Every position has an exact opposite
A position beside its negative, which is the same game with the players exchanged, and the sum of the two. The sum is worth zero every time — a second-player win — because the second player can answer each move with its mirror image. It is the fact that makes values a group, and it is what lets one position be subtracted from another.
19 essays call
negation-mirror. 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
62 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.
Sums and comparison
Turn the board through a right angle
A two-by-four Domineering board is worth something no number can express, and Right is ahead on it. Turn a second board through a right angle, put the two side by side, and the total is exactly zero. Every position has an exact opposite, and that single fact is what makes subtraction — and therefore comparison — possible at all.
Where it stops
One part that never ends
The game called `on` has one move and it is back to itself. Add anything to it — a star, a point, its own mirror image — and the whole board is drawn. So `off` is exactly the negative of `on` and their sum is not zero, which is the group law failing for a reason that has nothing to do with who is winning.
Where it stops
Misère play has no negatives
Put a position beside its own mirror image and answer every move with the mirror move. Under normal play the answerer wins and the sum is worth zero. Under misère the answerer still has every reply and loses because of it — so there is no zero, no subtraction, and no comparison, which is why the misère theory had to be rebuilt rather than adjusted.
Sums and comparison
Where the impartial theory stops
Sprague–Grundy gives every impartial position one number, and the number is complete. The moment the two players have different moves no number works at all — not a harder one to compute, none — and three positions here are compared against every nimber to show it.
Where it stops
Two misère outcomes are not enough
Knowing who wins each part does not say who wins the sum. Over 676 sums built from a pool of twenty-six positions, nine of the sixteen pairs of outcome classes settle the answer under normal play and not one of the sixteen settles it under misère — and the nine that work are theorems about a value being zero, which is exactly the thing misère play does not have.
Sums and comparison
Equal in every company
Two games are equal when no third game can tell them apart — a quantifier over every position there is, discharged by one finite test. A search over 184 contexts separates all 5,790 unequal pairs it is handed and still calls two different games the same, which is exactly why G − H = 0 is a theorem and an exhaustive search is not.
Sums and comparison
The values that are their own negatives
Every game satisfies G + (−G) = 0, so a game equal to its own negative satisfies G + G = 0 — it has order two in a group whose elements otherwise have infinite order. The nimbers do. So does ±1, on sight. Over the 1,474 values born by day three there are 30 of them and only four are nimbers, every one of the 900 sums of two is another, and the equality test and a symmetry of the written form agree 1,474 times out of 1,474.
Sums and comparison
What can be struck out
From G + X = H + X it follows that G = H, in one line, by adding −X to both sides. It is the shortest theorem here and the most used: it is what makes comparing two boards region by region legitimate. Over 10,648 triples the hypothesis fires 484 times and the conclusion holds 484 times — and the licence expires in three separate directions, each of which loses the same axiom in a different way.
Values
The birthday of a sum
Two values born by days m and n have a sum born by day m + n at the latest, which is the bound that stops a board made of many small parts from being unboundedly complicated. Over 231 pairs of day-two values the bound holds every time and is exact 163 times — and every pair it misses by three days or more has a sum that is a number or a nimber, so the slack is not noise but a measure of how much cancelled.
Where it stops
What a wider pool rescues
The misère outcome table has sixteen cells, and over a pool of ten positions fifteen of them hold fewer than four outcomes — which looks like structure and might be a shortage of positions. Thirteen values further on there is nothing left: every pair of outcome classes takes every outcome, so the near-misses were the pool, and the prediction the rung below made was right.
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.
Sums and comparison
The thirty that cancel themselves
Thirty values born by day three are equal to their own negatives, and every one of them has a mean of exactly nought and two stops that are exact opposites. Neither property comes close to picking them out — 496 values of the day have a mean of nought — and half of the thirty are hot, one of them the hottest value the day produces.
Where it stops
The company that is closed
Restricted equality licenses substitution only inside a company closed under addition, and none of the five companies this site computes in is closed — day two keeps a quarter of its own sums. Searching for companies that are closed finds seven, at one, two, four and eight members, and every member of every one of them is its own negative.
Sums and comparison
A self-negative value costs a day
The rung below placed the thirty values equal to their own negatives on the temperature scale and asked whether being self-negative forces anything about when a value can be born. It does, exactly: the earliest self-negative value of temperature t is born the day after t itself, which accounts for the four temperatures that carry one and the four that carry none. The guess it offered — that the first value of each temperature is a self-negative one — holds at four temperatures out of five and is not the shape of the answer.
Sums and comparison
At least five hundred and seventy-one
The rung below dated the self-negative values — the earliest of temperature t is born the day after t — and left the count to a day-four census nobody can run. The construction settles it instead: a value is its own negative exactly when its form is a mirror, so the subgroup can be built from subsets of the day below rather than sifted out of the day above. Day four supplies at least 571 against day three's 26, and the share of a day that is self-negative keeps falling.
Sums and comparison
What identifies two subsets
Every subset of a day gives a self-negative value by mirroring it, and 1,793 subsets of day two give thirty values. The collapse happens in two stages with different characters: domination takes the 1,793 to 96 antichains and is a theorem, and the rest of the reduction takes 96 to 30 and is concentrated almost entirely on two values — nought, which has an exact description, and star, which has none.
Sums and comparison
A mex with no impartial game in it
The rung below described the zero fibre of the mirror map and left star's fourteen undescribed. Star's fibre is 'some element is at least nought, and none is at least star' — and the two rules are one rule: the mirror value is the least nimber no element of the set reaches. That is a mex, in a construction built entirely from partizan values.
Sums and comparison
The case that was supposed to be hard
The mex rule for the mirror construction was to be proved by induction, and the step flagged as needing care was the one where an option is incomparable with the nimber. There is no induction: the argument is four lines, and incomparability is what makes two thirds of the cases go through — because a fuzzy sum is a first-player win and the first player is the opponent.
Sums and comparison
Twenty-six other values
The mex rule accounts for sixty-six of the ninety-six antichains and is silent on the other thirty. Every one of those thirty is worth a self-negative value born by day three — and the same mex, run over that family instead of over the nimbers, is exact on all ninety-six. The nimber rule is this one cut short after its fourth member.
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