Combinatorial Game Theory
who wins, and by how much
Series

Nogo — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Every empty NoGo board a build can solve. The empty boards, with the value the recursion returns and the outcome that follows from it. The one-row boards run 0, star, switch and repeat, which is a pattern with no reason behind it that survives past six squares.

    Every group must keep breathing

    NoGo is Go with no captures at all: a stone may be placed only if, afterwards, every group on the board still has a liberty. That makes a move's legality a fact about the whole board rather than about the squares it occupies — and a board therefore almost never breaks into independent parts. Of 117 boards here whose empty points fall into two regions, 24 are the sum of their regions and 93 are not.

    part 1 · positions
  2. 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.

    part 2 · positions
  3. 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.

    part 3 · positions
  4. 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.

    part 4 · positions

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