Two magicians A and B perform the following trick:

A leaves the room and B chooses 4 members from the audience at random. Each member chooses a card numbered from 1 to 100 (each chooses a different card) and after B has seen their cards he chooses a card from the remaining deck of cards. The 5 chosen cards are shuffled by an audience member and handed to A who just returned to the room. Prove that A is able to figure out which cards each member picked. Consider that the chosen members form a row and e.g. the leftmost member picks the first card and the rightmost member (B) picks the last card.

The 4 cards that A is to reorder can be in any of 4!=24 orders.

It would be easy enough to work out a system where B picks a card from 1 to 24 to indicate which card is whose.  There are two problems that must be overcome:

1.  The needed card may already be taken.
2.  If there are other cards form 1 to 24, A won't know which is B's card.

Problem 1. can be overcome by letting B pick a card over 24 and then A would just subtract 24 from it.  (Take the card Mod 24)  If that card is taken as well then add 48 etc.

Problem 2 is the big problem: How can a card indicate which one it is and also indicate the proper order?

Posted by Jer on 2007-05-11 13:20:12