Translation — the series · Combinatorial Game Theory

Every day-three value, moved by every quarter. Four claims counted over 3,000 translations — 120 values, each moved by every quarter from −3 to 3: that adding a number leaves the temperature alone, that it shifts both stops by exactly itself, that the interval between the negated stops is where the two players are confused, and that the only failures of the last are on its endpoints.

What a number does to a fight

Adding a number to a position moves everything and changes nothing: over three thousand translations the temperature never once shifted and both stops moved by exactly the number added, every time. What the number decides is whether the fight is worth having — and the interval where the two players are confused is exactly the open interval between the negated stops, right in all 2,890 cases away from its endpoints and wrong in 110 that are all on them.

part 1 · sums

Seven infinitesimals, added to a whole day. Each row adds one infinitesimal to every value born by day three. The stops never move — that is what being smaller than every number means — the temperature moves a handful of times, and the outcome class moves in a quarter of the additions.

What an infinitesimal does to a fight

Adding a number moves both stops by exactly itself. Adding something smaller than every number moves neither — across 10,318 additions to a whole day of values, not once — and the outcome class changes anyway, 2,622 times. It changes at exactly one kind of position: the ones with a stop sitting on zero, which is where the numbers have run out of things to say.

part 2 · sums

The stops stop working. Adding a switch to each of 120 day-three values, and asking whether the two stops of the sum are the sums of the two stops. They are on 330 of 720.

What a fight does to a fight

A number added to a position shifts both its stops by itself; an infinitesimal moves neither. A hot game does neither: over 720 sums the two stops add on 330 and are wrong on the rest. What survives is the mean, which adds on every one of the 720 — and the failure has a bound, since no stop is ever out by more than twice the smaller of the two temperatures, a bound 222 of the sums attain exactly.

part 3 · sums

The bound that needed no third number. The rung below's bound and the conjectured replacement, scored over 1,440 sums whose addend has a follow-up. The two-number bound holds everywhere and is attained; the three-number one fails.

A bound with one number too many

The rung below bounded how far a hot addend can drag a stop — twice the smaller of the two temperatures — over sums whose addends were all plain switches, and conjectured that an addend with a follow-up would need twice the smaller of three numbers. Over 1,440 sums with bent addends the two-number bound holds everywhere and is attained 358 times, and the three-number version fails on 66.

part 4 · sums

Which end, in four lines. The complete rule for where a sum's error lands, exact on every pair in the census. Three of its four cases are decided by the value being translated alone.

Which end a sum lands at

The rung below found the errors in a translated stop clustered at the two ends of the range its bound allows — 660 at nought and 358 exactly on the bound — and asked for a rule saying which end a given pair lands at. There is one, in four lines, exact on all 1,440 sums. Three of the four cases are decided by the value being translated alone, and the property that decides them is the bend the switches ladder found for a different question entirely.

part 5 · sums

One quantity, two currencies. Each bent value with how far its temperature falls short of its stop reading and how far its stop falls short of the translation bound. The second is exactly twice the first.

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.

part 6 · sums

The census, closed. The three classes of the translation census with the expression each obeys, scored over all 1,440 pairs.

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.

part 7 · sums

The bound survives and the rule does not. The census against the widened sweep, with the bound and the two-line rule scored separately on each.

One expression proved, and one withdrawn

The census closed with three expressions exact on 1,440 pairs, and the rung above asked for derivations. The cold one has a four-line proof. The straight one has a threshold the census cannot determine — any constant between 4/3 and 3/2 fits it — and eight more addends of the same family break it on 38 pairs while leaving the bound above it untouched.

part 8 · sums