The thread: The sum is the object — page 3
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.
What it costs
A catalogue that knows what it will meet
The rung below priced a catalogue of regions by its reach and found the coverage saturating, and asked what a catalogue ordered by frequency would cost instead. Eight shapes answer half the components a played Domineering board produces; a catalogue by size needs fifteen for the same, and 1,042 for what 119 chosen by frequency reach. Three quarters of a size-ordered catalogue never turns up in play at all.
Particular games
Read from the back forwards
The rung below found a two-run Push strip converging at a rate set by the back run and asked what a third run does — whether the rate is still the rearmost run's, or whether the rates compound. It is the rearmost run's, and for every gap: widen the front gap of a three-run strip, two whole runs away, and the value still dies at the last run's rate. Shove, the game one clause away, compounds.
Particular games
One domino every three cells
The rung below gave the optimistic packing count as a formula in odd runs and asked for the other end of the interval, expecting a formula in the even ones. Parity is the wrong arithmetic: the smallest maximal packing is a sum of ⌈(len−1)/3⌉ over the runs, exact on all 1,042 shapes. That makes the whole interval readable off a drawing — and shows it can never reach the value, because regions with the same runs have different values.
Temperature
Eight squares, and no hotter
The rung below found no Domineering position hotter than three halves on four boards and asked for the position that attains it. It is a region of eight squares, there are five of them up to symmetry, three are the hot core of an attaining board on every size swept — and the ceiling holds at nine and ten squares too, where the obvious extrapolation predicted seven quarters.
Sums and comparison
The same number in two currencies
The rung below found bent-walled values falling strictly inside the translation bound and asked how far. The shortfall is the value's own hottest follow-up's temperature — exactly, on 400 of 408 pairs, and twice it on the other eight — which makes the whole error one expression. And it is the switches ladder's constant: a half there and a whole here, because a temperature is half a stop gap.
What it costs
The catalogue a strong player needs
A Domineering catalogue built from random play faces an objection that could overturn it: random play is not play. A player that reads the board produces the same head — eight of the ten commonest shapes — and concentrates far harder: 114 entries answer nine tenths of what it meets, against 2,018. And a catalogue measured on random play over-serves it, while the reverse fails.
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.
Particular games
The short side is not in the lemma
The closed form for a two-sided Maundy Cake rested on one unproved statement: that no divisor beats the largest prime. Written out, that statement never mentions the short side — it is an inequality between a multiset of primes and a term count — and once it is stated that way it has a two-line proof, term by term. The ladder ends in a theorem rather than a grid.
Particular games
Where the runs meet
A Domineering region's value interval is a function of its run lengths and its value is not — twenty-one groups of shapes share a multiset and disagree. The crossing count separates none of them, and neither do ten other local statistics, fifteen sets of which agree on everything and differ in value. What separates nineteen of the twenty-one is where along its runs each crossing sits.
Sums and comparison
Where the value stops mattering
Fourteen straight-walled pairs missed the bound on a translated stop and had only a threshold to explain them. Their stops move by exactly the addend's temperature — a formula with the value nowhere in it — which turns the threshold into the boundary between two lines and closes a census of 1,440 pairs that has been open for four rungs.
Sums and comparison
A side about to lose its move
A fifth of the second reduction's work removes the last option a player had on a side, and no rule on the ladder had looked at one — because a deletion with no survivor has no pair in it. The recognisable object is not which option goes but whether the side is one an option can go from, and two comparisons on the parts decide it on all 525.
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.
Temperature
The ceiling was a plateau
Three halves of a move looked like a ceiling on a Domineering region's temperature: it held at eight squares, at nine and at ten, and the rise that had been a quarter every two sizes stopped. At eleven squares four regions reach seven quarters — and they contain the hottest eight-square shapes and are hotter than them, so the extra material is not cold.
What it costs
A catalogue that builds itself
A solver that stores every region it has to evaluate builds a catalogue out of its own games. After 650 games it holds 232 of the 1,042 shapes and is still growing — and the order things arrive in is nearly arbitrary while the order they are consulted in reproduces a census of a strong player's games almost exactly.
What it costs
The table that changes its mind
The advice that ten entries chosen by use serve nine lookups in ten was untested: it describes a table sorted after the fact rather than a solver that only ever held ten. A solver that only ever held ten gets 94.2 per cent — beating the best ten chosen with the whole run in view, because there is no best ten.
Where it stops
The price of asking what the parts are
The third licence lets a solver look up a region rather than a position, and the rung below priced it by the entries it stores. Priced by the work it costs, it saves between a third and two thirds of the expansions and pays for them with a flood fill at every node — six times the total. A square would have to be ten times cheaper than a table probe before it broke even.
Impartial games
The pairing the formula hides
Welter's closed form sums a function over every pair of coins and needs an extra term when the count is odd, which the rung below called a surprise. Read as a matching it is not: an odd number of coins cannot be paired, the left-over coin contributes its own square, and some matching gives the value on every position measured.
Sums and comparison
A cross in the table
Which pairs need a second reduction had never been asked. Sorted by the two temperatures the answer is a cross — the whole gap-of-a-quarter column and the whole colder-is-three-quarters row — and it is exactly necessary on all 431 with no exception, and wrong 477 times in the other direction.
Sums and comparison
The proof needs both reductions
The gift-horse theorem was to be proved by showing the added option dominated. It is, on 97.8 per cent — and the other 232 are reversible instead, with nothing left over. The case the proposal missed is almost entirely one follower: none under a positive number, 190 under a negative one.
Particular games
The obvious cut is the wrong one
Maundy Cake's rule was proved by restating its lemma so the short side vanished. Cutcake's collapses the same way — into a binary length instead of a multiset of primes — but the cut the argument needs is not the one the ladder predicted. Halving is wrong on a third of all cakes, and the smallest counterexample is six squares by two.
Temperature
The residues as a sequence
Four of the eight residue sequences never repeat, and a recurrence is not a description. There is a closed form and it is not for the game: the stops and the temperature of the n-th residue are periodic with period one, two or four on every sequence in the pool, while three of them produce a different game at every n.
Where it stops
Closing the wild side
The twenty-two wild genus symbols are not closed under addition, and the rung below offered two answers: a monoid nobody had guessed, or no algebra at any size. Neither. Five of the six games with wild heaps close at three or four heaps, with closures of two to five symbols, and the sixth is still growing.
What it costs
A wall an amazon can walk through
An arrow burns a square for good, so an Amazons board that has fallen into pieces should stay in pieces. Over 127,583 positions it does not: fifty-one thousand moves put two regions back together. Every one of them is a single diagonal step, and what is wrong is not the game but the rule used to find the regions — which was borrowed from a game whose pieces lie along the board's own lines.