Combinatorial Game Theory
who wins, and by how much
Theme

The thread: The parts do not decide the whole — page 3

Outcomes do not add. Neither do temperatures, atomic weights, misère outcomes or the value of an auction. Which quantities survive being added is the question every method here turns on.
Why a square holds fewer values than a line. The same number of squares laid out two ways, with the number of places a hop can start, the longest chain one can run, and the number of distinct values every arrangement of that shape produces. A hop needs three squares in a line, so a long row supplies more of them than a compact rectangle of the same area — and the value counts follow. The second dimension is not the way to reach the deeper values, which is the opposite of what the rung below expected. Out in the world

The second dimension is not the deep end

The rung below says a row of eight reaches every corner of the vocabulary and goes far into none of them, and that the narrowness is a fact about the board. So the obvious next move is a rectangle — and nine squares in a square hold twenty-five values where nine squares in a line hold fifty-eight. The geometry says why before any stone is placed.

 A row of files, valued rather than won. Dawson's pawn diagram on a single row of one to 5 files, with the value of the position under each capture rule beside the nimber ·137 gives the corresponding heap. The winners agree throughout; the values agree until five files, where the diagram is worth ∗ and the heap is ∗3. How it was found

A wall the pawns cannot cross and the rule can

Two rows of Dawson's diagram separated by a file with no pawn on it: 1,616 moves were examined and not one crosses the gap. With captures optional the rows add on every diagram checked. With captures compulsory they do not, because the compulsion is a rule about the whole board — and the game that is a sum is the one ·137 does not describe.

 One more row, and the correction comes back. Dawson's diagram of three files beside a row of one, then two, then three, each drawn with the difference between the whole board's value and the sum of its rows. The correction is ∗2, then 0, then ∗2: adding a row removes it and adding another restores it. How it was found

A difference the rows cannot predict

The diagrams that are not the sum of their rows have been counted and never priced. Priced over 50 diagrams and 63,408,981 positions, the difference takes three values and is a function of nothing a reader can see: seven diagrams whose rows are worth ∗ and ∗ split five to two on it, the third value arrives only at the ninth file, and the one rule that survives is a parity — all twenty-one diagrams of three, five and seven rows add, and every failure carries an even number of rows.

 The law, on a board rather than in a bag. The parity law applied to every position of a real board that has fallen into chains and loops, with the verdict computed independently from the board's own strings. The components are the ones the geometry produces rather than the ones a sweep constructs. Out in the world

A thousand positions and no exception

The parity law was fitted to constructed bags of chains and loops inside a string budget. A board's positions are a different population — the sizes are what the geometry allows, the components come correlated, and a six-box board holds exactly one position that is a loop of six. Tested on all 1,032 of them and all 160 of the four-box board's, the law is right every time, against a verdict computed from the strings by a walk that has never heard of a component.

 What control is worth, to a box. The margin the player who does not have to open nets, beside the formula that predicts it: the total less four boxes for every long chain after the first. Computed against the solver on every endgame of long chains in range. Out in the world

Four boxes for every chain after the first

Nimstring answers who is forced to open and says nothing about the score. The margin has a formula: the controller nets the total less four boxes for every long chain after the first — two surrendered and two not taken, each time control is kept. Checked against the solver on 175 endgames it is exact on 172, never too generous, and exact wherever it promises the controller anything at all. The three it misses are the three where it promises nothing.

 What a held pass can tell apart. Nim heaps, Kayles rows and heaps of Dawson's chess of sizes one to 8, grouped by whether any company of up to two of them gives a different outcome with a held pass on the board. The groups outnumber both the Grundy values and the pairs of Grundy value and held-pass value. Where it stops

What a component would have to carry

For a held pass to be decided by a summary of each component, the summary must separate every pair of components some company tells apart. The Grundy value does not — Nim 1 and Kayles 8 are equal games that a held pass separates beside a single Nim heap of two. Nor does the Grundy value with the component's own held-pass value: Kayles 3 and Kayles 6 agree on both and are split by a company of two Nim heaps. Over twenty-four components, fifteen classes against fourteen pairs, and the gap widens as the pool grows.

All themes