Sunday, 1 July 2012

A Card to identify

There are 5 cards randomly given to you. You throw one of them by your choice and arrange rest 4 of them in an order such that your friend can identify the card you threw by looking at the order?
Note: You can form a strategy with your friend

1. S-Spade, H-Heart, C-Club, D-Diamond. First assign an order like AS<2S<3S..<KS<AH<...<KD.
Since we have 5 cards atleast 2 of them will be of same suite. I choose one of these to throw.
Observation: arrange the cards of same suite in a circle (like a clock) then any two of them are at max at a distance of 6 from each other.
Now, the two out of 5 cards which are of same suite can be atmax at a circular distance of 6. The card which i choose to throw is the one from which i can travel to the other card, of same suite, on the circle by moving in clockwise direction using at max 6 steps.
Place the other card on the top of the pile. This tells us the suite of the card we have thrown. Now to know the number of the card we need to tell the circular distance from the card on top. This distance can be 6 at max.
We have 3 different cards and we have an order defined on them. There are 3! =6 combinations of these cards each corresponding to a different distance value.
We use this arrangement to tell the number of the card.
Using these two things we can identify both, the number and the suite, of the card.

2. suppose i have 5 cards a,b,c,d,e.. a&b are of the same suit
a) deciding suit
The suit can be decided pretty easily as you have 5 cards with you. I will choose one of the 2 cards from the same suit.
b) deciding the number.
we can first assign an order to the cards like S<H<C<D. so the other 3 cards I have(apart from the two of the same suit) can be arranged in 3! ways. so i can show my friend any number from 1 to 6. Of the remaining two cards(of the same suit, I'll treat them circularly) I will throw away the card b which I can form with a+i where i is any number from 1to 6. for eg. if a=A, b=8 I'll throw away a because I can show a as b+6 (circularly adds upto 1)