Collatz Conjecture Calculator
Explore the 3x+1 problem and generate hailstone sequences
Generate Collatz Sequence
Try any positive integer (e.g., 7, 11, 27)
Safety limit to prevent infinite loops
Collatz Sequence Results
Hailstone Sequence
✅ Conjecture Confirmed!
The sequence reached 1 after 16 steps, supporting the Collatz conjecture.
Famous Collatz Examples
Starting with 7
Sequence: 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
Length: 17 steps
Peak: 52
Starting with 11
Sequence: 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
Length: 15 steps
Peak: 52
Starting with 27
Sequence: 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
Length: 112 steps
Peak: 9,232
Starting with 31
Peak Value: 9,232 (after many steps)
Length: 107 steps to reach 1
This number demonstrates the wild "hailstone" behavior with dramatic ups and downs!
The Collatz Rules
Quick Facts
Also known as
3x+1 problem, Hailstone sequence, Syracuse problem
Conjecture
Every positive integer eventually reaches 1
Tested up to
2^95 ≈ 295 quintillion
Status
Unproven - still an open problem!
Key Terms
Stopping Time
Steps to reach a value smaller than starting number
Total Stopping Time
Steps to reach 1 for the first time
Hailstone
Name for the up-and-down pattern like hail in clouds
4-2-1 Loop
The final cycle: 4 → 2 → 1 → 4 → 2 → 1...
Understanding the Collatz Conjecture
What is the Collatz Conjecture?
The Collatz conjecture, also known as the 3x+1 problem, is one of the most famous unsolved problems in mathematics. It states that no matter what positive integer you start with, if you repeatedly apply the simple rules, you will always eventually reach the number 1.
The Rules
- If the number is even: Divide it by 2
- If the number is odd: Multiply by 3 and add 1
- Repeat: Continue until you reach 1
Why "Hailstone"?
The sequences are called "hailstone sequences" because they resemble the chaotic up-and-down movement of hailstones in storm clouds before finally falling to the ground (reaching 1).
The Mathematical Mystery
Simple Rules, Complex Behavior
Despite the simple rules, predicting the behavior of any given starting number is impossible without computing the sequence.
Extensive Testing
Every number tested so far (up to 2^95) has eventually reached 1, but no general proof exists.
Chaotic Nature
Small changes in starting numbers can lead to dramatically different sequence lengths and peak values.
Warning!
Many mathematicians have become obsessed with this problem. As the XKCD comic warns: "Don't spend too much time on it - it's a mathematical rabbit hole!"
Special Cases and Patterns
Powers of 2
Numbers like 2, 4, 8, 16, 32... have predictable, monotonically decreasing sequences.
16 → 8 → 4 → 2 → 1
Negative Numbers
Negative integers can enter cycles and never reach 1:
-1 → -2 → -1 → -2...
-5 → -14 → -7 → -20...
Record Holders
Some numbers have exceptionally long sequences or high peaks:
27 reaches 9,232 (112 steps)
63,728,127 has 949 steps
Formal Definition
The Collatz function C(n) is defined as:
⎩ 3n + 1 if n ≡ 1 (mod 2)
Conjecture: For any positive integer n, there exists some k ≥ 0 such that C^k(n) = 1, where C^k denotes applying the function C exactly k times.