Thought Toys · Cycles & change · Exhibit 122
Periodical cicadas spend thirteen or seventeen years underground, then come out all at once, in numbers no predator can eat through. The strange part is the number. A brood on a twelve-year clock, facing a predator whose own numbers peak every four years, walks into that predator at every single emergence it will ever have. Move the brood to thirteen and three emergences in four come up to an empty field. Then shift the predator by one year, and everything you just concluded turns out to be the wrong lesson.
A hundred and forty years, one row per cycle ● the brood emerges ● predators peak ● both at once
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The top strip is time. Amber diamonds are the years this brood comes up; cyan ticks are the years the predator's own population is at its peak. A red ring means both happened in the same year, and that emergence was eaten.
Set the brood to twelve years against a four-year predator. Every emergence is ringed. Not most — every one, for as long as both clocks keep running, because twelve is a whole number of fours and so every multiple of twelve is also a multiple of four. Now nudge the brood to thirteen. Three rings in four vanish. The predator did not change, the brood did not get faster or tougher, and the number of years underground went up by one.
The bar chart underneath is the whole landscape at once. Each bar is a life cycle the brood could have; its height is the share of emergences eaten. The tall bars are the cycles that share a factor with the predator's. And now press Shift the predator a year, which moves the predator's peak years without changing anything else.
Everything you just believed rearranges. The twelve-year brood, doomed a moment ago, is now eaten never — it steps over the predator's years for ever. The thirteen-year brood is still eaten one time in four, exactly as before. Shift again, and again: the prime never moves, and the composites keep flipping between catastrophe and complete safety.
The level cyan rule across the bars is the honest summary, and it is the whole exhibit. It marks each cycle's average over every head start the predator could have had, and it is flat. Every life cycle — prime, composite, 2 or 19 — is eaten exactly one time in p on average. Primes are not eaten less. They are eaten predictably. A brood that shares a factor is making a bet on where the predator happened to start: win it and you are untouchable, lose it and you are eaten at every emergence you will ever have, for ever, with no way back. A coprime brood does not take the bet. It accepts the average and is guaranteed it.
That is why this matters for an animal rather than only for a number. Evolution does not get to choose the predator's head start, and it cannot re-roll a bad one. A strategy whose payoff is the same on average but never catastrophic beats one that is sometimes free and sometimes fatal, because the fatal branch does not get a second turn.
Two more things to try. Press Add three rival predators: the advantage survives and shrinks honestly, from certain death to about 73% eaten, not to nothing — piling up enemies on different clocks is the strongest objection to the whole story. Then push the predator's cycle to 13 while the brood sits at 13. The prime brood is eaten every single time. So the mechanism was never primality. It was coprimality: sharing no factor with whatever you are running from. Being prime is simply the cheapest way to be out of step with everything smaller than you, and it stops working the instant something matches you exactly.
The rule, exactly. A brood emerging every L years and a predator peaking every p years land in the same year only at multiples of their lowest common multiple. So:
when both start in step: share eaten = gcd(L, p) / p years between eaten emergences = lcm(L, p)Give the predator a head start of φ years and this changes. Write g = gcd(L, p). The brood's emergence years only ever land on multiples of g in the predator's cycle, so:
eaten g/p of the time on p/g of head starts, and never on the rest — average over head starts = 1/p, for every LAn earlier draft of this page stated only the aligned case, and a review pushed on it correctly: stated that way the page implies primes are eaten less on average, which is false. They are not. The average is identical for every life cycle; coprimality buys zero variance, not a lower mean. The correction is now the point of the exhibit rather than a footnote to it.
The predator is treated as a switch that is either at a peak or not, and "eaten" means the emergence meets a peak. "Safest" everywhere on this page means one thing only: the lowest share of emergences meeting a peak, under this binary overlap model, against the predators currently set. The timeline above is a 140-year window; every percentage is computed over the full repeat cycle, which for some pairs is far longer than the window shows. Only the watched predator takes the head start — the three rivals stay aligned — so the button has exactly one meaning. Real periodical cicadas also use predator satiation — emerging in numbers so vast that the local predators are full long before the brood is gone — and the two ideas work together: satiation needs the whole brood to arrive on one day, which needs a shared clock, and a shared clock is what makes the arithmetic here apply.
Verified in node (improve/verify/122-cicada-primes.js, 17 checks, all exact
rather than sampled — the first twelve before this page existed, the last five added
the same day when the review above forced the correction):
Which leaves the honest gap. This page explains why a periodical cicada's cycle should be prime. It does not explain why the surviving broods are 13 and 17 rather than 7 or 11, and neither does the arithmetic — the leading accounts add a minimum development time in a cold climate and, importantly, the avoidance of hybridising with neighbouring broods, since two broods on coprime cycles also rarely meet each other. The prime-cycle hypothesis (Lloyd and Dybas, 1966; Goles, Schulz and Markus, 2001; Yoshimura, 1997) is a live and contested idea, not settled fact, and the part that is not in dispute is the arithmetic above.
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