Combinatorial Game Theory
who wins, and by how much
Concept

Correlation — where it appears

A number between −1 and 1 saying how closely two measurements rise and fall together, used here only to refuse a proposed explanation rather than to offer one. A correlation with the wrong sign is a refutation; one with the right sign is an invitation to hold the third variable fixed and look again.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The birthday is a floor. How many more pieces a value's cheapest exhibit needs than the value has days. It is never fewer, on any of the 728 non-number values, and it is exactly none on 476 of them.

The birthday is a floor

The rung below measured a correlation of 0.73 between a value's birthday and the size of its cheapest exhibit, and asked which values are dearer than the birthday suggests. The relation is not a trend. Over all 728 non-number values the exhibit is never smaller than the birthday and is exactly the birthday on 476 of them, and the excess on the other 252 belongs to the game rather than to the value — the ruleset accounts for 40 per cent of its variance.

values · Realisability The trend, running backwards. The excess grouped by the width of the value's canonical form. Values with wide option lists are exhibited nearer their birthdays than narrow ones, which is the opposite of what the proposal predicted.

Wider costs less

The rung below found the cheapest exhibit of a value never smaller than its birthday, exactly equal on two thirds, and the ruleset explaining 40 per cent of the rest. The variable it proposed for the remainder was the width of the form. Width and excess correlate at −0.39: the wider the value, the closer to its birthday it is exhibited, and inside a ruleset the relation cannot even agree on a sign.

values · Realisability 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

Named alongside it

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

EnumerationValueBirthdayCanonical formCounterexampleEnd-NimHackenbushRealisabilityAmazonsApproximationBoardBorn on day

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