Generator

Knowing who wins is not enough

Three pairs of positions, every one of which is in outcome class N on its own. Their sums are not all the same, and not all in the same outcome class — so the outcome of a sum cannot be worked out from the outcomes of its parts, and that is why the theory needs values.

10 essays call outcomes-do-not-add. 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.

Where it is called

Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.

A position is the sum of its parts. Four separate Hackenbush sprigs. A move is a move in one of them, so the position is their disjunctive sum, and its value is the sum of their values. Which part to play in is the entire decision, and the values are what makes it decidable.

Sums and comparison

The sum is the object

Real positions come apart into independent regions, and a move happens in exactly one of them. That operation — the disjunctive sum — is what the whole theory is built to survive, and it is the reason values exist at all.

Four things a position can be. Every position falls into one of four outcome classes, and only three of them correspond to a comparison with zero. The fourth — first player wins — is a position confused with zero, neither greater, smaller nor equal, and it is where the subject departs from arithmetic.

Values

Who moves last

The player who cannot move loses. That single convention generates the whole theory — and it produces four outcomes rather than three, because a position can be confused with zero rather than greater, smaller or equal to it.

Sums and comparison

Outcomes do not add

Knowing who wins each part of a position tells almost nothing about who wins the whole. Counted over every sum of two values born by day two, six of the outcome table's ten entries are settled and four are not — and every settled one is settled by the order rather than by anything about outcomes. Two first-player wins reach all four classes between them.

Sums and comparison

Which part to move in

The value of a sum is the sum of the values. The move in a sum is not the move in any part, and there is no rule that reads it off the values — in the smallest interesting example, the only winning move is in the component worth nothing.

on + off: what the backward analysis settles. A position graph in which the moves can lead back to where they started. The labels are the order in which a backward analysis settles each position, starting from the ones where a player has already run out of moves. Positions the analysis never reaches are drawn — and there is no test for that; being unreachable is what a draw is.

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.

One position, three ways of writing it, and only one of them adds. The same positions as a sentence about who wins, as a description of the position itself, and in the notation Winning Ways introduced. The first two columns carry identical information and support no operation whatever. The third column can be added — and the sums below it are values that no manipulation of the first two columns could reach, because two of these pairs start from the same two outcomes and finish differently.

How it was found

The notation was the argument

Up, star and the brace form are not abbreviations for case analyses. They are the claim that these objects add — and the arithmetic they support is arithmetic that no table of outcomes could ever produce, because two positions with identical outcomes can have different sums.

The values born by day three 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. The nimbers do, and they are not the only ones: a switch symmetric about zero is unchanged by negation, and so is anything whose Left options are the negatives of its Right options. Each row carries the value, whether it is a nimber, and its outcome.

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.

What each heap is worth. The value of a single heap of each size. Nothing here repeats: the forms grow deeper as the heap grows, which is what stops the impartial theory's periodic table from having an analogue.

Particular games

Two players, two lists

Give each player their own list of how many counters they may take and the impartial theory stops applying. What survives is the outcome: it settles into a repeat, for every pair of lists, and that is a theorem. What does not survive is the value — on four of six pairs swept it has no repeat inside sixty heaps, and the birthdays are still climbing at the edge of the window.

What the stops settle. The four open entries of the outcome table over every sum of two day-two values, read by the four inequalities on the stops of a sum. The stops settle 176 of 369 and are never wrong; the other 193 all have a bound exactly on nought.

Sums and comparison

What a part would have to report

Outcome classes leave four entries of the addition table open. Give each part its two stops as well and four inequalities settle 176 of the 369 open day-two sums, never wrongly — and every sum they leave has a bound sitting exactly on nought. Used as a lookup table instead, even the whole thermograph with the outcome beside it leaves 96 sums undecided. Four outcomes do what no thermograph can: who wins a day-two position alone, beside −1, beside 1 and beside a star names it, and no three questions do.

A number from the mast does not finish the job. Eight reports a part could make, from its two stops to its value, with the number of sums no lookup table indexed by the report can decide. Beside day three: stops 19,036, stops and outcome 16,402, stops and who is to move when each ends 17,088, the atomic weight of each foot 14,924, the feet 5,842, the feet with who chooses 4,962, and with who is to move 3,164; the value leaves none. On day two the atomic weight leaves 159, more than the 136 left by the stops and the outcome.

Sums and comparison

A foot is a menu, not a number

When a sum's fight ends level on nought, the winner is decided by what is left at the foot of each part's fight, and the obvious number to report from there is the atomic weight. Over 32,428 sums of a day-two and a day-three value it leaves 14,924 undecided, and on day two it does worse than a single outcome letter. What sits at a foot is a choice, made by one player, with someone to move when it is made. Report all three and every sum of two cold parts is decided; 3,164 sums are still not, and each holds a hot part whose own fight is not the one played in company.

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