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.
3 essays call
partizan-strip. 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
42 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.
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.
Particular games
A sequence with a rule and no period
The values of the subtraction game where Left takes one or two and Right takes one or three never repeat — thirty-one heaps, thirty-one different values. They are nonetheless completely described: three seeds and the rule v(k + 3) = {0 | v(k)} generate every one of them, which is what a pattern without a period looks like.
Particular games
The condition that survived the wider sweep
Which pairs of subtraction lists have a value sequence that repeats? Over the 49 pairs drawn from one, two and three, two conditions answer it identically — a translation and all-odd — and both are exactly right. Over the 961 pairs drawn from one to five, all 83 translations still repeat with no exception and four all-odd pairs do not, at heap ninety with a period as long as forty-two. Neither condition is necessary: 104 pairs repeat that satisfy neither.
The whole library · The position index · The figures that play back