Optimize the team's strategy in this lottery
- From: Francois Grieu <fgrieu@xxxxxxxxxxxx>
- Date: Sun, 26 Feb 2006 15:11:48 +0100
[follow-up to rec.puzzles, this is mildly related to crypto]
A team of 10 persons plays a strange lottery.
Material used:
- 10 balls marked 0 to 9;
- 10 boxes marked A to J on the outside;
- 5 identical peas.
The lottery organizer secretly and randomly puts one
ball in each box, and one pea in 5 randomly selected
boxes.
Each team member is assigned a number in range
0 to 9, and plays in sequence. He is isolated from
the others while playing. Those that have played are
isolated from those that have not.
When it is his turn, a player is allowed, 6 times, to
examine the content of one box he selects. After theses
examinations he commits a bet on if the ball with his
assigned number is in a box with a pea, or not.
The teams wins if and only if ALL 10 bets are correct.
Beyond initially defining a common strategy, no
communication can occur between team members during a
game. Bets are kept secret until the end of the game.
Boxes, balls and peas can't be moved.
What is a strategy optimizing the odds of the team ?
For the combinatoric inclined: what are the odds of
the best strategy in the general case of n
balls/boxes/persons, p peas, m boxes opened ?
Extra: after playing a number of games, the team sees
their sucess rate fall sharply. They find the organizer
understood their strategy and now cheats in preparing
the game. How do they repair their strategy, without
cheating ?
François Grieu
Simplified following a remark by dgates <dgates@xxxxxxxxxxxxxxxx>
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