Puzzle time - binaries
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Consider a string of binary numbers s, such as s = 1011.
Now consider a machine that works on s. It performs the following three steps:
- Append 1 to the right end of s.
- Add the new s from step 1 to the old s with binary addition
- Remove all trailing 0s.
The machine iterates these steps and stops only when s = 1.
For instance, for s = 1011 we have s = 10111 after step 1, s = 10111+1011 = 100010 after step 2, s = 10001 after step 3.
After the next iteration we get s = 01101, then 0101, then 1. The machine stops.
Can you find an s such that the machine doesn't stop? Or conversely, can you make an argument why the machine will always stop for any s?
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