n → n / 2
Collatz Conjecture Explorer — The 3n + 1 Problem
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Slow
300ms
Famous starting numbers
Live sequence & trajectory
3n+1
Type a number and hit Start
Watch the hailstone sequence climb and crash to 1. Try 27 for a 111-step ride to 9,232.
n → 3n + 1
when n = 1
268 ≈ 2.95 × 1020
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What is the Collatz Conjecture?
The Collatz Conjecture — also called the 3n + 1 problem, hailstone sequence, or Syracuse problem — is one of the most famous unsolved problems in mathematics. Proposed by Lothar Collatz in 1937, it states that for any positive integer, repeatedly applying a simple rule will always eventually reach 1.
If n is even: n → n / 2
If n is odd: n → 3n + 1
Repeat until you reach 1.
If n is odd: n → 3n + 1
Repeat until you reach 1.
Despite its simple formulation, no one has been able to prove this is true for all positive integers. It has been computationally verified for all numbers up to 268 (approximately 295 quintillion), yet a general proof remains elusive.
Famous Collatz sequences
27 — the classic
111 steps, peaks at 9,232. A small number with a surprisingly long and dramatic trajectory.
871 — high peak
178 steps, peaks at 190,996. Reaches extreme heights before descending to 1.
6,171 — long journey
261 steps to reach 1. One of the longest sequences for numbers under 10,000.
63 — deceptively long
108 steps from a two-digit number. Shows how small inputs produce long sequences.
Why is it unsolved?
The Collatz Conjecture resists proof because the sequence behaviour appears chaotic and unpredictable. The interplay between multiplication (3n + 1) and division (n / 2) creates orbits that seem random, making it extremely difficult to establish any general pattern that would apply to all integers.
In 2019, Fields Medalist Terence Tao proved that almost all Collatz orbits attain almost bounded values — the strongest partial result to date. A complete proof covering every positive integer remains out of reach.
"Mathematics is not yet ready for such problems." — Paul Erdős
Frequently asked questions
What is the Collatz Conjecture?
The Collatz Conjecture (also called the 3n+1 problem or hailstone sequence) states that for any positive integer, if you repeatedly apply the rule divide by 2 if even or multiply by 3 and add 1 if odd, you will always eventually reach 1. Despite being verified for all numbers up to 268 (about 295 quintillion), no general proof exists. It remains one of the most famous unsolved problems in mathematics, proposed by Lothar Collatz in 1937.
Why is the number 27 famous in the Collatz Conjecture?
The number 27 is a classic example because its sequence is surprisingly long and dramatic. Despite being a small starting number, it takes 111 steps to reach 1 and climbs to a peak value of 9,232 before descending. This illustrates the unpredictable nature of Collatz sequences, where small inputs can produce unexpectedly complex trajectories. Try it in the calculator above to see the full animated sequence.
What is a stopping time in the Collatz Conjecture?
The stopping time (or total stopping time) is the number of steps it takes for a Collatz sequence to reach 1 from a given starting number. For example, starting from 6 the sequence 6, 3, 10, 5, 16, 8, 4, 2, 1 has a stopping time of 8 steps. Some numbers have very long stopping times relative to their size. The number 6,171 takes 261 steps, making it one of the longest sequences under 10,000.
Has the Collatz Conjecture been proven?
No. As of 2025, the Collatz Conjecture remains unproven despite decades of effort. It has been verified computationally for all starting numbers up to approximately 268. In 2019, Fields Medalist Terence Tao proved that almost all Collatz orbits attain almost bounded values, which is the strongest partial result to date. Paul Erdős famously said mathematics is not yet ready for such problems.
What are hailstone numbers?
Hailstone numbers refer to the values in a Collatz sequence because, like hailstones in a cloud, they go up and down unpredictably before eventually falling to the ground (reaching 1). The sequence rises when odd numbers are transformed via 3n+1 and falls when even numbers are halved. This turbulent behaviour is what makes the conjecture so fascinating and difficult to prove.