Eventual periodicity — where it appears
Named by 16 essays across 5 fields — each of them below, with the objects they name alongside it.
Naming a game with a number
An octal code is a rule table compressed into an integer. It turns "which game" into something that can be counted through, which is how the family was swept — and how the games nobody can solve were found.
The period is small and the proof does not say so
Every subtraction game repeats eventually — that is a theorem, and its proof gives a bound of sixteen thousand for a three-move set. Over 112 sets the longest period measured is twenty-two. The proof and the fact are four orders of magnitude apart, and the rule of thumb that closes the gap is broken by one set in the sweep.
The values that keep arriving
A Grundy sequence that repeats uses finitely many values and stops needing new ones. Six thousand heaps into ·007 the count of distinct values is 187 and still climbing, and the share of heaps carrying something outside the twenty-two commonest rises from 32% in the first thousand to 85% in the sixth. The rare values a periodicity argument needs to thin out are getting commoner.
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.
A period with a constant added
An octal code says what a player may do when removing k counters, in three bits; a hexadecimal code adds a fourth — leave three heaps — and the digits run to fifteen. Over twenty-two codes swept to six hundred heaps, five hexadecimal ones repeat with a fixed amount added each time round and no octal one does. Their values climb for ever and never repeat, so a search that looks only for repetition reports them unsettled.
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.
A code that climbs by three
Five hexadecimal codes were known to repeat with a constant added, and every one of the five constants was a power of two — either a fact about exclusive-or or a coincidence over five cases. Sweeping all 255 two-digit codes settles it: seventy-one climb, seventy of them by 1, 2, 4 or 16, and one by three. The exception is ·3f, whose values are 3⌊n/6⌋ + (n mod 3) on every heap to twelve hundred.
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.
Two counters, and one displaced term
The rung below found four Grundy sequences in the odd-saltus class and asked which term each displaces and whether the digits predict it. They do — but there are two base-three counters and not one, chosen by whether a heap of one can be taken away. And there are three sequences rather than four: the fourth is the third with three isolated values, and was counted separately because its period had not settled.
A set with three descriptions, and a function with none
Wythoff's cold positions can be written three ways that share no arithmetic — an irrational constant, a greedy rule, a condition on Fibonacci digits — and all three are exact. The same game's Grundy values have no closed form at all. Both facts are about one table, and the gap between them is the subject.
What the arithmetic cost in 1956
The rung below ends by respecting a hand computation without pricing it. Priced in the operations a person actually performs, ·137's certificate is 7,919 of them — and the same sweep says ·47's is sixty-three times that, that a splitting move is what makes the cost quadratic, and that seventeen of sixty-four codes have no certificate at any price.
The formula is a limit
Cap the take in Lasker's Nim at k counters and the game is a finite rule table, 4.33…3, whose Grundy sequence repeats with period k + 1 rounded up to even and follows Lasker's formula until the cap bites. The formula is what those periods converge to. And the same column of codes, with a free split in front, holds Kayles itself: the rule 4.4 on a heap of n + 1 is Kayles on a row of n.
Three heaps and a pass
Nim with a single pass that may not end the game is easy on one heap and on two: a heap swaps each odd size with the even one above it, and two heaps lose exactly at (2k − 1, 2k). On three heaps the losses are known only as a list. Fix the smallest heap and each slice of the list settles into a pattern after an irregular start — period 4, 8, 10, then 160 at a smallest heap of ten, and nothing visible from eleven.
A rare class exclusive-or respects
The rare values of ·007 were measured by rank — everything outside the twenty-two commonest — and found getting commoner. A sparse-space argument needs a different split: a rare class closed under nothing and a common class closed under exclusive-or, which is to say a parity of bits. Over every parity of every bit, ·007 and ·6 have no class that thins. Across the 79 growing octal codes of up to three digits, one does: ·354, whose odd values fall from 288 in its first two thousand heaps to seven in its last ten thousand.
The odd values keep coming
·354 is the one growing octal code of up to three digits whose values have a class that thins: 288 odd values in the first two thousand heaps, seven between ten and twenty thousand, the last of those at 18,908. A sparse-space argument needs that class to stop. Taken to a hundred thousand heaps it does not: 48 more odd values arrive after twenty thousand, every block of ten thousand holds some, and the last in range is at 99,600. They keep no schedule a remainder can see, their gaps are spread like arrivals, and most of them are one number, 37 — each one a heap where every split that could have reached 37 misses it.
Two takes, one line
A subtraction game with two legal takes, a and b, has a Grundy sequence that fits in one sentence: cut the heap's remainder modulo a + b into blocks of a, alternate nought and one, and turn every even block at or past b into a two. The proof is three cases long and holds for every pair. Add a third take and the sentence has no successor — but it does not vanish. The period of a three-take game is a sum of two of its takes six times in seven, which sum is decided by the third take's remainder modulo the first two's period, and the long irregular periods cluster on the one arrangement where the third take is the other two added together.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchPeriodicityGrundy sequencesGrundy valueClosed formOctal gameMexImpartialTake-and-breakUnsolved gameCounterexampleSubtraction game