Combinatorial Game Theory
who wins, and by how much
Concept

Nogo — where it appears

Go with captures forbidden: a stone may be placed only if every group of both colours still breathes. Removing the captures removes the scoring and puts the game inside normal play, at the cost of a board that rarely splits into a sum.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Where running out of moves is permanent. Eleven rulesets, each walked position by position from three small boards, with every position at which a player has no move examined for whether any continuation gives them one back. Nothing here is evaluated: dead-ending is a property of the rules, and two boards worth the same value can differ on it. 9 of the 11 are dead-ending and 2 are not.

Nobody comes back

There is a class of games in which running out of moves is permanent, and it is the setting almost every modern misère result is stated in. Nine of this site's eleven rulesets belong to it across 5,334 positions; the two that do not are Toads and Frogs and Amazons, and Toads and Frogs loses the property to a single clause — delete the hop and it joins the list.

limits · Dead-ending When the regions add. Every board in the independence census — 117 positions whose empty points fall into two or more regions — tested against a stated criterion and against the guess it replaces. The criterion is that no stone group has liberties in two different regions, which makes a move in one region unable to change what is legal in another. It holds on 9 boards and the regions add on every one of them.

When the regions add

The rung below described the NoGo boards whose regions add as the ones with symmetric walls, and said the description was a guess made from six examples. It is wrong: fourteen symmetric boards do not add and sixteen that add are not symmetric. What replaces it is a criterion about liberties — sound on all 117 boards, provable in a line, and complete on only nine of the twenty-four.

positions · Nogo 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 A criterion that does not lift. The NoGo separation criterion run on strips and on three-row boards. On strips it holds on nine boards and explains all nine independent ones. On 227 three-row boards it holds on none at all.

A wall that bends

On a NoGo strip, two empty stretches add when no group breathes into both — a wall of two stones of different colours does it, and the criterion explains nine of ninety-three boards and all nine that it covers. On a three-row board it explains none of 227, and not because it is less accurate. A wall across a board has to bend, a stone at the bend sees empty squares on both sides by itself, and every one of the 227 has a group breathing into both regions. The condition is unsatisfiable.

positions · Nogo

Named alongside it

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

CounterexampleDecompositionGoIndependenceRegionCriterionDisjunctive sumEnumerationExhaustive searchGroupLibertyAdditivity

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