Collatz Conjecture Calculator

Generate the Collatz sequence (3n+1 problem) for any positive integer and see steps to reach 1.

Enter Starting Number

Result

Steps to Reach 1
111
steps (stopping time)

Step-by-Step Solution

If even: n/2 | If odd: 3n + 1

Understanding the Collatz Conjecture

The Collatz conjecture, also known as the 3n + 1 problem, the Ulam conjecture, or the hailstone sequence, is one of the most famous unsolved problems in mathematics. It was first proposed by Lothar Collatz in 1937 and has remained unproven despite extensive study by mathematicians for nearly a century.

The conjecture states that for any positive integer n, the following process will always eventually reach 1: if n is even, divide it by 2; if n is odd, multiply it by 3 and add 1. Despite its simple formulation, no one has been able to prove that this is true for all positive integers.

The Collatz Rules

Even Number Rule

If the current number is even, divide it by 2.

n -> n / 2

Odd Number Rule

If the current number is odd, multiply by 3 and add 1.

n -> 3n + 1

Stopping Condition

The sequence terminates when it reaches 1 (entering the 4-2-1 cycle).

1 -> 4 -> 2 -> 1 (cycle)

Stopping Time

The number of steps needed to reach 1 from the starting number.

Steps until n = 1

Maximum Value

The highest value reached during the sequence before eventually decreasing to 1.

max(sequence)

Hailstone Numbers

The sequence values are called "hailstone numbers" because they rise and fall like hailstones in a cloud.

Values go up and down

Notable Examples

The Number 27

The number 27 is a famous example because its Collatz sequence is surprisingly long. Starting from 27, the sequence takes 111 steps to reach 1 and reaches a maximum value of 9,232. This demonstrates how even small starting numbers can produce dramatically long and high-reaching sequences.

Powers of 2

Any power of 2 reaches 1 very quickly by simply dividing by 2 at each step. For example, 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1 takes just 6 steps. Powers of 2 have the shortest possible stopping times for their magnitude.

Current State of Research

The Collatz conjecture has been verified computationally for all starting values up to approximately 2.95 * 1020 (as of 2020). However, no one has been able to prove it mathematically for all positive integers. The famous mathematician Paul Erdos said about the conjecture: "Mathematics may not be ready for such problems."

Why Is It So Hard to Prove?

  • Unpredictable behavior: The sequence can increase dramatically before eventually decreasing, making it hard to establish uniform bounds.
  • Mixing of multiplication and division: The interplay between the 3n+1 and n/2 operations creates complex dynamics that resist standard mathematical techniques.
  • No obvious pattern: Stopping times and maximum values do not follow simple, predictable patterns related to the starting number.
  • Connection to deep mathematics: The conjecture has connections to number theory, dynamical systems, and computability theory.

Fun Facts

  • The conjecture is also called the "simplest impossible problem" in mathematics.
  • In 2019, Terence Tao proved that the Collatz conjecture is "almost always true" in a measure-theoretic sense, but a full proof remains elusive.
  • Some mathematicians have joked that the Collatz conjecture is a "mathematical virus" because anyone who hears about it will spend time trying to solve it.
  • Jeffrey Lagarias collected the various names for this problem in a 2010 survey: there are over 20 different names used in the literature.