Collatz Conjecture Calculator
Free Collatz conjecture Calculator for sequences. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Collatz Conjecture Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: If n is even: n -> n/2 | If n is odd: n -> 3n + 1
Worked example — Stopping time: 111 steps | Peak value: 9,232 | Peak/Start ratio: 341.93x
Formula
If n is even: n -> n/2 | If n is odd: n -> 3n + 1
Starting from any positive integer, repeatedly apply these two rules. The Collatz conjecture states that this process will always eventually reach 1, regardless of the starting number. The sequence of values is also called hailstone numbers because they rise and fall like hailstones in a cloud.
Worked Examples
Example 1: Classic Example: Starting at 27
Problem:Compute the Collatz sequence starting at 27. How many steps does it take to reach 1, and what is the maximum value?
Solution:Starting at 27: 27 -> 82 -> 41 -> 124 -> 62 -> 31 -> 94 -> 47 -> 142 -> 71 -> ... The sequence climbs to a peak of 9,232 at step 77 After the peak, it descends through powers of 2 The sequence finally reaches 1 after 111 steps Total odd steps: 41 | Total even steps: 70
Result:Stopping time: 111 steps | Peak value: 9,232 | Peak/Start ratio: 341.93x
Example 2: Power of 2: Starting at 64
Problem:Compute the Collatz sequence starting at 64 (which is 2^6).
Solution:Starting at 64: 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1 Since 64 = 2^6, the sequence simply halves 6 times All steps are even-number divisions No odd steps occur, so the value never increases This is the shortest possible sequence for a number of this magnitude
Result:Stopping time: 6 steps | Peak value: 64 | All 6 steps are even divisions
Frequently Asked Questions
What is the Collatz conjecture and why is it famous?
The Collatz conjecture, also known as the 3n+1 problem, is one of the most famous unsolved problems in mathematics. The rule is simple: start with any positive integer, if it is even divide by 2, if it is odd multiply by 3 and add 1. The conjecture states that no matter what starting number you choose, the sequence will always eventually reach 1. Despite its simple formulation, no one has been able to prove this is true for all numbers, even after decades of effort by brilliant mathematicians. Paul Erdos famously said that mathematics is not yet ready for such problems.
How does the Collatz sequence work step by step?
Starting with any positive integer n, apply two rules repeatedly. Rule 1: if n is even, divide it by 2 (n becomes n/2). Rule 2: if n is odd, multiply by 3 and add 1 (n becomes 3n+1). Continue until you reach 1. For example, starting with 6: 6 is even, so 6/2=3. Then 3 is odd, so 3x3+1=10. Then 10/2=5, 5x3+1=16, 16/2=8, 8/2=4, 4/2=2, 2/2=1. The sequence is 6, 3, 10, 5, 16, 8, 4, 2, 1, taking 8 steps. The path can be surprisingly long and unpredictable even for small starting numbers.
What is the stopping time in the Collatz sequence?
The stopping time (also called total stopping time) is the number of steps required for the sequence to reach 1 from the starting number. Different starting values produce vastly different stopping times. The number 1 has a stopping time of 0. The number 2 takes 1 step. The number 27 is famous because despite being relatively small, it takes 111 steps and reaches a maximum value of 9,232 before finally descending to 1. There is no known formula to predict the stopping time without actually computing the sequence. This unpredictability is part of what makes the conjecture so challenging to prove.
What are some notable numbers in Collatz sequences?
The number 27 is perhaps the most famous example because it takes 111 steps and climbs to 9,232 before reaching 1, demonstrating how a small starting number can produce a long, wild trajectory. Powers of 2 have the shortest sequences since they simply halve down to 1 (for example, 64 takes only 6 steps). The number 9,663 takes 184 steps. Among the first million numbers, 837,799 has the longest sequence at 524 steps. The number 77,031 reaches a peak value over 21 million. These examples illustrate the chaotic and unpredictable nature of the sequences generated by this deceptively simple rule.
