# Tokens in buckets

Discrete Mathematics Level 5

$$n$$ buckets are arranged in a circle, each containing 1 token. Starting with one of the buckets, you take out a token 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?

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