Combinatorial Game Theory
who wins, and by how much
Concept

Coupon — where it appears

A move worth a fixed amount, drawn from a stack that cools by a fixed step, standing in for the rest of a board. An environment of them turns a question about a position in company into a question about which coupon a player leaves the stack at.

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

When the players stop taking coupons. Every pair of fights from a pool of nine, played beside a coupon stack, with the coupon standing when somebody first plays on the board. Sixty of the eighty-one leave exactly when the coupon falls to the board's temperature.

When to leave the environment

A Go player's question is not which fight to take but when to stop taking the small stuff. Put two fights beside a stack of coupons and the orthodox answer — leave when the coupon falls to the hottest temperature on the board — is exact on sixty of eighty-one pairs. All twenty-one departures have a fight with a follow-up in them, and every pair of plain switches leaves on time.

temperature · Coupons Thirteen cells, thirteen scores. The rule scored in every cell of the unit interval on the designed pool, at three components.

A pool built to have an answer

The coefficient in the rule score a component by t − λa scored identically for every λ in the unit interval, because the rule reads an ordering and that pool's orderings changed at three places. A pool designed to have twelve crossings turns the interval into thirteen different rules, and all three board sizes agree on one cell: between a quarter and a third.

temperature · Coupons Three orders, one of them right. The holder's rank above the crossover under three different orderings of the options.

Which top is the top

The crossover law's proof rests on the walls above the crossover being governed by the top two options, and the check was never run. Run on 23,586 heights it holds exactly — but only when the options are ranked by mean value. Ranked by the temperatures the law is stated in, it fails on a fifth of them.

temperature · Sente

Named alongside it

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

CounterexampleEnumerationMean valueSenteTemperatureApproximationStrategyAmbientDay threeDisjunctive sumEnvironmentFollow-up

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