Answer to the previous puzzle
Note that when one round is done, there are 50 people left and the knife comes back to 1 and the next cycle starts.
After next round, 25 people are left and the knife once again comes back to 1.
However, the next time, it does not come back to 1.
In general, if the number was always divisible by 2 at every step, it would always come back to 1.
Which means if the starting number is a power of 2, 1 (or whoever starts with the first move) will be the sole surviving person.
Now let’s deal with 100. The nearest power of 2 is 64. So, as the process starts, one by one, the count is being eliminated downwards from 100. Let’s see what is happening when it gets down to 64.
For 64 to be left, (100-64=) 36 would have been dead. In the sequence of dying the numbers are 2,4,6,…. Thus in the 36th killing, person number 72 died. Which means 71 killed 72 and handed the knife to 73.
So, 73 has the knife and there are 64 (a power 2) left in the circle.
By our logic above, 73 will be the sole surviving person.