5 cards trick
This is a magic trick performed by two magicians, Alice and Bob, with one shuffled deck of N unique cards. (Nothing is mentioned about suits: you may consider these cards to be simply enumerated from 1 to N.) Alice asks a member of the audience, Carol, to randomly select 5 cards out of a deck. Carol then returns her chosen 5 cards to Alice. After looking at the 5 cards, Alice picks one of the 5 cards and gives it back to Carol. Alice then arranges the other four cards in some way, and gives them to Bob in a neat, face-down pile. Bob examines these 4 cards and determines what card is in Carol's hand (the missing 5th card). Carol is astonished!
What is the largest number of cards N that the deck can contain before the trick is no longer performable? Prove it.
How specifically do you execute the trick on a deck of maximal size N?

Equation: N = P(M,M) + (M-1)
Explanation: Permutation of M is P(M,M), that is the information M cards can code, plus M-1 (because M-1 will be shown)

For M=5;
P(5,5) + (M-1) = 120 + 4 = 124.

For M=3;
P(3,3) + (M-1) = 6 + 2 = 8.

we have P(2,8) = C(3,8), and C(3,8) = C(5,8) if we can find a map from P(2,8) to C(5,8) and the 5 out 8 cards do not contain the 2 out 8 cards, we are done.
- posted by Pop @ 8:16 AM