$\textbf{Cookies}$
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:
- Have one person get more cookies than himself, and one person get fewer cookies than himself.
- 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?
$\textbf{Answer}$
Steve: $334$, Tony: $666$, Bruce $0$.
$\textbf{Analysis}$
Assuming Steve takes at least $335$ cookies, then Tony will take just one less than Steve takes (or all the remaining if Steve takes at least $501$ cookies). In this case, Steve will certainly end up with the most cookies, and Bruce will end up with the fewest cookies. Thus, Tony becomes the modest person, not Steve.
Now suppose that Steve takes $334$ cookies. Then it can be shown that there is no way for Tony to finish as the second place. In this case, he will simply take all and leave nothing to Bruce. This is the best that Tony can do which, at the same time, makes Steve the modest person. Hence, it will be the end result.
In order to show that Tony cannot finish as the modest person if Steve takes $334$ cookies, let's first consider the case if Tony takes $333$ or more cookies. In this case, there are at most $333$ cookies left, so there is no way for Bruce to achieve the first objective. Consequently, Bruce will simply takes all the remaining cookies. This results in a loss to Tony as he has achieved no objective. Meanwhile, if Tony takes no more than $332$, Bruce can take $333$ cookies to secure him as the modest person which is also a loss to Tony. Either way, Tony cannot become the modest person if Steve takes $334$ cookies.
$\textbf{Note}$
It is somewhat intuitive to start investigating the number around one third of the total number of cookies because Steve wants to achieve the first goal. Then, it is just a matter to test numbers around $333$.