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        riddles >> medium >> Another Sum-Product Variant
        
(Message started by: ThudanBlunder on Jun 24^th, 2008, 3:38pm)

Title: Another Sum-Product Variant
Post by ThudanBlunder on Jun 24^th, 2008, 3:38pm

Three people V, C, and X are joined by M who holds sixteen playing cards which they can see:

SPADES: J, 8, 7, 4, 3, 2
DIAMONDS: A, 5
CLUBS: K, Q, 6 ,5, 4
HEARTS: A, Q, 4

M secretly selects a card at random and tells V its value and C its colour.
In X's hearing, he then asks them if they know which card he chose.
The following conversation ensues:
V: I don't know what the card is.
C: I knew that you didn't know.
V: I know the card now.
C: I know it, too.

X then thinks for a moment and correctly deduces what M's card is.

How is this possible?

Title: Re: Another Sum-Product Variant
Post by SMQ on Jun 24^th, 2008, 6:51pm

From each statement X (and we) can deduce:

V: [hide]The card is a 4, 5, Q, or A (because the others only appear in a single suit)[/hide]
C: [hide]The card is a Diamond or Heart (because the other suits have unique cards)[/hide]
V: [hide]The card is not an Ace (because an Ace appears in both Diamonds and Hearts)[/hide]
C: [hide]The card is the Five of Diamonds (because there are still two possible Hearts)[/hide]

--SMQ

Title: Re: Another Sum-Product Variant
Post by Eigenray on Jun 25^th, 2008, 1:45am

Colour == suit?

Title: Re: Another Sum-Product Variant
Post by SMQ on Jun 25^th, 2008, 5:33am

on 06/25/08 at 01:45:59, Eigenray wrote:

Colour == suit?

No, just color. Does that pose a problem for my above reasoning?

--SMQ

Title: Re: Another Sum-Product Variant
Post by Eigenray on Jun 25^th, 2008, 5:46am

Then I don't see how C is able to determine the card.

Title: Re: Another Sum-Product Variant
Post by rmsgrey on Jun 25^th, 2008, 8:11am

on 06/25/08 at 05:46:19, Eigenray wrote:

Then I don't see how C is able to determine the card.

Me neither:

4H, 5D and QH all look to be possible given the information available - we know, and C knows, that the card is red (were it black, C couldn't have known that V didn't know its value) and, from V being able to identify it from its colour and value, we know it's not an Ace, but I don't see any way to narrow down V's knowledge further than 4,5,Q

Of course, if C knows the suit, rather than just the colour, we can get 5D out of it all.

Title: Re: Another Sum-Product Variant
Post by ThudanBlunder on Jun 25^th, 2008, 6:00pm

on 06/25/08 at 05:46:19, Eigenray wrote:

Then I don't see how C is able to determine the card.

I agree. But the book does say 'colour', not 'suit'. As 'suit' allows X to deduce 5D, I guess the book must be wrong. (No answer is given.)