Combinatorial Game Theory
who wins, and by how much
Generator

The 22 values born by day two, and the order they form

The 22 values born by day two, and the order they form
The 22 values born by day two, and the order they form. Each value sits above everything it is greater than, joined to what it covers. The order has 36 covering relations and is nine levels deep, and 52 of its 253 pairs are incomparable — and it is still a lattice: every pair has a least upper bound and a greatest lower bound among the same 22 values. Two values are marked, together with their join and their meet.

Each value sits above everything it is greater than, joined to what it covers. The order has 36 covering relations and is nine levels deep, and 52 of its 253 pairs are incomparable — and it is still a lattice: every pair has a least upper bound and a greatest lower bound among the same 22 values. Two values are marked, together with their join and their meet.

5 essays call value-lattice. 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

50 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
↑ ↑ L A number and a fight · The class where nobody runs out first · Below zero · Cooling adds and heating does not · Cooling by exactly one · Equal in every company · Turn the board through a right angle · Infinitesimals · Misère play has no negatives · Nobody has to move · Nobody wants to move here · One part that never ends · Three ways to add the same games · Outcomes do not add · The birthday of a sum · The company that is closed · The fight never runs backwards · The simplest game above both · The sum is the object · Toads and Frogs · What a number does to a fight · What a wider pool rescues · What an infinitesimal does to a fight · What can be struck out · What is left when the small change is thrown away · When the ups add · Which part to move in · Who moves last
↑ ∨ ∗, inside day three ⇑ L How rare it is to be bigger · The simplest game above both
↑ ∨ ∗, inside day two 1/2 L How rare it is to be bigger · The simplest game above both
∗ ∗ N A move that must be answered · A number and a fight · A rule with no promise at all · A self-negative value costs a day · At least five hundred and seventy-one · Below zero · Cooling adds and heating does not · Cooling by exactly one · Equal in every company · Equal in this company · Turn the board through a right angle · Fifty-two errors and seven sizes · How hot a day gets · How rare it is to be bigger · Infinitesimals · Misère play has no negatives · Nobody has to move · Nobody wants to move here · Nothing worth fighting over · Numbers avoid numbers · One part that never ends · Three ways to add the same games · Outcomes do not add · The birthday of a sum · The company that is closed · The fight never runs backwards · The first theorem, and the winner it declines to name · The other way to move a row · The simplest game above both · The thirty that cancel themselves · The values that are their own negatives · Three players and no answer · Toads and Frogs · What a number does to a fight · What a wider pool rescues · What an infinitesimal does to a fight · What can be struck out · What is left when the small change is thrown away · When the ups add · Where the impartial theory stops · Where the order and the sum disagree · Which part to move in · Who moves last · Start at the end and work backwards
∗ ∧ ↑ ↓∗ N The simplest game above both
∗ ∧ ∗2 ↓∗ N The simplest game above both
∗ ∧ 1 | −1 0 | −1 N Where the order and the sum disagree
∗ ∨ ↑ 1/2 L The simplest game above both
∗ ∨ ∗2 ↑∗ N The simplest game above both
∗ ∨ ∗2, inside day three {0, ∗ | 0, ↑∗} N Where the order and the sum disagree
∗ ∨ ∗2, inside day two ↑∗ N Where the order and the sum disagree
∗ ∨ 1 | −1 1 | 0 N Where the order and the sum disagree
∗ ∨ 1 | −1, inside day three {1 | ↓} N Where the order and the sum disagree
∗ ∨ 1 | −1, inside day two 1 | 0 N Where the order and the sum disagree
∗ against {1 | 0, ∗} {{1↓∗ | ↓∗}, ∗ | {↓ | −1↓}} N Fifty-two errors and seven sizes
∗2 ∗2 N A move that must be answered · At least five hundred and seventy-one · Equal in every company · Equal in this company · Turn the board through a right angle · Misère play has no negatives · The fight never runs backwards · The simplest game above both · The sum is the object · The thirty that cancel themselves · The values that are their own negatives · Two misère outcomes are not enough · When the ups add · Where the impartial theory stops · Which part to move in
0 0 P A rule with no promise at all · A self-negative value costs a day · An option nobody would take · At least five hundred and seventy-one · Equal in every company · Fifty-two errors and seven sizes · How rare it is to be bigger · Nobody has to move · Nothing worth fighting over · The fight never runs backwards · The first theorem, and the winner it declines to name · The operator that puts the star back · The other way to move a row · The simplest game above both · The thirty that cancel themselves · The values that are their own negatives · Three players and no answer · Two people, four years apart, one theorem · What is left when the small change is thrown away · When a switch is not a switch · Where the order and the sum disagree · Who moves last · Start at the end and work backwards
0 | −1 0 | −1 N One of four questions · The fight never runs backwards · Where the order and the sum disagree
0 | −1 ∧ 1 | 0 0 | −1 N One of four questions · Where the order and the sum disagree
0 | −1 ∨ 1 | 0 1 | 0 N One of four questions · Where the order and the sum disagree
0 ∧ ↓∗ −1/2 R Fifty-two errors and seven sizes · How rare it is to be bigger · Where the order and the sum disagree
0 ∧ ∗ −1/2 R Fifty-two errors and seven sizes · How rare it is to be bigger · The simplest game above both · Where the order and the sum disagree
0 ∧ ∗2 ↓ R The simplest game above both
0 ∧ 0 | −1 −1∗ R Fifty-two errors and seven sizes · How rare it is to be bigger · Where the order and the sum disagree
0 ∧ 1 | −1 {∗ | −1} R Fifty-two errors and seven sizes · How rare it is to be bigger · Where the order and the sum disagree
0 ∧ 1 | 0 ↓ R Fifty-two errors and seven sizes · How rare it is to be bigger · Where the order and the sum disagree
0 ∨ ↓∗ ↑ L Fifty-two errors and seven sizes · How rare it is to be bigger · Where the order and the sum disagree
0 ∨ ∗ 1/2 L Fifty-two errors and seven sizes · How rare it is to be bigger · The simplest game above both · Where the order and the sum disagree
0 ∨ ∗, inside day three {0 | {0, ∗ | −1}} L How rare it is to be bigger · The simplest game above both
0 ∨ ∗, inside day two 1/2 L How rare it is to be bigger · The simplest game above both
0 ∨ ∗2 ↑ L The simplest game above both
0 ∨ ∗2, inside day three {0 | ↑, ∗} L How rare it is to be bigger · Where the order and the sum disagree
0 ∨ ∗2, inside day two ↑ L How rare it is to be bigger · Where the order and the sum disagree
0 ∨ 0 | −1 ↑ L Fifty-two errors and seven sizes · How rare it is to be bigger · Where the order and the sum disagree
0 ∨ 1 | −1 {1 | ∗} L Fifty-two errors and seven sizes · How rare it is to be bigger · Where the order and the sum disagree
0 ∨ 1 | 0 1∗ L Fifty-two errors and seven sizes · How rare it is to be bigger · Where the order and the sum disagree
0 against {0, ∗ | −1} {{1↑∗ | ↑} | {↑∗ | −1↑}, ∗} N Fifty-two errors and seven sizes
0 against {1 | 0, ∗} {{1↓ | ↓∗}, ∗ | {↓ | −1↓∗}} N Fifty-two errors and seven sizes
1 | −1 1 | −1 N A self-negative value costs a day · At least five hundred and seventy-one · The fight never runs backwards · The thirty that cancel themselves · The values that are their own negatives · What a number does to a fight · Where the order and the sum disagree
1 | 0 1 | 0 N One of four questions · The fight never runs backwards · Where the order and the sum disagree
1 | 0 ∨ 0 | −1, inside day three 1 | 0 N The simplest game above both
1 | 0 ∨ 0 | −1, inside day two 1 | 0 N The simplest game above both
1 ∨ ∗, inside day three 1 L The simplest game above both
1 ∨ ∗, inside day two 1 L The simplest game above both
1/2 ∨ ∗2, inside day three 1/2 L Where the order and the sum disagree
1/2 ∨ ∗2, inside day two 1/2 L Where the order and the sum disagree
the discrepancy at 0, ↓∗ {−1/2 | −1/2↑} R Fifty-two errors and seven sizes
the discrepancy at 0, ∗ ∗ N Fifty-two errors and seven sizes
the discrepancy at 0, 0 | −1 {↑∗ | −1↑∗} R Fifty-two errors and seven sizes
the discrepancy at 0, 1 | 0 {1↓∗ | ↓∗} L Fifty-two errors and seven sizes

