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One rule, four words long: halve it if even, triple it and add one if odd. Every number anyone has ever tested eventually falls to 1, machine search has checked every start below 2^68, and nobody can prove it always happens. This is the shortest route on the platform from "I understand the rule completely" to "and nobody knows why it works", and it needs no physics, no calculus and no trust in a model — every number here is an exact integer you could check by hand.
Before you start — None whatsoever. If you can halve an even number and triple an odd one, you have the prerequisites. The student tier uses averages and a little probability; the advanced tier writes out every formula the code computes.
Take 27. It is odd, so it becomes 82; even, so 41; odd, so 124. Simple enough to do on paper — and it climbs to 9232 before it comes down, taking 111 steps to reach 1. Nothing about 27 announces that. The number beside it, 26, is finished in ten.
Do — Open the classic map and read the outcome counts under the chart: sixty-four starts, sixty-four of them reached 1. Then read Convergence — it says 100, and the lab will keep telling you that 100 is evidence and not a proof.
Stopping time — how many steps a start takes to reach 1 — is not smooth in the starting number. It depends on the exact binary expansion, so two consecutive integers can differ by a factor of ten with nothing to warn you. That irregularity is not noise on top of a pattern; as far as anyone can tell, it IS the pattern.
Do — Open 26 and 27 together. Ten steps against 111, and the Spread gauge reads 91. Now widen the span and watch how little the picture settles down: more numbers do not make it tidier.
An odd number always produces an even one, so every tripling is followed by at least one halving — on a random-looking odd number, two of them on average. Three times a number, halved twice, is three quarters of what you started with. That is the heuristic every mathematician reaches for: the typical orbit drifts DOWN, by a factor of about 3/4 per tripling. It explains the behaviour of the average orbit and says nothing whatever about any particular number, and closing that gap is the whole difficulty.
Do — Open the record holder — 837799, the longest stopping time below a million, at 524 steps. Read the Odd-share gauge: about a third, the same as everywhere else. Then hold that number in mind for the next step.
Multiply by 5 instead of 3 and the same argument runs backwards: five times a number, halved twice, is five quarters of what you started with, so the typical orbit drifts UP. Orbits climb until the lab stops following them. This is what makes 3n+1 special — and note where the specialness lives: not in the shape of the rule, but in whether 3/4 is less than 1.
Do — Open the 5n+1 map. Convergence collapses; the outcome counts show orbits leaving the window instead of arriving at 1. Check the Odd-share gauge — it has barely moved. The ratio of halvings to triplings is not what changed.
There are exactly two ways the conjecture could be false: an orbit that climbs forever, and an orbit that falls into a loop which does not contain 1. The second one is not hypothetical in general — change the addend from 1 to 5 and the loops are right there, four or eight numbers long, catching starts that will never reach 1 no matter how long you wait. A proof of the Collatz conjecture has to rule out BOTH, for every integer, forever. Nobody has ruled out either.
Do — Open the 3n+5 map and read the outcome counts: most starts end in a cycle rather than at 1. Follow one orbit in the chart and watch it come back to where it has already been. Then ask the honest question the whole lab is built around — what, exactly, would you have to prove to be sure this never happens for 3n+1?
How many of the numbers you tried ended up at 1. For the ordinary 3n+1 rule this reads 100 every time — every number anyone has ever tested reaches 1. Nobody has been able to prove that it always will, and that is the open problem: the gauge is full, and the question is still open.
Try — Leave the start where it is and move the multiplier from 3 to 5. Convergence falls off a cliff, and the outcome counts underneath tell you why: some orbits now loop forever and some climb until the lab stops following them.
Out of every hundred steps, how many were the "multiply" kind rather than the "halve" kind. It comes out near 33 almost whatever you do — roughly two halvings for every multiplication. That ratio is the reason the ordinary rule tends downward: multiplying by 3 and then halving twice leaves you smaller than you started.
Try — Read this gauge with the multiplier at 3, then at 5, then at 7. It barely moves. The ratio of halvings to multiplications is not what changes between a map that falls and a map that climbs — what changes is how far each multiplication throws you.
Neighbouring numbers do not behave like neighbours here. 26 reaches 1 in ten steps; 27 takes 111. This measures how far apart the quickest and slowest starts in your survey were — high means the numbers you tried had almost nothing in common, which is the normal state of affairs.
Try — Set the span to 2 and start at 26. Two consecutive integers, ten steps against 111, and the gauge reads 91. Nothing about 26 and 27 predicts that difference.
How far the highest flyer in your survey climbed above where it started, counted in binary digits. 27 starts as a 5-digit binary number and reaches a 14-digit one — it climbs to 9232 before it ever comes down. Measured in digits rather than in plain size, so a small number that grows enormously scores higher than a large number that grows a little.
Try — Start at 27 with a span of 1 and read the altitude, then start at 704511 — the highest-climbing start below a million in absolute terms. Its altitude score is LOWER, because it was already a big number. That is the gauge working as intended.
How much of your survey was retreading ground. Once two starting numbers land on the same value, they follow exactly the same path from then on — so orbits merge, like streams joining a river, and most of the work is done more than once. A high number here is the shape of the problem: everything seems to pour into the same channel.
Try — Widen the span from 1 to 128 and watch coverage climb: the more neighbours you survey, the more of them are already downstream of one another. Then switch the addend to 5 and watch what merging does when the tree has more than one root.
Surfacing sources you can verify is a deliberate anti-pseudoscience measure, not a bibliography. Nothing on this page asks to be taken on trust.
Glossary — every term used above, defined once.