100 Boxes
Can you create a strategy to win the million dollar prize money?
A group of 100 players are offered the chance to play the following game for the possibility of a one million dollar prize ($10,000 apiece). But the entry fee is steep: $1000 from each player. Hmm. We’d better take a close look at the odds.
Here’s how the game goes. First of all they are each given a different number (integer) between 1 and 100. At one point they will be assembled and will be admitted to a room one at a time in numerical order, player 1 first, followed by player 2, etc.
In the room there is a long table with a row of 100 wooden boxes numbered in order from 1 to 100. Each box can be opened and inside it is a card with a number on it such that each number between 1 and 100 appears on exactly one card. These numbered cards have been distributed randomly among the 100 boxes, one card in each box.
Once a player steps inside the room, he will have no further contact with any player who has not yet been in the room. While the player is in the room he is allowed to open and look inside 50 boxes—any 50 that he wants. Following that he must close the boxes leaving them in exactly the same state as he found them. After he has opened his 50 boxes he leaves the room. This goes on until all 100 players have passed, one by one, through the room.
Now here’s how they win. They get the million dollars if every player opens the box that contains his own number. If at least one of the players fails to find his number, they lose and there will be no prize.
This might seems a bit hopeless, but here’s a ray of light. The players are allowed to get together before-hand and work out a strategy. What should they do. What are their odds of winning the prize? Would you invest $1000 to be part of the team?