BrainTeaser NSA Intermediate

Problem - 4706


Steve, Tony, and Bruce have a plate of $1,000$ cookies to share according to the following rules. Beginning with Steve, each of them in turn takes as many cookies as he likes (but must be at least $1$ if there are still cookies on the plate), and then passes the plate to the next person (Steve to Tony to Bruce to Steve and so on). They all want to appear to be modest, but at the same time, want to have as many cookies as possible. This means that they all try to achieve:

  1. Have one person get more cookies than himself, and one person get fewer cookies than himself.
  2. Have as many cookies as possible.

The first objective takes infinite priority over the second one. If all of them are sufficiently intelligent and can choose the best strategy for themselves, what will be the end result?

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