3x+1
3x+1 is a wonderful thing, I wasn't aware of it this morning but now I'm thinking about it. I'm certainly not going to be the person who solves it, but looking at it I was immediately seeing a relationship to binary.
With the loop end at 4, 2, 1, being powers of two, makes it a natural realization and I am quite certain I'm not the first person to consider it in this way.
The rules of 3x+1 in binary are the same, except dividing by 2 in binary is extremely simple. An even numbers in 0, to divide by 2, you drop the last 0.
So for example, I quickly jotted out the paths of 1-7 in binary:
- 1 -> 100 -> 10 -> 1 -> L (the loop)
- 10 -> 1 -> L
- 11 -> 1010 -> 101 -> 10000 -> 1000 -> 100 -> 10 -> 1 -> L
- 100 -> 10 -> 1 -> L
- 101 -> 10000 -> ... -> 1 -> L
- 110 -> 11 -> 1010 -> 101 -> 10000 -> ... -> 1 -> L
- 111 -> 101010 -> 10101 -> 100010 -> 110100 -> 11010 -> 1101 -> 101000 -> 10100 -> 1010 -> 101 -> 10000 -> ... -> 1 -> L
Mostly did this as a mental exercise and to explore the tie I noted with binary, but it is interesting. I am certain I'm not the first person to explore it this way, so I'm retreading well trodden ground, but it is still a nice stretch out of my normal day-to-day thinking.
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