Pick any positive whole number. If it is even, halve it; if it is odd, triple it and add 1. Repeat. The Collatz conjecture says you will always reach 1, whatever number you start with. A ten-year-old can check it, computers have confirmed it for every number below (about 2.4 sextillion), and still nobody can prove it.
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The rule
Starting from 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. That is 8 steps. Once you reach 1, the rule sends you around the loop forever.
Conjecture (Lothar Collatz, 1937): every starting number eventually reaches 1.
The famous case of 27
Small numbers usually finish fast, which makes 27 a shock. It takes 111 steps to reach 1, and on the way it climbs as high as 9,232 before falling back down. Its neighbor 28 needs only 18 steps. Paths look random: they rise and fall with no visible pattern, which is why the sequence is nicknamed the “hailstone” sequence.
Why is it so hard to prove?
To prove the conjecture you would have to rule out two kinds of failure:
- A different loop: a starting number that cycles forever without ever hitting 1.
- Escape to infinity: a starting number whose path keeps growing forever.
A heuristic suggests why paths usually shrink. After an odd step, is even, so it is always halved at least once. On average an odd number gets multiplied by about per round trip, so the numbers tend to drift down. But “on average” is not a proof: the conjecture is about every number, and averages say nothing about a single exceptional one.
The strongest result so far is due to Terence Tao (2019): almost all starting numbers, in a precise statistical sense, reach values that are almost bounded. It is the closest anyone has come, and it still does not settle the question. The mathematician Paul Erdős reportedly said that mathematics may not be ready for problems like this one.
“I think I found a counterexample”
Every claimed counterexample so far has been a mistake. Before believing one, check:
- Size: David Barina’s computer search, published in 2025, verified every starting number below . A counterexample must be larger than that.
- Arithmetic: a single wrong step, often an odd number treated as even, makes any path look strange. Recompute with exact integer arithmetic, not floating point, which rounds large numbers.
- What you actually found: a real counterexample is either a loop that avoids 1 (show the full loop) or a proof that a path grows forever. “It is still going after a million steps” is neither.
Try these
Puzzle 1
How many steps does 7 take to reach 1, and what is the highest number it reaches?
Show Answer
7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1. That is 16 steps, with a highest value of 52.
Puzzle 2
Which starting numbers reach 1 in exactly 4 steps?
Show Answer
Work backward from 1. A number can come from (always) or from (when that is an odd whole number). Going back 4 steps: . The only other branch would be , which is not a new number. So the only answer is 16.
Puzzle 3
Show that a starting number of the form (with ) drops below its starting value within 3 steps.
Show Answer
is odd, so the next value is . That is even, giving , and again even, giving . Since for , the path drops below the start after 3 steps. This is the kind of argument computer searches use to skip most numbers.
Related
Frequently asked questions
Has the Collatz conjecture been proven?
No. It is still open. It has been checked by computer for every starting number below , and Terence Tao proved in 2019 that it holds in a weaker sense for almost all numbers.
Is there a prize for solving it?
Paul Erdős offered a 500-dollar prize, and later private prizes have been announced, but it is not one of the Clay Millennium Prize Problems.
Why is it called the 3n + 1 problem?
Because of the odd step. It also goes by the Syracuse problem, Ulam’s problem and the hailstone problem.