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Overview

The Collatz Conjecture

April 10, 2024
1 min read
unsolved-math sequences

The Simple Rule

Start with any whole number nn.

  1. If nn is even, divide it by 2 (n/2n/2).
  2. If nn is odd, multiply it by 3 and add 1 (3n+13n + 1).
  3. Repeat.

Example Sequence

Start with 12:

  1. 12 is even →6\rightarrow 6
  2. 6 is even →3\rightarrow 3
  3. 3 is odd →(3×3)+1=10\rightarrow (3 \times 3) + 1 = 10
  4. 10 is even →5\rightarrow 5
  5. 5 is odd →(3×5)+1=16\rightarrow (3 \times 5) + 1 = 16
  6. 16 is even →8\rightarrow 8
  7. 8 is even →4\rightarrow 4
  8. 4 is even →2\rightarrow 2
  9. 2 is even →1\rightarrow 1

The sequence is: 12,6,3,10,5,16,8,4,2,112, 6, 3, 10, 5, 16, 8, 4, 2, 1.

The Unsolved Mystery

The German mathematician Lothar Collatz conjectured that every starting number eventually reaches 1. However, no one has been able to prove this is true for every number, nor has anyone found a number that doesn’t reach 1.

Note

Try it: Start with 7. The path is quite long! 7→22→11→34→17→52→26→13→40→20→10→5→16→8→4→2→17 \to 22 \to 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.