Math
Collatz Conjecture Calculator
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 always reach 1, and nobody has proved it. This calculator follows the whole hailstone sequence for your number, counts the steps, finds the peak, and charts the ride.
Any positive whole number up to 40 digits. Try 27, 97, 871, or 6171.
Try an example
Result
Steps to reach 1
—
- Highest value
- —
- Odd steps
- —
- Even steps
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
The rule
Starting from 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1, eight steps. Starting from 27 takes 111 steps and climbs as high as 9,232 before falling back to 1. The up-and-down path is why these are called hailstone sequences.
What the conjecture claims
Lothar Collatz proposed in 1937 that every positive whole number eventually reaches 1. Computers have verified it for every starting value below about 2^68 (roughly 3 × 10^20), yet no proof exists. Once you reach 1 the sequence cycles 1 → 4 → 2 → 1 forever, so the process stops there.
Notable starting values
| Start | Steps to 1 | Peak |
|---|---|---|
| 1 | 0 | 1 |
| 6 | 8 | 16 |
| 7 | 16 | 52 |
| 27 | 111 | 9,232 |
| 97 | 118 | 9,232 |
| 871 | 178 | 190,996 |
| 6,171 | 261 | 975,400 |
| 77,031 | 350 | 21,933,016 |
Reading the results
- Steps (also called total stopping time) counts every halving and every 3n + 1 move until 1 appears.
- The peak is the largest number visited. Odd steps push the value up; even steps pull it down.
- On average a 3n + 1 step is followed by at least one halving, so sequences drift downward overall, which is the heuristic reason the conjecture is believed.
- The chart uses a log scale so a peak of millions and a finish at 1 fit on one picture.
How to use this calculator
- Type a positive whole number (commas are fine).
- Read the step count, peak, and odd/even breakdown.
- Scroll the full sequence; the peak is highlighted.
- Open the work to see the first steps written out with the rule that produced each one.
Common mistakes
- Starting from 0 or a negative number: the conjecture is about positive whole numbers only.
- Counting the starting number as a step: 1 takes 0 steps and 2 takes 1 step.
- Assuming bigger starts take more steps: 27 takes 111 steps but 28 takes only 18.
Worked examples
The famous 27
111 steps, peak 9,232
Steps to reach 1
111
Short run
6 reaches 1 in 8 steps
Steps to reach 1
8
Already 1
Zero steps
Steps to reach 1
0
Power of 2
1024 just halves 10 times
Steps to reach 1
10
Long climb
6171 takes 261 steps
Steps to reach 1
261
Large start
A 15-digit number
Steps to reach 1
290
Invalid
Zero is not allowed
Steps to reach 1
—
Frequently asked questions
What is the Collatz conjecture?+
The claim that repeatedly applying n → n/2 (even) or n → 3n + 1 (odd) to any positive whole number eventually reaches 1. It is unproven but verified for enormous ranges.
How many steps does 27 take?+
111 steps, reaching a peak of 9,232 along the way, which is remarkable for such a small start.
Why is it called the hailstone sequence?+
Because the values rise and fall repeatedly, like hailstones tossed up and down in a storm cloud, before finally dropping to 1.
Has anyone found a number that does not reach 1?+
No. Every starting value tested, up to about 2^68, reaches 1. But testing is not a proof, and the general case remains open.
What happens after reaching 1?+
1 is odd, so 3(1) + 1 = 4, then 4 → 2 → 1. The process cycles forever, which is why calculators stop at the first 1.
Do powers of 2 have the shortest sequences?+
Yes. 2^k only halves, taking exactly k steps with no 3n + 1 moves.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026