Question

Our professor did this magic trick that goes like this:
He is blindfolded at the beginning. His assistant (call her Pat) asked us (the class) to pick 5 cards from a well-shuffled ordinary deck of 52 cards. One of the 5 cards is returned to the deck (either Pat or the class chooses this card, doesn't really natter) and the other 4 are laid down the table in front of our teacher. He removes his blindfold, reads the 4 cards on the table, and gets the 5th card from the deck! We did this exercise 3 times, and he got it every time! O_O

Answer 1

Our teacher said there was a mathematical explanation to his trick, but I suck at probability, so I can't decipher the trick. I ask you now, people of OpenStudy, do you have any idea how this is done?

Answer 2

Do you mean like this: http://cotpi.com/p/44/ ? It's a neat trick indeed.

Answer 3

I don't understand parts of the proof.
Why does s+c_i=i mod(5)?
Why does c_i=-s+i mod(5) imply that there are 24 possibilities for c_i?

Answer 4

S is the sum of the cards shown(only 4), c_i is the picked card. i is the sum mod 5 of all the picked cards (5). S + c_i is congruent to i mod 5. They are the same number, except that i has the mod.

Answer 5

Okay, so I get that but then why does c_i=-s+i (mod 5) mean that there are only 24 possibilities for c_i?

Answer 6

Because c_i = -s for the remaining cards, numbered 0-119. There are 4! ways of changing s, so there are 24 possible outcomes. It has been a long time since I last did some probability, so I'd wait for someone with more practice to explain it better than me :-)

Answer 7

Check http://courses.csail.mit.edu/6.042/spring10/cardTrick.pdf it's a lot more in depth and easier to read :-)

Answer 8

Wow. Thanks very much for this reference. This made me understand. =))

Answer 9

No problem, mate. Glad I could be of any help :-)