Where it is called

Changing this generator changes every one of these figures.

How often one value is above another. The partial order counted on two successive days. The proportion of pairs that can be compared at all falls sharply, and so does the proportion of values that can be compared with zero — which is the proportion of positions whose winner does not depend on who moves. Sums and comparison

How rare it is to be bigger

Values are partially ordered, and 'partially' does most of the work. On day two, 179 of 231 pairs can be compared and 13 of the 22 values can be compared with zero. One day later the shares are 60% and 29%, and the largest set of mutually incomparable values found rises from four to at least twenty-three. Comparison is the exception; confusion is what values normally do to one another.

 The 22 values born by day two, and the order they form. Each value sits above everything it is greater than, joined to what it covers. The order has 36 covering relations and is nine levels deep, and 52 of its 253 pairs are incomparable — and it is still a lattice: every pair has a least upper bound and a greatest lower bound among the same 22 values. Two values are marked, together with their join and their meet. Values

The simplest game above both

Values sit in a partial order, and a partial order is entitled to be ragged: two things with no least thing above them. The 22 values born by day two are not ragged at all. Every one of their 253 pairs has a least upper bound and a greatest lower bound among the same 22, and the order is distributive on all 10,648 triples — so it is a lattice, and the join of zero and star is one half.

 The identity that would join the order to the addition. Every pair of the twenty-two values born by day two, asked whether the join plus the meet equals the sum. It holds on all 201 comparable pairs, where the join is the larger and the meet the smaller and it cannot do otherwise, and on none of the 52 incomparable ones. 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.

 Fifty-two errors, put to four instruments. The fifty-two discrepancies the lattice identity leaves on day two, counted by what distinguishes them. As values no two are the same; as pairs of stops there are seven; as means three and as temperatures three. Not one of them is a number, and only three are values born by day two. Values

Fifty-two errors and seven sizes

Day two is a lattice and a group and not a lattice-ordered group, and the fifty-two incomparable pairs it fails on leave fifty-two different error terms. Measured rather than listed, the fifty-two collapse: seven pairs of stops, three means, three temperatures, and a rule that predicts the temperature from the pair on forty-four of them.

 The order one day out. Whether the values born by day three still form a lattice. Twice as many pairs are incomparable as at day two, and every incomparable pair still has a least upper bound and a greatest lower bound — so the order becomes more tangled without becoming ragged. Values

One of four questions

Three rungs of this ladder rest on sweeps of day two — 22 values, 253 pairs. Day three is 1,474 values and over a million pairs, and only one of the four questions can be asked of it. The order can: twice as many pairs are incomparable and every one of 1,606 sampled still has a least upper bound and a greatest lower bound, none of them a value day two already had. The other three compare sums of day-three values, which are born on day six, and sixty of those exhausted an eight-gigabyte heap.

The whole library · The position index · The figures that play back