A “conjecture” is an idea that is believed to be true, but has not yet been proven. They are authentic unanswered questions for students to explore.
Definition
The Collatz Conjecture (from 1937) states that, no matter what number you start with, if you follow two simple rules, you will always (eventually) end up at 1. If you can’t wait, jump to the interactive version.
The Rules
- If it’s odd, change n to 3n + 1
- If it’s even, change n to n ÷ 2
Examples
Start With 5
- 5 is odd so it becomes 3 × 5 + 1 or 16
- 16 is even so we halve it to 8
- 8 is even
- 4 is even
- 2 is even
- 1
Start With 3
- 3 is odd so it becomes 3 × 3 + 1 or 10
- 10 is even so we halve it to 5
- 5 is odd
- 16 is even
- 8 is even
- 4 is even
- 2 is even
- 1
Start With 15
15 → 46→ 23→ 70 → 35 → 106 → 53 → 160 → 80 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
No number has been found that breaks this pattern, but the conjecture hasn’t been proven for all numbers. Here’s an interactive version I created for you or your students to play with.