An easier puzzle (but still medium difficulty).
You start with 4 integers $(a,b,c,d)$ and play a game. At each step you replace $(a,b,c,d)$ with $(|a-b|, |b-c|, |c-d|, |d-a|)$, i.e. you replace the numbers with the absolute value of differences with the adjacent numbers (adjacent when placed on a circle).
You only stop when the numbers all become zero.
For example, say you start with $(1,2,3,4)$
After one step, it becomes $(1,1,1,3)$, then $(0,0,2,2)$ then $(0, 2, 0, 2)$, then $(2,2,2,2)$ and then $(0,0,0,0)$ after which you stop.
Are there games (i.e. some initial choice of the four integers) where you will never stop?
What about that case if you start with five integers?
Hard: Characterize the $n \gt 1$ such that if you play the game with $n$ integers, you will always stop, irrespective of the initial choice of the $n$ integers.
[Solution to part one]
[Solution to part two]
You start with 4 integers $(a,b,c,d)$ and play a game. At each step you replace $(a,b,c,d)$ with $(|a-b|, |b-c|, |c-d|, |d-a|)$, i.e. you replace the numbers with the absolute value of differences with the adjacent numbers (adjacent when placed on a circle).
You only stop when the numbers all become zero.
For example, say you start with $(1,2,3,4)$
After one step, it becomes $(1,1,1,3)$, then $(0,0,2,2)$ then $(0, 2, 0, 2)$, then $(2,2,2,2)$ and then $(0,0,0,0)$ after which you stop.
Are there games (i.e. some initial choice of the four integers) where you will never stop?
What about that case if you start with five integers?
Hard: Characterize the $n \gt 1$ such that if you play the game with $n$ integers, you will always stop, irrespective of the initial choice of the $n$ integers.
[Solution to part one]
[Solution to part two]
No comments:
Post a Comment