Collatz Steps Calculator
The Collatz process, also called the 3n plus 1 problem, applies a simple rule repeatedly: if the current number is even, halve it; if it is odd, multiply by three and add one. Starting from any positive whole number and following the rule, the sequence appears always to reach one, though no one has ever proved it must for every starting value. This calculator takes a starting number and counts how many steps the process takes to first reach one. Each application of the rule is one step, so the count is the length of the path down to one. For a start of six the path is six, three, ten, five, sixteen, eight, four, two, one, which is eight steps. The calculator also shows the peak value the sequence reaches, since trajectories can climb high before falling. Mathematicians, programmers and puzzle lovers explore the Collatz problem because of its baffling mix of simple rule and unproven behavior. Enter a starting whole number to get the step count immediately; the input is a plain integer, so no decimal formatting applies. Every figure here is computed deterministically from the rule shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
The Collatz rule halves even numbers and does 3n + 1 on odd ones. Starting at 6, the sequence reaches 1 in 8 steps.
Collatz Steps formula
if n is even: n -> n / 2
if n is odd: n -> 3n + 1
repeat until n = 1
each application is one step
conjecture: every start reaches 1
Apply the rule over and over: halve even numbers, triple-plus-one odd numbers. Count each application as a step until the sequence first reaches one.
Worked example
Count the Collatz steps starting from 6.
- 6 is even -> 3 (step 1); 3 is odd -> 10 (step 2); 10 -> 5 (step 3)
- 5 -> 16 (step 4); 16 -> 8 (step 5); 8 -> 4 (step 6)
- 4 -> 2 (step 7); 2 -> 1 (step 8). Total: 8 steps
Starting from 6 it takes 8 steps to reach 1. This is the calculator's default input, so the result above matches the widget exactly.
Collatz steps for small starts
Steps to first reach 1.
| Start | Steps | Peak |
|---|---|---|
| 6 | 8 | 16 |
| 7 | 16 | 52 |
| 11 | 14 | 52 |
| 27 | 111 | 9,232 |
| 97 | 118 | 9,232 |
Computational reference: US National Institute of Standards and Technology (NIST).
Collatz Steps Calculator: frequently asked questions
What is the Collatz conjecture?
It is the unproven claim that starting from any positive whole number and repeatedly halving when even or tripling-and-adding-one when odd, you always eventually reach one. It has been checked for enormous ranges of numbers, but no general proof exists, which is why it remains a famous open problem.
How are steps counted?
Each application of the rule counts as one step, whether it is a halving or a triple-plus-one. The count is how many applications it takes to first reach one. Starting from six, eight applications bring the sequence to one, so the step count is eight.
What is the peak value?
It is the largest number the sequence reaches before falling to one. Collatz trajectories often climb well above the starting value before descending. The calculator reports this peak alongside the step count, since the climb can be dramatic even for small starts.
Does every number really reach one?
It does for every number ever tested, which is an astronomically large range, but no one has proved it must hold for all numbers. The conjecture could in principle fail for some untested value by entering a loop or growing without bound, though that is considered unlikely.
What is the Collatz rule?
Halve the number if it is even, or multiply by three and add one if it is odd, repeating until you reach one. Starting at six the path is 6, 3, 10, 5, 16, 8, 4, 2, 1, which is eight steps.
Official sources
- Mathematical functions and integer-sequence reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.