The thread: The parts do not decide the whole — page 3
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.
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.
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.
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.
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.
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.
Where it stops
One value more than Nim
Top Entails broke the nim-sum: nine of thirty-six two-heap positions came out wrong when a move could compel the reply. The repair is one extra value. Read each heap off the heap itself, and every heap of Top Entails and of a second entailing game is either a nimber or loony — a win for the mover in any company. One rule then decides every sum of those games and Nim together: 4,016 positions, and it is never wrong. A loony heap is a pass, and the compulsion is what buys it.
Sums and comparison
What a part would have to report
Outcome classes leave four entries of the addition table open. Give each part its two stops as well and four inequalities settle 176 of the 369 open day-two sums, never wrongly — and every sum they leave has a bound sitting exactly on nought. Used as a lookup table instead, even the whole thermograph with the outcome beside it leaves 96 sums undecided. Four outcomes do what no thermograph can: who wins a day-two position alone, beside −1, beside 1 and beside a star names it, and no three questions do.
Where it stops
The table closes, until heaps of four
Classes that no company can tell apart always combine, so the question left by the held pass is not whether it has an algebra but whether the algebra is finite. Computed the way a misère quotient is, it is: four classes for heaps of one where misère play needs two, eleven for heaps up to two, fourteen up to three, nineteen for Kayles rows up to five. Admit Nim heaps of four and the table never closes — a single line of losses, 2k + 3 heaps of two beside 2k heaps of four, gives every count of twos a class of its own. And at the bottom of every closed table sits the normal-play group, with the loss moved to nim-sum one: on a crowded board the held pass is a heap of one.
Sums and comparison
The pairs stand still and the order spreads
The share of pairs of game values the order can compare falls from 77.5 per cent on day two to 60 on day three and then stops: built twenty times under two recipes and calibrated, day four sits within two standard errors of day three. That looked like a floor under the order. Measured beyond pairs it is not. Chains of three keep falling, triples with no comparable pair keep rising, and the largest set of mutually confused values in a sample of 120 grows from about 20 to about 30 — every one of forty seeds wider than day three's mean. The floor belongs to one statistic, not to the order it summarises.
Temperature
Why cooling has to add
Cooling came through a sweep of 1,768 sums untouched while heating failed 263 times, and the sweep's own account was cautious: cooling is not additive in general, and a richer pool would break it. Aimed exactly where that account said to look — infinitesimals hidden under low temperatures, at taxes of a quarter and a half — 107,816 more pairs break nothing, because there is nothing to break. Cooling distributes over every sum of short games. Its freezing clause fires on one side of the equation and not the other in 6,032 of those pairs, and costs nothing every time, because what it hands back is the mean, and means add.
Sums and comparison
A foot is a menu, not a number
When a sum's fight ends level on nought, the winner is decided by what is left at the foot of each part's fight, and the obvious number to report from there is the atomic weight. Over 32,428 sums of a day-two and a day-three value it leaves 14,924 undecided, and on day two it does worse than a single outcome letter. What sits at a foot is a choice, made by one player, with someone to move when it is made. Report all three and every sum of two cold parts is decided; 3,164 sums are still not, and each holds a hot part whose own fight is not the one played in company.
How it was found
The rows that cancel
Every one of Dawson's diagrams that failed to be the sum of its rows had an even number of rows, and twenty-one with an odd number all added. One file past the edge of that sweep, four files beside three beside one fails: three rows, worth ∗ against ∗3 by its rows. What the parity was tracking is two deletions a sum allows — a row worth nought, and two identical rows — which the compulsory board turns out to allow as well. Each of 44 diagrams is worth exactly what is left after them, and in the fifty-diagram sweep that remainder happened to have as many rows, odd or even, as the diagram it came from.
Where it stops
A compulsion that points elsewhere
Top Entails and entailing Nim are decided by one value more than Nim because the compelled reply stays in the heap that compelled it. Point it at a different heap instead and the two games part company. Entailing Nim does not notice — its heaps are worth exactly what Nim's are, because Nim's winning reply was never in the heap just moved. Top Entails loses its loony heaps and then every value of the Nim kind: no assignment of nimbers explains its sums, because a heap of three beside two heaps of one is a win and beside two heaps of two is a loss.
How it was found
What a stable set knows about a draw
On a graph with cycles, outcome labelling calls a position lost, won or drawn, and stops. The stable sets go further in exactly one place. On every game of up to five positions they hold every lost position, no won one, and among the drawn positions are exactly the stable sets of the drawn part taken alone. So a stable set is a certificate that the player who keeps returning to it never gets stuck. It can distinguish draws that labelling cannot, and two parts with no stable set can have one between them.