Collatz Conjecture Calculator

Explore the mysterious 3n+1 problem and watch numbers journey to 1.

Last updated: 2026-05-24T22:58:31.873Z

Starting Number

Maximum Steps

Steps to 1

111

Max Value

9,232

Sequence (first 50 terms)

n=0:27×3+1

n=1:82÷2

n=2:41×3+1

n=3:124÷2

n=4:62÷2

n=5:31×3+1

n=6:94÷2

n=7:47×3+1

n=8:142÷2

n=9:71×3+1

n=10:214÷2

n=11:107×3+1

n=12:322÷2

n=13:161×3+1

n=14:484÷2

n=15:242÷2

n=16:121×3+1

n=17:364÷2

n=18:182÷2

n=19:91×3+1

n=20:274÷2

n=21:137×3+1

n=22:412÷2

n=23:206÷2

n=24:103×3+1

n=25:310÷2

n=26:155×3+1

n=27:466÷2

n=28:233×3+1

n=29:700÷2

n=30:350÷2

n=31:175×3+1

n=32:526÷2

n=33:263×3+1

n=34:790÷2

n=35:395×3+1

n=36:1,186÷2

n=37:593×3+1

n=38:1,780÷2

n=39:890÷2

n=40:445×3+1

n=41:1,336÷2

n=42:668÷2

n=43:334÷2

n=44:167×3+1

n=45:502÷2

n=46:251×3+1

n=47:754÷2

n=48:377×3+1

n=49:1,132÷2

... and 62 more steps

What is the Collatz Conjecture?

The Collatz conjecture, also known as the 3n+1 problem, is one of mathematics' most famous unsolved problems. It states that for any positive integer, if you repeatedly apply a simple rule, you will eventually reach the number 1. Despite its simplicity, no one has been able to prove this is true for all numbers.

The rule is: if the number is even, divide it by 2. If it's odd, multiply by 3 and add 1. For example, starting with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. The conjecture has been verified for all numbers up to 2^68 (about 295 quintillion), but a general proof remains elusive.

The Collatz Algorithm

The Rules

If n is even:

n → n ÷ 2

If n is odd:

n → 3n + 1

Repeat until n = 1

Key Observations

• Numbers eventually reach a cycle: 4 → 2 → 1 → 4
• The path can be extremely long (e.g., 27 takes 111 steps)
• Values can spike dramatically before descending
• No mathematical proof exists yet (Millennium Prize Problem)
• Verified up to 2^68 ≈ 295 quintillion

Worked Example: Starting with 6

Follow the sequence step by step:

Start: n = 6 (even)

Step 1: 6 ÷ 2 = 3 (now odd)

Step 2: 3 × 3 + 1 = 10 (now even)

Step 3: 10 ÷ 2 = 5 (now odd)

Step 4: 5 × 3 + 1 = 16 (now even)

Step 5: 16 ÷ 2 = 8 (now even)

Step 6: 8 ÷ 2 = 4 (now even)

Step 7: 4 ÷ 2 = 2 (now even)

Step 8: 2 ÷ 2 = 1 (reached goal!)

Reached 1 after 8 steps

Frequently Asked Questions

Is the conjecture proven?

No, despite being studied for decades, no proof exists. The Collatz conjecture remains an open problem in mathematics.

Why is it important?

While simple, it reveals deep complexities about iteration and unpredictable behavior in simple systems. It has connections to chaos theory.

Can very large numbers break the pattern?

Not found yet. All numbers tested (up to 2^68) eventually reach 1, supporting but not proving the conjecture.

What's the longest sequence found?

For starting numbers under 1 billion, 63728127 takes 949 steps to reach 1 with a maximum value

Does the starting number matter?

Yes. Some numbers reach 1 quickly (e.g., 5 takes 5 steps), while others take hundreds or thousands of steps.

What if we change the rule?

Mathematicians have studied variants like 5n+1, which exhibits different behavior and often doesn't always reach 1.

Is this related to fractals?

Yes! The Collatz sequence has fractal-like properties and is studied in chaos theory research.

Why start with positive integers?

Negative integers and zero have different behaviors. The conjecture specifically applies to positive integers > 0.

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