Big Maths Ideas · Beta

The Collatz conjecture

Trace the exact 3n + 1 journey of a positive integer.

If the number is even, halve it. If it is odd, multiply by 3 and add 1. Repeat with the new number.

n\mapsto\begin{cases}n/2&n\text{ even}\\3n+1&n\text{ odd}\end{cases}

This start reaches 1 after 111 steps.

Inspected step: 0
Exact current value: 27
Next operation: Odd: 3 × 27 + 1 = 82
Completed steps: 111
Largest exact value: 9232

Exact sequence values

27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → 137 → 412 → 206 → 103 → 310 → 155 → 466 → 233 → 700 → 350 → 175 → 526 → 263 → 790 → 395 → 1186 → 593 → 1780 → 890 → 445 → 1336 → 668 → 334 → 167 → 502 → 251 → 754 → 377 → 1132 → 566 → 283 → 850 → 425 → 1276 → 638 → 319 → 958 → 479 → 1438 → 719 → 2158 → 1079 → 3238 → 1619 → 4858 → 2429 → 7288 → 3644 → 1822 → 911 → 2734 → 1367 → 4102 → 2051 → 6154 → 3077 → 9232 → 4616 → 2308 → 1154 → 577 → 1732 → 866 → 433 → 1300 → 650 → 325 → 976 → 488 → 244 → 122 → 61 → 184 → 92 → 46 → 23 → 70 → 35 → 106 → 53 → 160 → 80 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

The idea behind the experiment

The Collatz conjecture concerns every positive integer and remains unproved. Values here are exact integers; only the logarithmic graph is approximate. Reaching 1 for one start gives evidence, not a proof for all starts.

Emory Maths on Collatz

Predict, test and explain

Step 1 of 3

Predict

Starting at 6, how many steps does the Collatz rule take to reach 1?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Starting at 6, how many steps does the Collatz rule take to reach 1?

$6\to3\to10\to5\to16\to8\to4\to2\to1$

Hint

Apply the rule to the current value at each step.

Reveal answer

6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1 has eight transitions.

$8\text{ steps}$

Build an explanation

Hint 1

Apply the rule to the current value at each step.

Hint 2

Odd numbers may first increase before later decreases.

Hint 3

A computation stopped at its cap is not a proven counterexample.

Worked example

This example uses fixed values, separate from the diagram controls.

Trace the journey from 6.

6 is even, giving 3; 3 is odd, giving 10.
Continue: 10, 5, 16, 8, 4, 2, 1.
Count transitions rather than displayed values.

Answer: Eight steps.

Connect this idea

Watch out: There are nine displayed values but eight steps between them.