,830 on the largest board — and it is sound, and it is worth one node in a thousand to a solver. It can never fire on an empty rectangle at all, which is why the ladder's whole subject is the half turn.

impartial · Pairing One board, two answers to how many pieces it is in. Every position reachable from a small Amazons opening, counted by depth, under two ways of deciding whether two squares are in the same region. Counting only edge neighbours, a third of all positions are in pieces; counting corners too, an eighth are.

A wall an amazon can walk through

An arrow burns a square for good, so an Amazons board that has fallen into pieces should stay in pieces. Over 127,583 positions it does not: fifty-one thousand moves put two regions back together. Every one of them is a single diagonal step, and what is wrong is not the game but the rule used to find the regions — which was borrowed from a game whose pieces lie along the board's own lines.

complexity · Decomposition Why a square holds fewer values than a line. The same number of squares laid out two ways, with the number of places a hop can start, the longest chain one can run, and the number of distinct values every arrangement of that shape produces. A hop needs three squares in a line, so a long row supplies more of them than a compact rectangle of the same area — and the value counts follow. The second dimension is not the way to reach the deeper values, which is the opposite of what the rung below expected.

The second dimension is not the deep end

The rung below says a row of eight reaches every corner of the vocabulary and goes far into none of them, and that the narrowness is a fact about the board. So the obvious next move is a rectangle — and nine squares in a square hold twenty-five values where nine squares in a line hold fifty-eight. The geometry says why before any stone is placed.

applied · Kōnane The only two moves in Kōnane that are not captures. A filled Kōnane board with every opening Black may play marked on it, and the value of the position White's best reply leaves. The opening is the one place in the game where a player removes a stone rather than capturing with one, so it is played under a different rule from everything after it — and the choice is worth a computed amount rather than nothing.

The two moves that are not captures

Kōnane begins from a full board and the first two moves lift stones rather than take them, which is the only time in the whole game anybody does. Nothing on this site applies to them, and the choice is not free — on a 3 × 5 board two of Black's eight openings leave a position a whole move worse than the other six, and on a 3 × 3 board none of the five does.

applied · Kōnane How much of a board the endgame theory reaches. Every subset of a board's strings, counted by whether the surviving coins fall into chains and loops. The share is taken over the positions with no free box on the table, since a position with a capture available is one a player takes rather than chooses from.

The endgame theory arrives late

Every component the chain-and-loop theory names has coins of degree two, so a position it can read is one where every surviving coin holds exactly two strings. Over a six-box board that is 1,033 of the 28,028 positions with no free box on the table — 3.7 per cent — and more than half of them only after twelve of the board's seventeen strings have been cut.

applied · Dots and Boxes The law, on a board rather than in a bag. The parity law applied to every position of a real board that has fallen into chains and loops, with the verdict computed independently from the board's own strings. The components are the ones the geometry produces rather than the ones a sweep constructs.

A thousand positions and no exception

The parity law was fitted to constructed bags of chains and loops inside a string budget. A board's positions are a different population — the sizes are what the geometry allows, the components come correlated, and a six-box board holds exactly one position that is a loop of six. Tested on all 1,032 of them and all 160 of the four-box board's, the law is right every time, against a verdict computed from the strings by a walk that has never heard of a component.

applied · Dots and Boxes

Named alongside it

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

Exhaustive searchDecompositionEnumerationRegionCounterexampleDomineeringImpartialIndependenceSymmetryCanonical formDisjunctive sumInvariant

All concepts