Combinatorial Game Theory
who wins, and by how much
Concept

Up-bracket — where it appears

A pair of integers between which a position's atomic weight must lie, found by comparing it against multiples of up. It is what comparison alone can establish, and a star in the value is what turns the pair into a band rather than a point.

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

What an approximation is worth. Every pair of Clobber rows up to six squares, judged twice: by their up-brackets and by the comparison itself. The bracket is never wrong where it speaks, and most of what it declines to answer has no answer.

When the bracket decides

A Clobber row's value is an all-small game nobody can hold in their head, so the practical answer is the up-bracket: a pair of integers between which its atomic weight must lie. As an approximation it is worth exactly what it settles — 1,585 of 7,875 pairs of rows are ordered by it, every one of those orders is right, and of the 6,290 it declines, 4,222 have no answer either.

positions · Clobber Which end of the interval is open. Every value born by day 3 compared with each of its own two stops — 2,948 comparisons, each one a search over the difference. The two rows are mirror images because the day is closed under negation, and the small number in each row is the exception class: the 352 values whose two stops coincide, for which the left stop is the right stop and the law has nothing to bite on.

Which end of the interval is open

The confusion interval is open at both ends, and the two ends are not the same kind of open. At its own left stop a position can be below the number, confused with it or equal to it, and — over 2,948 comparisons — above it exactly thirty-three times, every one of them a value whose two stops are the same number and whose left end is therefore also its right one.

values · Stops

Named alongside it

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

All-smallAtomic weightComparisonConfusedExhaustive searchInfinitesimalStar (∗)ApproximationBoundCanonical formClobberConfusion interval

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