n buckets are arranged in a circle, each containing 1 token. Starting with one of the buckets, you take out a coin and move it into the next bucket counterclockwise. You then take two tokens out of this bucket and move them into the next one. This process is repeated over and over, alternating moving one token and then two tokens. When all tokens are removed from a bucket, it is removed from the circle. For how many positive integer values of n<1000 will all the tokens eventually end up in a single bucket?
20 (no of the form 2^k+1 and 2^k+2 between 1 and 1000)
To solve this problem, we need to observe the pattern that emerges as we go through the process described.
Let's start with a small example and see if we can find a pattern. Suppose we have 4 buckets arranged in a circle.
Initially: 1 1 1 1
After moving one token counterclockwise: 2 1 1 0
After moving two tokens counterclockwise: 2 1 1 0
After moving one token counterclockwise: 3 1 0 0
After moving two tokens counterclockwise: 3 0 1 0
After moving one token counterclockwise: 4 0 0 0
As we can see, in this case, all tokens eventually end up in a single bucket.
Now let's try with 5 buckets:
Initially: 1 1 1 1 1
After moving one token counterclockwise: 2 0 1 1 2
After moving two tokens counterclockwise: 2 0 1 1 2
After moving one token counterclockwise: 3 1 1 0 2
After moving two tokens counterclockwise: 3 1 1 0 2
After moving one token counterclockwise: 4 1 0 0 3
After moving two tokens counterclockwise: 4 1 0 0 3
After moving one token counterclockwise: 5 0 0 1 3
After moving two tokens counterclockwise: 5 0 0 1 3
After moving one token counterclockwise: 5 0 0 1 3
After moving two tokens counterclockwise: 5 0 0 1 3
In this case, all tokens also eventually end up in a single bucket.
Based on these examples, we can make an observation. If the number of buckets n is odd, then the tokens will eventually end up in a single bucket. However, if n is even, the tokens will be distributed across two buckets and will never end up in a single bucket.
Therefore, there are 500 positive integer values of n < 1000 for which all the tokens eventually end up in a single bucket.