Hailstone Number — Definition, Formula & Examples

A hailstone number is any number in the Collatz sequence — a sequence generated by repeatedly applying a simple rule: if the number is even, divide by 2; if odd, multiply by 3 and add 1. The name comes from how the values rise and fall like hailstones in a cloud before eventually reaching 1.

Given a positive integer $n$ , the hailstone sequence (Collatz sequence) is defined recursively by $a_0 = n$ and $a_{k+1} = a_k / 2$ if $a_k$ is even, or $a_{k+1} = 3a_k + 1$ if $a_k$ is odd. Each term $a_k$ in this sequence is called a hailstone number. The Collatz conjecture states that for every positive integer $n$ , the sequence eventually reaches 1.

Key Formula

a_{k+1} = \begin{cases} a_k / 2 & \text{if } a_k \text{ is even} \\ 3a_k + 1 & \text{if } a_k \text{ is odd} \end{cases}

Where:

$a_k$ = The current term in the hailstone sequence
$a_{k+1}$ = The next term in the sequence

How It Works

Pick any positive integer and apply the rule: divide by 2 if even, or compute

3n + 1

if odd. Repeat the process on each result. The sequence bounces up and down unpredictably — sometimes climbing to values much larger than the starting number — but every starting value ever tested eventually falls to 1. Once the sequence hits 1, it cycles through

1 \to 4 \to 2 \to 1

forever. Despite its simplicity, no one has proven that every positive integer eventually reaches 1, making this one of the most famous unsolved problems in mathematics.

Worked Example

Problem: Generate the hailstone sequence starting from 6.

Start: Begin with 6. Since 6 is even, divide by 2.

6 \div 2 = 3

Step 1: 3 is odd, so compute 3(3) + 1.

3 \times 3 + 1 = 10

Step 2: 10 is even, so divide by 2. Then continue: 5 is odd, so 3(5)+1 = 16. Then 16 is even, and we keep halving.

10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1

Answer: The complete hailstone sequence for 6 is: 6, 3, 10, 5, 16, 8, 4, 2, 1 (8 steps to reach 1).

Visualization

Why It Matters

The Collatz conjecture is one of the most accessible unsolved problems in number theory — easy to state yet resistant to proof. Exploring hailstone sequences builds intuition about parity, recursion, and iterative processes, skills used in computer science and algorithm analysis.

Common Mistakes

Mistake: Applying the odd rule (3n + 1) when the number is even, or vice versa.

Correction: Always check parity first. If the number is divisible by 2, divide by 2. Only use 3n + 1 when the number is odd.