The thread: It depends on the company — page 2
Where it stops
A function with no formula
The rung below's composition rule is exact on tame pairs and wrong on all fourteen wild ones, which looked like an exact boundary. Two heaps further it is wrong on 34 of 35 and right on one — Kayles' five and nine — so the boundary was a boundary of the pool. What survives is stronger and stranger: the pair of symbols still determines the sum on the wild side, and no rule of that shape describes it.
Where it stops
A product against a sum
A company closed under both addition and options licenses a solver to rewrite any subposition, and the rung below found that exactly the nimber groups have both closures. Priced on Cram boards, that licence is the difference between walking a product of position sets and walking their sum — four to twenty-five times on two components, twenty-four to a hundred and sixty-one on three — and it is available to impartial games because their class representative is a heap rather than a form.
Where it stops
Half a licence is nearly all of it
The rung below priced the substitution licence a restricted universe gives a solver and asked what half of one is worth — the licence to rewrite components but not subpositions. It is worth nearly the whole saving. Rewriting components collapses a million and a half states to three thousand six hundred; rewriting subpositions collapses those to eight hundred and eighty-four, and splitting the pieces takes it to fourteen.
Particular games
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.
Temperature
The worst value in its own interval
The rung below scored a component by its temperature less its hottest answer's and asked what rate the answer should really be charged at. Every weight strictly between nought and one scores the same and beats the rung below's choice of one at every board size — because a ranking rule's score is a step function of its own coefficient, and one is exactly where two components tie.
Where it stops
One half multiplies, the other adds
The rung below priced the two halves of a substitution licence on sums of two Cram boards and predicted that the first half's saving would grow with the number of components while the second's would not. It is right, and both halves have closed forms: the component licence saves s^(k−1)/k and the subposition licence k·s over a shape count that never moves.
Temperature
The premises an induction would need
The rung below settled by a grouping test that a position's crossover depends on its own temperature and its answer's and on nothing below them, and asked for the induction. The four paragraphs are not written here; the checking they would rest on is. The law holds at five levels, survives translation, heating and cooling — and none of that is the step.
Particular games
An effect that changes sign
Which squares two Amazons share turns out to matter about as much as how many — three shared squares in a line run at 0.63 where three scattered run at 2.51. But the effect of clumping is hotter at one distance and colder at the next, so the arrangement predicts well and describes nothing, which is not what the four rungs below it produced.
Temperature
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.
Where it stops
The licence that weighs nothing
The third substitution licence is constant in the number of components, exactly as predicted, and it saves under two times where the first saves seventy-six million. Priced by its table instead of by its saving it is the only one of the three whose cost does not run away — which reverses the order three rungs of this anchor have put them in.
Where it stops
A quotient that identifies nothing
The dead-ending class is famous for quotients rather than comparisons, so the matched pair was asked the question its own subject is about. Neither quotient identifies a single pair of positions, and both are separated by exactly five addends — because a quotient is small when its universe is poor, which is a choice of company and not a property of a class.
Sums and comparison
A floor, and not a decline
Comparability fell eighteen points from day two to day three and the next day cannot be enumerated. It can be built — and the construction's bias measured one day lower, where the truth is known. Corrected, day four comes to 60.6 per cent against day three's 59.7: the fall was a one-day event.
Out in the world
A ko is won somewhere else
The rung below shows the ko rule buying finiteness by deleting one edge. What it buys with the same edge is a fight nobody can settle by looking at it — the prohibition forces a player to spend a threat, threats are counted on the rest of the board, and every decided cell of the sweep goes to whoever is ahead on a quantity that is not in the picture.
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.
Sums and comparison
The split slips one day deeper
The reversal case of the gift-horse theorem was described in one line — the follower's own move reverses every gift horse, whenever the follower has one — and tested only where it was found. In the mirror it holds exactly, with 1 and −1 trading places. One day deeper it fails: under ↑ and ½, three gift horses on built day-four bases are not reversed by the follower's move. All three are dominated, so the theorem stands; the clean split by follower does not.
Sums and comparison
Twenty draws and a second recipe
Day four's comparability was reported as 60.6 per cent against day three's 59.7, from one built sample and one calibration. Built twenty times with each of two recipes whose biases differ by six points, and calibrated against all 1,474 day-three values rather than a quarter of them, the corrected figure spreads over twelve points from seed to seed and the two recipes agree within one standard error. The floor survives; the decimal was one draw.
Out in the world
A cancelling pair is a zero
Two cancelling coin rows side by side cancel against their two negatives — all 190 pairs from rows of two, four and five coins, the odd row that cancels without pairing off included. And a cancelling row beside its negative is invisible next to any other row: in 741 tests against every row of one to three coins, neither score of the context moves. A non-cancelling pair moves a score in 452 of 780. Scoring games have no inverses in general; this class has them, and they behave as inverses must.
What it costs
Two heaps of testing are enough
A misère quotient is computed by testing positions against positions, and the universe used to find twelve classes of Dawson's chess was every position of up to four heaps tested against every other — 511,225 outcomes. Varied one size at a time, the count stops growing at tests of two heaps and positions of three: 12,100 outcomes find the same twelve classes. The narrower universe the earlier essay drew did not merge anything; it held fewer positions. And the corner that is enough moves: for Kayles at heap twelve, two-heap tests miss a class.
What it costs
Twelve classes, seven questions
Twelve misère classes of Dawson's chess were found by testing 715 positions against 715 others. Seven of those tests are enough to tell every class from every other — a greedy choice against a floor of four, since each test is one yes-or-no question. Kayles needs nine of 715 and Nim sixteen. The seven cost almost nothing to use and cannot be found without the whole closure, and they do not carry: the tests found with heaps up to seven tell apart only seven of the twelve classes with heaps up to nine.