Math Is Fun Forum

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

The Prisoners and the Cards

Whilst wandering aimlessly around the internet, I found a puzzle kind of similar to this one, and it took me a good while to solve so I thought I would share it.

Two prisoners are given a chance to escape jail by winning a card game that works as follows:

The guard randomly selects five cards out of a normal deck and gives them to prisoner #1 (say, Alyson).
She must then give four of the cards, one at a time, to the guard. After getting each card, the guard then delivers it to prisoner #2 (Brian).

After receiving the four cards, Brian must then determine what Alyson's 5th card is.

They're given the day prior to discuss strategy, etc., but while the game is going on they can communicate only with the cards. Alyson can't bend the cards or mark them somehow, and she can't delay giving cards to the guard in order to make some kind of timer signal.
To be generic, if she thinks of any way to communicate with Brian other than the cards, then the guard will think of a foolproof way of stopping it.

So, what's their game plan?

(PS. It's an evil jail and the two prisoners are fighters for righteousness, so we want them to escape. )

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: The Prisoners and the Cards

Suit and face value?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: The Prisoners and the Cards

Yup. It's a standard deck of 52 cards, and Brian needs to find out exactly which of the 52 Alyson has kept.

For a bonus challenge, you could extend it to be a 54-card deck with 2 jokers (and for a bonus bonus challenge, make those jokers red and black), but I can't think of solutions to either of those at the moment so they might not be possible.

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Why did the vector cross the road?
It wanted to be normal.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: The Prisoners and the Cards

The suit of the 5th card can be easily communicated, but I've no idea how you could send the value

Last edited by bossk171 (2007-07-24 06:56:28)

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: The Prisoners and the Cards

I would say you're on the right track, but then your example seems to contradict your reasoning. Is that a typo?

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Why did the vector cross the road?
It wanted to be normal.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: The Prisoners and the Cards

I would say you're on the right track, but then your example seems to contradict your reasoning. Is that a typo?

No, I'm just a bit of an idiot, it's fixed now.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: The Prisoners and the Cards

Are we going to get an answer to this? It's really driving me nuts...

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: The Prisoners and the Cards

Oh, go on then.

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Why did the vector cross the road?
It wanted to be normal.