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Math • Collatz Conjecture Calculator

Collatz Conjecture Calculator

Compute the hailstone (Collatz, 3n+1) sequence for any starting integer, including step count and peak value.

Stopping Time (Steps to 1)
111 steps
Peak Trajectory Value
9,232
Trajectory 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
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Calculation Guide & Reference

Collatz Conjecture (3n+1) Mathematics

Collatz conjecture calculator computes the hailstone (3n+1) sequence for any starting integer, showing the step count and peak value.

Standardized Mathematical Formula
n → n/2 if even, n → 3n+1 if odd

Starting from any positive integer, repeatedly halve it if even, or multiply by 3 and add 1 if odd. The (unproven) Collatz conjecture claims this sequence always eventually reaches 1, for every starting number.

Variables:
n:The current value in the sequence
How It Works (Step-by-Step)
  • 1Enter a starting positive integer.
  • 2View the full hailstone sequence, the number of steps to reach 1, and the peak value reached along the way.
Real-World Numerical Example
Starting at 27

Compute the Collatz sequence starting from 27, a classic example known for its long, high-flying trajectory.

•Apply the rule repeatedly: 27 → 82 → 41 → 124 → 62 → 31 → ... continuing until reaching 1.
•The sequence takes 111 steps to reach 1.
•Along the way it peaks at 9,232 — over 340 times the starting value.
Result: Starting from 27, the sequence takes 111 steps to reach 1, peaking at 9,232.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

Also called the 3n+1 problem or hailstone sequence: for any positive integer, repeatedly apply "halve if even, triple-and-add-1 if odd," and the conjecture claims you always eventually reach 1.

27 takes 111 steps to reach 1 and climbs as high as 9,232 along the way — an unusually long and dramatic trajectory for such a small starting number, making it a popular illustrative example.

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