Collatz Lab

n → n/2 · n → 3n + 1

Binary digits of each value along the orbit of 26. Height is size, and the climb in the middle is what a proof would have to rule out.

Steps to finish, per starting integer. Neighbours are not neighbours here.

Outcomes

Reached 1: 2
Fell into a loop: 0
Left the window: 0
Still running at the cap: 0

Highest value reached: 9232, from 27.

The survey

First integer

Integers surveyed

The map

Odd-step multiplier

Odd-step addend

Metrics

Total298 / 500

CONVERGENCE100

What is convergence?

Computational experiment

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.

ODD SHARE36

What is odd-step share?

Established physics

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.

Lagarias — "The 3x+1 problem and its generalizations" (Amer. Math. Monthly 92, 1985) and the annotated bibliography

SPREAD91

What is spread?

Established physics

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.

Lagarias — "The 3x+1 problem and its generalizations" (Amer. Math. Monthly 92, 1985) and the annotated bibliography

ALTITUDE64

What is altitude?

Established physics

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.

Lagarias — "The 3x+1 problem and its generalizations" (Amer. Math. Monthly 92, 1985) and the annotated bibliography

COVERAGE7

What is coverage?

Established physics

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.

Convergence 100 means every start in this survey reached 1. That is the evidence for the conjecture, not a proof of it — the same search has been pushed to 2^68 and the question is still open.

Loading lab…

One rule, no exceptions found, no proof either.

Take any positive integer. If it is even, halve it. If it is odd, triple it and add one. Repeat. Lothar Collatz asked in 1937 whether this always reaches 1, and the answer is still unknown — despite the sequence being simple enough to explain to a child and despite every one of the first 2⁶⁸ integers having been checked by machine. Paul Erdős said of it: “Mathematics is not yet ripe for such problems.”

There are exactly two ways the conjecture could be false, and this lab can show you both — not for 3n+1, where nobody has found either, but in its neighbours. Change the multiplier to 5 and orbits climb out of the window instead of coming down: that is what a divergent trajectory looks like. Change the addend to 5 and orbits fall into loops that do not contain 1: that is what a rogue cycle looks like. A proof of the Collatz conjecture has to rule out both, for every integer, forever, and the fact that the same rule shape produces both a few clicks away is the reason the problem is hard.

Every number here is an exact integer and every operation is addition, multiplication, halving or a comparison. There is no seed, no step size and no floating-point model, so two runs of the same configuration are identical everywhere, which is what lets the server re-derive your score from your parameters alone. The lab stops following any orbit past 2⁴⁸ and reports that as leaving the window rather than as divergence — it cannot tell an orbit that would come back at 2⁶⁰ from one that never will, and neither can anybody else.