Combinatorial Game Theory
who wins, and by how much
Concept

Saltus — where it appears

The constant a Grundy sequence gains each time it repeats, which is what makes an arithmetic period arithmetic rather than plain. A saltus of nought is ordinary periodicity; anything else means the values never repeat and the sequence is nevertheless solved.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Climbing is the ordinary case. The two-digit and three-digit hexadecimal families, each swept for exact and arithmetic periodicity. Seven in ten of the settled three-digit codes repeat with a constant added.

The third digit

The rung below found 71 of the 255 two-digit hexadecimal codes repeating with a constant added rather than exactly, and asked whether the same share holds one digit wider. It rises. Of the 4,095 three-digit codes, 1,433 climb and 617 repeat exactly — seven in ten of the settled ones — so a saltus is the ordinary way a hexadecimal game settles and the exact repetition the octal survey was built to find is the special case.

impartial · Hexadecimal The digits they share. The condition satisfied by eighteen of the nineteen codes that climb by three. It says that splitting a heap into three is available on exactly one take and buys nothing else.

The only way to split into three

Nineteen three-digit hexadecimal codes climb by three, and the rung below asked whether they share a form and what digits they have in common. The digits are exact: on eighteen of them the only way to split a heap into three is by taking exactly three counters, and taking three counters can do nothing else. The form is not shared — the eighteen carry four distinct sequences, and exactly one of the four counts in base three.

impartial · Hexadecimal Two counters, not one. The four periods of the odd-saltus class against the two base-three counters, with which each follows.

Two counters, and one displaced term

The rung below found four Grundy sequences in the odd-saltus class and asked which term each displaces and whether the digits predict it. They do — but there are two base-three counters and not one, chosen by whether a heap of one can be taken away. And there are three sequences rather than four: the fourth is the third with three isolated values, and was counted separately because its period had not settled.

impartial · Hexadecimal Not rare at all. How many hexadecimal codes whose sequence settles have a stretch of heaps before the pattern begins.

A pattern that has not started yet

A pre-period was supposed to be rarer in this family than a defect. Two hexadecimal codes in five have one, 321 have a pre-period longer than their own period, and the code the rung below found slow takes fifty-four heaps to settle rather than two blocks — which is also the account of three defects the rung below recorded and could not explain.

impartial · Hexadecimal One of the two quantities is inert. The candidate predictors of a pre-period's length, each scored by correlation against the measured length.

The quantity that carried nothing

The rung below proposed predicting a pre-period's length from the saltus and the period. The saltus correlates with it at −0.03, which is nothing; the period correlates at 0.77 with a coefficient of one, so a pre-period is about one period long. The digit that predicts whether there is one predicts nothing at all about how long.

impartial · Hexadecimal

Named alongside it

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

EnumerationImpartialOctal gamePeriodicityGrundy valueHexadecimalCounterexampleGrundy sequencesInvariantCodesExhaustive searchApproximation

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