Take any whole number. If it is odd, triple it and add one. If it is even, halve it. Repeat. Every number ever tested crashes down to 1, and nobody can prove that all of them must. This page is an atlas of that storm: what the rule does, and the strange structures it leaves behind.
Watch a single number ride the rule. Amber steps are the odd moves,
where the number roughly triples. Cyan steps are the even moves, where it halves.
The height axis is logarithmic: each gridline is a power of two, so tripling and halving become
straight climbs and drops. Seed 27 is the classic surprise: it needs 111 steps and
climbs to 9,232 before it dies.
This is the entire unsolved problem. No number theory course is required to state it, which is exactly why Paul Erdős said mathematics is not yet ready for such questions.
Run the rule backwards and all flights fuse into a single tree rooted at 1: every number
n has the child 2n, and when n ≡ 4 (mod 6) it also has the child (n − 1)/3.
Draw the tree with one bending rule, lean one way for a doubling and
the other way for a (n − 1)/3 branch, and pure arithmetic grows
something organic. The balance of the two turns is the balance of halvings and triplings inside every
flight; the coral's overall curl is the 0.415-bit drift of Chapter 3, made visible as posture.
A visual form made famous by the mathematician Edmund Harriss. Nudge the two turn angles and the species changes: kelp, fern, smoke. The arithmetic underneath never changes. Trace a seed and its whole forward flight lights up as one path sliding down the branches to the root, because running the rule forward is exactly walking this tree toward 1.
Every seed up to 1,000 flies here on the same log-scale axes. Most die in a few dozen
steps, but a rare seed breaks the flight-time record and is drawn in amber.
The heuristic behind the downward pull: each tripling is followed, on average, by two halvings,
so a typical flight loses about 2 − log₂3 ≈ 0.415 bits per odd step.
On average everything sinks. The conjecture is that nothing escapes the average forever.
Record flights up to 1,000: seed 27 holds out for 111 steps, then 54, 73, 97, 129, 171, 231, 313, 327, 649, 703 and finally 871, which flies for 178 steps and peaks near 190,996.
Now run every seed at once. Each drop sits at its seed's position on the horizontal axis and at its current value's height on the log axis. On every tick, all drops apply the rule together: a flash of amber when one triples upward, cyan as it halves down. Drops that reach 1 fall out of the sky. Watch the population thin: half the sky is gone within about 30 steps, while a few stubborn cohorts, the descendants of 27's storm, hang in the air past step 170.
The survival curve at the bottom is the distribution of total stopping times, drawn live. Its long thin tail is the whole difficulty of the conjecture: proving the tail never becomes infinite.
One dot per seed: horizontal position is the seed, height is how many steps it takes to reach 1. A hundred thousand dots do not fall randomly. They condense into drifting bands and rays, because trajectories merge: once two seeds funnel into the same number, their flight lengths differ by a constant forever after. Each visible ray is a family of seeds sharing one downstream destiny. Amber marks the record breakers.
Hover to read any seed's flight length. The longest flight below 100,000 belongs to 77,031, which takes 350 steps.
Wind the integers into an Ulam spiral: 1 at the center, counting outward turn by turn. Then launch every seed up to 10,000 and let each cell glow by how many flights pass through it. The traffic map that emerges is the conjecture's road network. The blazing core is the 4 → 2 → 1 drain every flight must use. Powers of two burn bright as the off-ramps of that drain, and the diagonal textures are arithmetic families that share highway segments. Switch modes to color each cell instead by its own flight length.
Hover any cell to read its number, its traffic count, and its own flight length. Traffic is drawn on a logarithmic brightness scale; the busiest cells near the center are crossed nearly 10,000 times while the dim mist is crossed once or twice.
In binary, the rule becomes a machine. Halving just deletes the last bit: trivial.
But 3n + 1 is n + (n << 1) + 1: the number is added to its own shifted
copy, and carries ripple through the bits. Each row below is one step of a flight, least
significant bit on the right. Amber rows are odd and about to triple;
cyan rows are even and about to shrink. The unsolved question, restated: prove
that the carry ripples can never keep a bit pattern wide forever.
Seed 255 is eight ones in a row: watch tripling stretch the solid block before the collapse. Hover a row to read its decimal value and the move it is about to make.