3n+1 always comes home. Swap in a 5, and it mostly doesn't.

Take any number. If it's even, halve it; if it's odd, triple it and add one. Repeat. Every number ever tested — up to 300,000 of them — eventually reaches 1. Keep the exact same rule but swap that 3 for a 5, and most numbers never come home at all.

What you're seeing

The rule is deliberately tiny: if the current number is even, halve it; if it's odd, multiply it by k and add one. Start from n₀ and repeat. The top chart traces exactly that, one step at a time, value on a log scale so both the small numbers and the occasional huge spike stay visible on the same chart. At k=3 — the famous "Collatz" rule — the path can wander a long way before it commits: 27 claws up to 9,232, 342 times its own starting value, before it finally tumbles down to 1 after 111 steps. Try the neighboring numbers 26 and 27 back to back: 26 reaches 1 in a mere 10 steps, 27 takes 111. Two numbers one apart, wildly different journeys — non-monotonic, in the same way a coastline is jagged rather than smooth.

The bottom chart zooms out to a whole population: one thin bar for every starting number from 2 to 150, height equal to its stopping time. At k=1 and k=3, every single bar is cyan — every tested start reaches 1, some quickly, some (like 27, or 129 at 121 steps) only after a long climb. Switch to k=5 and watch the chart turn mostly red: bars that reach the very top of the chart are starting numbers that never make it to 1 at all. Within this window of 149 starting numbers, only 26 (17%) still reach 1. Another 23 (15%) fall into one of several exact repeating loops that skip 1 entirely. The remaining 100 — most of them — climb straight past a trillion within the step budget and show no sign of stopping.

That's the whole point of the toy: the k=3 rule looks so simple that "does it always come back to 1?" seems like it should have an easy answer, and every number anyone has ever checked — far more than the 300,000 checked here — says yes. But nobody has proved it for every number, and this exhibit doesn't pretend otherwise: verified over a range, not proven in general, is an honest and very different claim. Nudge one digit of the rule from 3 to 5, keeping everything else identical, and "always comes back" stops being true even in the tested range. Two lessons, then: "verified so far" and "true for a reason" are not the same guarantee — and a one-character change to a simple rule can flip its entire long-run behavior.

The rule, exactly. For odd multiplier k, step(n) = n/2 if n even, else k·n + 1 iterated from n₀ until it first reaches 1 (or is shown not to). Verified in node (improve/verify/77-collatz.js, 15 checks):

Every start from 1 to 300,000 reaches 1 under k=1, and provably fast: an odd number is always immediately followed by an even one, so it can never "bump" twice running.
Every start from 1 to 300,000 also reaches 1 under k=3. The longest stopping time found is 442 steps, at n=230,631; the largest peak-to-start ratio is 107,860.7×, at n=159,487.
The landscape really is non-monotonic: 10 steps at n=26, then 111 at n=27, then straight back down to 18 at n=28.
The known-bad case, k=5. A start under 2,000 is found whose trajectory never reaches 1 — not "hasn't yet," but a genuine closed cycle. Confirmed two ways: replaying 10 steps from 26 returns exactly to 26, and the cycle {26, 13, 66, 33, 166, 83, 416, 208, 104, 52} provably excludes 1.
Across 1 to 2,000 under k=5, only 114 starts (5.7%) reach 1. 143 (7.2%) fall into a non-1 cycle and 1,743 (87.2%) escape past 10¹² inside the step budget — the same split the smaller on-page window shows.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 77.