We are playing a game of cards in which two people belong to a team. One of them gets five random cards, of which she displays four to her team member while she puts the fifth one face down. Her teammate is of course expected to guess the card which is hidden. Now, suppose these two have decided beforehand to win by cheating. What is the strategy that they will use to convey the hidden card to each other?
A few ‘of course’ answers to questions about the puzzle:
You can decide the order in which a player displays his cards
The order in which you put your cards down can not be decided. (eg. start with the last card, then go to the second one, etc)
No, he can’t talk or use anything other than his cards to communicate.
Coutesy: (Hari)krishnan Ramani
okay, we’re dealing with 48 choices for the partner to guess. The first step is to narrow down the suit, so using pigeon hole principle, the first card that she shows the partner should be of the same suit as the card that she decides to keep. Now, its 12 choices.
Now comes the tricky bit, the cards of a suit have values in form of a circle, with 13 points. So, if we know one of the 13, the max distance between it and any other card will be just 6. But of course, there’s distance and “direction” to be specified, so they can always decide to stick to 2->3->4..K->A->2 direction, oh this also means, she shows the “lower” card to the partner.
Now that direction is taken care of, we just need to specify the distance, which can range from 1 – 6, using just 3 cards. Again, if we specify an well ordering to the cards, we can come up with a highest, middle and lowest card. And get a 3!, which turns out to be 6. So, here, successful cheating!
For a more exhaustive version of the above solution,
http://techpathak.com/index.php?option=com_content&view=article&catid=42&id=127&Itemid=73
@Prakash and Chetan: Thanks for the solutions!