Today's puzzle is about contestants in a game show trying to win. It was once given to a different type of contestant, who was applying to study joint philosophy degrees at Oxford university.
The teenagers wanted to study PPE, philosophy and economics, and computer science and philosophy. The interviewer may have given hints and asked probing questions during the back-and-forth discussion in which they were set the puzzle in their admissions interviews. The interviewer wanted to know how the candidates solved the puzzle as much as the solutions they gave.
It is an interesting puzzle that relates to fundamental issues in logic and computer science.
The game show is on.
You are a contestant on a game show. A person is in a different room. Either you win or you lose the game. You can assume that the other contestant is just as logical as you are because you have never met them before.
The game starts with round 1 and goes on for as long as necessary. Each contestant has two choices.
Either I tell the host that I end the game and announce a colour, or I don't.
To send a message to the other contestant.
The messages are sent at the same time, crossing in transit.
To win the game, you have to announce the same colors at the same time. You lose if only one of you ends the game and the other ends it in a different colour.
The first round is about to start. What do you do?
You don't end the game on the first round. That is a bad strategy. If you end the game, you have to say a color. The other contestant must decide to end on the first round and also announce red if they want to win. Don't do it. Dialogue between you and the other contestant is the best strategy.
A simpler variation of the puzzle will help us understand what the main puzzle is asking you to do. The set-up is the same, except that in each round only one contestant sends a message. In round 1 you send a message, in round 2 the other contestant sends you a message, and you continue alternating between them.
A simple strategy is presented in this version. If you are more cautious, you could say: "I will declare red on round 2; if you also do we will win." The other contestant will follow your lead. You will both win by round 3.
This strategy doesn't work in the original puzzle when you have to send messages at the same time. Imagine your round 1 message is "let's declare red in round 3, please confirm in round 2." The other contestant is just as smart and logical as you, so they may have had the same idea. Let's say their message is: "Let's declare blue in round 3, please confirm in round 2." Where does that leave you both? What colour is in round 2?
The crux of the puzzle is to understand that if you send the other contestant the same message, they may send you a different one. You have to find a way to break the stalemate.
When this puzzle was used in Oxford admissions interviews, the contestant was not expected to come up with a perfect answer immediately. Since the effectiveness of any message will depend on the message you receive, there is no way to know in advance what the best message will be. The open-ended discussion of the issues involved came from the puzzle. Some strategies are vastly better than others.
These two variations may have been introduced by the tutor.
The contestants have three choices each round. They can either end the game and announce a colour, or they can send a message, as in the standard version. The messages are not delivered if both players send a message. Each player gets an error message saying the message was not delivered. What do you do?
The simplified version of the pigeon variation is that only one contestant sends a message per round. You start in round 1 and the other contestant starts in round 2. You can't be sure that the messages arrive because you are a long distance from each other. What do you do?
The variations begin to explain what the puzzle has to do with computer science: they are analogies of the problems that arise when computers talk to each other.
I will be back at 5pm UK with solutions to all three variations of the puzzle and a discussion.
Discuss your favourite game shows. Suggest new logic puzzles.
Today's puzzles were written by Joel David Hamkins. He was the Professor of Logic and the Sir Peter Strawson Fellow in Philosophy at University College, Oxford, until last December, when he became the Ohara Professor of Philosophy and Mathematics at the University of Notre Dame.
I made the puzzle more suitable for a newspaper column.
On Mondays, I set a puzzle here. I always look for great puzzles. Email me if you would like to suggest one.
I'm the author of several books of puzzles, most recently the Language Lovers Puzzle Book. I give school talks about math and puzzles. Please contact your school if you are interested.
I will be giving a puzzles workshop on April 21. You can sign up here.
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