Saltus — where it appears
Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.
The third digit
The rung below found 71 of the 255 two-digit hexadecimal codes repeating with a constant added rather than exactly, and asked whether the same share holds one digit wider. It rises. Of the 4,095 three-digit codes, 1,433 climb and 617 repeat exactly — seven in ten of the settled ones — so a saltus is the ordinary way a hexadecimal game settles and the exact repetition the octal survey was built to find is the special case.
The only way to split into three
Nineteen three-digit hexadecimal codes climb by three, and the rung below asked whether they share a form and what digits they have in common. The digits are exact: on eighteen of them the only way to split a heap into three is by taking exactly three counters, and taking three counters can do nothing else. The form is not shared — the eighteen carry four distinct sequences, and exactly one of the four counts in base three.
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 pattern that has not started yet
A pre-period was supposed to be rarer in this family than a defect. Two hexadecimal codes in five have one, 321 have a pre-period longer than their own period, and the code the rung below found slow takes fifty-four heaps to settle rather than two blocks — which is also the account of three defects the rung below recorded and could not explain.
The quantity that carried nothing
The rung below proposed predicting a pre-period's length from the saltus and the period. The saltus correlates with it at −0.03, which is nothing; the period correlates at 0.77 with a coefficient of one, so a pre-period is about one period long. The digit that predicts whether there is one predicts nothing at all about how long.
Growing is not climbing
The octal codes whose values grow were expected to be arithmetically periodic — repeating with a constant added each period — and to fool a reader with false climbs. Read for that pattern, the 220 octal codes of up to three digits with no certified period produce 246,048 readings in 1,200 heaps and not one holds for more than six periods. The false climbs are exactly as short as on the hexadecimal codes where the climb is real; the difference is that no right reading ever arrives, and the largest values grow more slowly than any constant climb allows.
Named alongside it
The objects these essays reach for when they reach for this one.
Octal gamePeriodicityEnumerationGrundy valueHexadecimalImpartialCounterexampleExhaustive searchGrundy sequencesInvariantCodesApproximation