256 ways of writing a position, 22 values between them
Every game whose options come from the four born on day one — 256 of them, counting each choice of Left and Right option sets separately. Reduced to canonical form they carry 22 distinct values, and the classes are nothing like equal in size: the largest holds a quarter of all the forms and the smallest holds four.
11 essays call
form-census. The drawing above is what it returns with no arguments at all; every
call below passes it something, because a placement that passes nothing draws whichever
member of the family the generator happens to default to rather than the one its essay argues
about.
The positions it draws
22 distinct positions, harvested by running this generator again at the options each essay passed it.
Where it is called
Changing this generator changes every one of these figures.
What it costs
What counts as the same position, and what that is worth
Folding a 4×4 Domineering board by its symmetries takes the table from 5,700 entries to 1,522 — a saving of 3.75, against a ceiling of exactly 4. An orbit cannot be larger than the group acting on it, so this is the one saving in the subject that can never change an exponent.
Values
Two hundred and fifty-six ways to write twenty-two things
Every game whose options come from the four born on day one — there are 256 of them, and between them they carry 22 values. The reduction that collapses one to the other has choices in it at every step, and uniqueness is the claim that none of the choices matters.
Values
How old a value is
A form's depth bounds the birthday of the value inside it, and reducing to canonical form attains the bound — for all 22 values born by day two, with no exception. Twenty-four of the 256 forms are older than what they are worth. The same reduction that makes the bound tight is what puts day three within reach: 98 option sets a side instead of four million, 9,604 forms, 1,474 values, a quarter of a second.
Values
The fight never runs backwards
Left's stop is never below Right's — in every one of 1,780 distinct values, computed twice by two independently written routes, with nothing that disagreed anywhere. The inequality is what makes a mean value well defined and a fight a fight; and where it collapses to equality, 433 of the 460 cold positions turn out not to be numbers at all.
Values
An option nobody would take
Every reduction of a form deletes. The gift horse principle adds: a move may be handed to a player for nothing, provided it is one they would never choose. Over all 484 additions to the values born by day two, 283 leave the value exactly where it was and the 201 that move it are precisely the ones the condition forbids — with the boundary at *not better*, which is a weaker demand than *worse*.
Values
The values nobody's game produces
The construction hands down 1,474 values by day three. Seventeen rulesets on this site, swept to eleven thousand positions, produce 1,193 — and only 116 of those are on the construction's list. Two of the twenty-two values born by day two are produced by no position of any game here, and 1,077 of the values that are produced are born later than day three. A value's birthday and a value's reachability have almost nothing to do with each other.
Values
How wide a form can get
Bypassing a reversible option replaces it with a whole option list, so a form grows in the middle of its own reduction. Whether it can come out wider than it went in is the question that leaves standing, and over 64,515 forms built from day-two options the answer is no, not once — the growth is real, it is transient, and the widest canonical form reached is exactly as wide as the widest form that reaches it.
Values
What a value costs to write down
The canonical form is the smallest form of its value, and it is smallest in the one currency the reduction happens to spend: options. Counted in symbols it is nothing of the kind — the widest value born by day three is not the longest, the longest has six options rather than seven, and every canonical form on the day except the seven integers writes some position out twice.
Values
Where the order and the sum disagree
Day two is a lattice, and day two is a group, and it is not a lattice-ordered group. The one identity that would join the two structures — the join plus the meet equals the pair — holds on exactly the 201 pairs where it cannot fail and on none of the other 52, and the errors split thirteen high, thirteen low and twenty-six confused.
Values
The same position, written once
Writing out the canonical forms of day three takes 24,940 nodes. Naming each distinct subposition once inside each form takes 10,102, and naming each distinct subposition once across the whole day takes exactly 1,474 — one per value, because nothing appears inside a canonical form that is not itself a value of the day.
Sums and comparison
What the colon respects
The ordinal sum reads the form of its base rather than its value, which is why the colon principle is stated for positions and not for values. Built over 9,604 forms it turns out to read the value on 636 of the 640 values that have more than one form, and the four it can tell apart are zero, one, minus one and star — the values born by day one, and no others.
The whole library · The position index · The figures that play back