The Simplest Impossible Problem: The Collatz Conjecture
Mathematics is usually about proving things. $a^2 + b^2 = c^2$ is proven. The area of a circle is proven. But there is a child-like problem involving simple arithmetic that has baffled the greatest minds in history for nearly a century. It's called the Collatz Conjecture (also known as the $3n+1$ problem). A famous mathematician, Paul Erdős, once said about this problem: "Mathematics may not be ready for such problems." Our calculator allows you to experiment with this mystery yourself, generating the sequence for any number you choose.
This guide explains the rules of the game, why it's so fascinating, and the strange patterns known as "hailstone numbers."
The Rules
Pick any positive integer $n$.
1. If $n$ is even, divide it by 2 ($n/2$).
2. If $n$ is odd, multiply it by 3 and add 1 ($3n+1$).
3. Repeat.
The Conjecture: No matter what number you start with, you will eventually reach 1.
Let's Try It: Starting with 6
- 6 is even → $6/2 = 3$
- 3 is odd → $(3 \times 3) + 1 = 10$
- 10 is even → $10/2 = 5$
- 5 is odd → $(5 \times 3) + 1 = 16$
- 16 is even → $16/2 = 8$
- 8 is even → $8/2 = 4$
- 4 is even → $4/2 = 2$
- 2 is even → $2/2 = 1$
- Done! We reached the "4-2-1" loop.
Why Are They Called "Hailstone Numbers"?
The numbers in the sequence are like hailstones in a storm cloud. They inevitably fall to the ground
(reach 1), but before they do, they are often swept up high into the atmosphere by strong winds
($3n+1$).
Example: Start with 27.
Most numbers drop quickly. But 27 is unruly. It bounces up to 9232 before finally crashing down
to 1 after 111 steps. It hangs in the air for a long time.
Has Anyone Disproved It?
No. Computers have checked numbers up to $2^{68}$ (that's roughly 295 quintillion), and every
single one has eventually reached 1.
However, in mathematics, checking quintillions of examples is not a proof. There might be one
number, somewhere out there, that either:
- Goes to infinity (never stops growing).
- Gets stuck in a different loop (not 4-2-1).
To date, no such number has been found.
Conclusion
The Collatz Conjecture is a reminder that complexity can arise from simplicity. Use our calculator to explore the path of different numbers. Try your birth year, your phone number, or random digits, and watch as they all—eventually—obey the rule of 1.