Realisability — where it appears
Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.
How long a row a value needs
Add a third colour that either player may topple and a row of seven dominoes reaches 1,047 distinct values where two colours reach 149. That makes the length of the shortest row worth a value into a measure of the value's complexity — one a reader can hold in their hand — and it is not the birthday: 1↑ is born on day three and needs seven dominoes.
The cheapest way to show a value
Eleven thousand positions from fifteen rulesets reach 1,193 values, and for each of them there is a smallest board that shows it. Set against the birthday the two measures agree hardly at all — until the numbers are taken out, at which point they agree rather well, and the whole apparent independence turns out to be a fact about integers.
The birthday is a floor
The rung below measured a correlation of 0.73 between a value's birthday and the size of its cheapest exhibit, and asked which values are dearer than the birthday suggests. The relation is not a trend. Over all 728 non-number values the exhibit is never smaller than the birthday and is exactly the birthday on 476 of them, and the excess on the other 252 belongs to the game rather than to the value — the ruleset accounts for 40 per cent of its variance.
Wider costs less
The rung below found the cheapest exhibit of a value never smaller than its birthday, exactly equal on two thirds, and the ruleset explaining 40 per cent of the rest. The variable it proposed for the remainder was the width of the form. Width and excess correlate at −0.39: the wider the value, the closer to its birthday it is exhibited, and inside a ruleset the relation cannot even agree on a sign.
The entry fee was the cap
Two rungs measured how much bigger a position has to be than the value it exhibits, and attributed what was left to the ruleset — Toads and Frogs paying 2.25 squares on everything, green Hackenbush paying nothing. Neither number is a property of the rules. Inside every ruleset the excess falls as the birthday rises, because the sweep's size cap censors exactly the values that would pay most — and three squares past the cap, Toads and Frogs exhibits values born later than the strip is long.
The rate was the alphabet
The rung below asked for a quantity a size cap cannot censor and proposed the rate: how many new values a ruleset produces per extra square. The rate is honest and it measures the notation — every ruleset grows at close to the number of symbols its positions are written in, and the seven span less than a factor of two. What separates them is the yield, which spans a hundred and nineteen.
The mirror was the floor
Toppling Dominoes' share of genuinely new values had fallen from one to a half over eight sizes, and the rung below could not tell a floor from a slow fall. Four more sizes settle it: the distance above a half halves every two sizes. And the half is not a shortage of values but a symmetry — a row played from the other end is the same game, and the ruleset is as injective as that allows.
Three groups, and three yields
The conjecture was that each ruleset's yield tends to the reciprocal of its symmetry group's order. Three rulesets have a trivial group and predicted yields of one; they measure 1.000, 0.531 and 0.204. And every colliding value in Push and Shove — all 175 of them — has two rows no symmetry relates.
A recipe instead of a census
Counting a thousand values in seven dominoes suggests Toppling Dominoes reaches every short game, and a count is not a construction. The obvious construction — lay the two options either side of a Left domino and a Right one — is exact on day one, right on a third of day three, and cannot be applied to nine in ten values at all.
Named alongside it
The objects these essays reach for when they reach for this one.
EnumerationBirthdayValueCanonical formHackenbushCounterexampleRulesetToppling dominoesEnd-NimExhaustive searchApproximationBorn on day