Yes! You've finally been chosen to participate
in the "Great Geek Game-show 3000". This is
the moment you've been waiting for, ever since
you puzzled out how to maximise your chances
of winning. You will nally be rich, and able
to buy all the algorithm books you want! Of
course you will have to share the winnings with
the other contestants, but since your algorithm
is vastly superior to the randomness of all the
other numb-skulls you are certain you will be
able to keep most of the prize money for yourself,
in exchange for telling them how you can all maximise your chances of winning.

The rules of the game are the following: There is a stage with N boxes, each containing
the name of one of the N contestants, and such that each contestant has their name in
exactly one box. The contestants enter the stage one at a time, and each one is allowed
to peek inside K of the boxes. If they nd their own name inside one of these boxes they
can get o the stage, and the game continues with the next contestant. If all contestants
nd their own name, everyone wins. But if one contestant fails to nd theirs, everyone
loses. After the game has begun, no communication between the contestants is allowed.

However you are all allowed to agree upon a strategy before the game begins, and this is
where you explain to all the others that the algorithm of everyone choosing K boxes at
random is a very bad one, since it gives a chance of winning only equal to (K/N)^N. Instead
you suggest the following algorithm:

Assign to each player and each box a unique number in the range 1,...,N. Then each
player starts with opening the box with the same number as themselves. The next box the
player opens is the box whose number is found inside the rst box, then the box whose
number is found inside the second box, and so on. The process goes on until the player
has opened K boxes, or found their own number.

Now to bring home your point of how superior your algorithm is, you will need
to calculate the exact odds of winning if all the contestants follow your directions.
Unfortunately, this is the only thing you haven't gured out yet...