Hail — 9x

What you are looking at

Pick a whole number. If it is even, halve it. If it is odd, triple it and add one. Repeat.

Start at 6 and you get 6, 3, 10, 5, 16, 8, 4, 2, 1. Start at 27 and you get a number that climbs to 9,232 before coming down, taking 111 steps to reach 1. The values rise and fall like hailstones in a thundercloud, which is where the name comes from.

The Collatz conjecture says that every starting number eventually reaches 1. It has been checked by computer for every starting value past 2⁶⁸ — roughly 2.95 × 10²⁰ — without a single exception. It is also completely unproved, and the tools that would normally attack it do not seem to apply. Paul Erdős, who paid cash for solutions to problems he liked, said of this one: "Mathematics is not yet ready for such problems," and offered $500 anyway.

The coral view is the one worth sitting with. Take every number from 1 up to your limit, compute its path down to 1, then reverse it, so every path starts at 1 and grows outward. Draw each path as a walk: one fixed step forward per term, bending slightly one way on an even term and the other way on an odd one. Because thousands of numbers share the tail end of their journey, the strands lie exactly on top of each other near the root and only separate as the numbers get rarer. What grows is not a diagram of a tree. It is a tree, and it is made entirely out of arithmetic.

The orbit view is the plainer one: a single number's trajectory plotted against time, on a logarithmic scale, with the peak and the step count called out. It is worth flipping between the two. The coral shows you why the problem looks tractable; the orbit shows you why it is not.

Try this

In orbit view, try 27. Then 703, 871, 6171, 77031. Each is a record-holder: the longest path of any number below it. They look like nothing special.
Try 27 and then 28. One takes 111 steps. Its neighbour takes 18. There is no continuity here at all.
Any power of two falls straight to 1 in a clean staircase. Try 1024, then try 1023.
In coral view, push the count to ten thousand and lean the even and odd angles apart. The thing turns into seaweed. Bring them close together and it becomes a fan.
Set the odd angle to zero. The tree collapses into a spiral, because now only the halvings turn — and you are looking at a picture of how much of the journey is just dividing by two.

Notes

The problem is usually credited to Lothar Collatz around 1937, and it has collected an unusual number of names — the 3n + 1 problem, the Syracuse problem, Kakutani's problem, Ulam's conjecture, the Thwaites conjecture — which is what happens when several people independently find a problem irresistible.

The strongest result to date is Terence Tao's, from 2019: almost all Collatz orbits attain almost bounded values. That is a genuine and hard-won theorem, and note carefully how far it is from the conjecture. "Almost all" leaves room for exceptions. "Almost bounded" leaves room for growth. After ninety years, that is the shape of the ground gained.

A heuristic for why it probably is true: an odd step multiplies by 3 and adds 1, then the result is necessarily even, so it is immediately halved. Net effect over an odd-then-even pair, roughly ×3/2. Even steps are ×1/2. If odd and even values turned up at random with equal frequency, the average multiplier per step would be √(3/2 × 1/2) ≈ 0.866, comfortably less than one, so orbits should shrink geometrically. That argument is not a proof and never will be — it assumes the very randomness that would have to be established — but it is why almost everyone believes the conjecture.

What you are looking at

Try this

Notes

Further reading