Has the Collatz conjecture been verified by computers?
Yes, computers have verified the Collatz conjecture for all starting numbers up to approximately 2.95 x 10 to the 20th power (as of recent computations). This means every number tested so far does eventually reach 1. However, computational verification, no matter how extensive, cannot constitute a mathematical proof because there are infinitely many positive integers. It remains possible, though increasingly unlikely, that some extremely large number could form an infinite loop or diverge to infinity. The computational evidence strongly supports the conjecture, but mathematics requires absolute proof for certainty.
Why does the 3n+1 operation tend to increase the value while n/2 decreases it?
When n is odd, the operation 3n+1 roughly triples the value, which always produces an even number that immediately gets halved. So the combined odd step effectively multiplies by about 3/2 or 1.5. Meanwhile, each even step halves the number. Statistically, about half the numbers in any sequence are even and half are odd, which means the average step multiplies by roughly (3/2)^(1/2) x (1/2)^(1/2), which is approximately 0.866. This is less than 1, suggesting sequences should generally decrease over time. This heuristic argument supports the conjecture but falls far short of a rigorous proof.
What mathematical approaches have been tried to prove the Collatz conjecture?
Mathematicians have attacked the Collatz conjecture from many angles without success. Terrence Tao proved in 2019 that almost all Collatz orbits attain almost bounded values, the strongest result to date. Other approaches include ergodic theory, which studies statistical properties of the iterative mapping. Algebraic approaches examine the problem in different number systems like p-adic numbers. Graph theory approaches model the sequence as a directed graph. Some researchers have studied generalizations like 5n+1 or other variants to understand the broader class of such problems. The conjecture resists all these methods, suggesting a fundamentally new technique may be required.
What patterns exist in the Collatz sequence graphs?
When plotted, Collatz sequences reveal several interesting visual patterns. Most sequences follow a characteristic shape: an initial chaotic phase where values rise and fall unpredictably, followed by a descent phase once the sequence drops below the starting number. The descent to 1 often follows a geometric decay pattern. When all sequences are plotted as a tree structure (working backwards from 1), the resulting graph has a fractal-like branching pattern. The ratio of even to odd steps tends toward 2:1 for most sequences. Plotting stopping times against starting numbers reveals a scattered cloud with no obvious formula, though clusters and patterns emerge at various scales.
Are there variants or generalizations of the Collatz problem?
Yes, several generalizations have been studied. The 5n+1 problem replaces 3n+1 with 5n+1, and interestingly, it has known counter-examples: the number 13 enters an infinite cycle 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13. The generalized form considers qn+r for various integers q and r. Some variants operate in the negative integers, where the original Collatz rules produce cycles like -1, -2, -1 and -5, -14, -7, -20, -10, -5. The Syracuse function is a streamlined version that only records odd values in the sequence. Studying these variants helps mathematicians understand what makes the 3n+1 case special.
What is the relationship between Collatz sequences and binary representation?
The Collatz sequence has elegant connections to binary arithmetic. Dividing by 2 is equivalent to removing the trailing zero in binary (a right shift). The 3n+1 operation on an odd number (which ends in 1 in binary) can be viewed as a binary transformation that shifts, adds, and carries. When you track the sequence of even/odd steps as a binary string, certain patterns emerge. For example, every sequence ending in 1 must pass through powers of 2, which are single 1-bits in binary. The Collatz tree, when organized by binary structure, reveals that numbers sharing similar binary patterns often have related sequence behaviors, though no exploitable regularity has yielded a proof.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.
🧮Area of Crescent Calculator
Calculate area of crescent with inputs, formulas, and instant results.
🧮Center of Mass Calculator
Calculate center of mass with inputs, formulas, and instant results.
🧮Centroid Calculator
Calculate centroid with inputs, formulas, and instant results.
🧮Chord Length Calculator
Calculate chord length with inputs, formulas, and instant results.
🧮Conic Sections Calculator
Calculate conic sections with inputs, formulas, and instant results.