Free Collatz Conjecture Calculator

The Collatz Conjecture: A Simple Yet Unsolved Problem

The Collatz conjecture, widely known as the 3x+1 problem, is one of the most famous unsolved puzzles in mathematics. Its rules are straightforward: take any positive integer, and at each step, if it is even, divide it by two; if it is odd, multiply it by three and add one. Despite this simplicity, no one has been able to prove that this process always ends at 1 for every possible starting number. A Collatz sequence calculator makes it easy to explore this fascinating behavior and see the patterns for yourself.

Defining the 3x+1 Rules

For a given starting integer $n_0$, the sequence follows the recurrence:

a_{k+1} = \begin{cases} \dfrac{a_k}{2} & \text{if } a_k \equiv 0 \pmod{2} \$$6pt] 3a_k + 1 & \text{if } a_k \equiv 1 \pmod{2} \end{cases}

The notation $a \equiv b \pmod{2}$ simply checks whether the number is even (0 mod 2) or odd (1 mod 2). Once the value reaches 1, it enters the cycle $4 \rightarrow 2 \rightarrow 1$ and never leaves. The conjecture asserts that this is true for all positive integers.

Witness the Chaotic Nature: Hailstone Sequences

When you generate a Collatz sequence, the numbers often bounce up and down in a seemingly random fashion. For example, starting with 6 yields:

$$6 \to 3 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1$$

Here, the oscillation is mild. A more dramatic case is the starting number 11:

$$11 \to 34 \to 17 \to 52 \to 26 \to 13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1$$

The path rises to 52, then falls to 26, climbs again to 40, and finally descends. This rise-and-fall pattern resembles the way hailstones grow inside a thundercloud, which is why these sequences are also called hailstone sequences. A dedicated hailstone sequence calculator can plot these undulating paths and show the exact stopping time.

The behavior is fundamentally unpredictable. Mathematicians consider the Collatz problem an undecidable problem in the sense that no computer program can decide ahead of time whether a given starting number will reach 1, without actually running through the steps. This forces researchers to rely on brute‑force computation to test ever‑larger numbers.

What Has Been Tested So Far?

Because the conjecture remains unproven, mathematicians have turned to computational checks for counterexamples—numbers that might diverge or enter a different loop. To date, every integer up to approximately

$$295\,147\,905\,179\,352\,825\,856 \quad (\approx 2.95 \times 10^{17})$$

has been verified to fall to 1. An even more extreme test involved the number $2^{100,000} - 1$, which has 30,000 digits. After nearly 1.5 million steps, it too reached 1. No counterexample has ever been found, but that does not prove the conjecture: a formal proof still eludes mathematicians.

For negative integers, the situation is more complex. The Collatz‑like rules applied to negative numbers produce three distinct loops (starting at −1, −5, and −17), and why these loops exist remains an open question.

Using the Collatz Problem Solver

The 3x+1 calculator on this page provides an intuitive way to interact with the conjecture. Enter any integer (positive or negative) and instantly obtain:

• The full sequence of terms from the starting number down to 1 (or the cycle for negatives).
• The stopping time—the number of steps required to reach 1 for the first time.
• A graphical chart that displays the value at each step, revealing the overall shape of the sequence.

The tool also supports custom rule modifications. You can change the multiplier (e.g., from 3 to something else) or the addend, allowing you to explore related problems—though such modifications may not retain the known properties of the Collatz conjecture.

Whether you are a student meeting the 3n+1 problem for the first time or a researcher looking for insights, this Collatz Problem Solver offers a convenient way to test numbers and visualize the chaotic dynamics that have baffled mathematicians for almost a century.