Mathematics in a Deck of Cards

I would like to look at some mathematically-based activities with a deck of cards,

and understand what underlies them. Here are a couple that you might think about:

(a) ten pairs of cards are placed faced upon a table, and the subject is asked to select

one of them, but to keep the selection a secret; the cards are then picked up and dealt

into four rows of five cards each. The subject is asked to indicate the row(s) in which

the two selected cards appear. Thereupon, the dealer is able to identify the selected pair.

(b) The cards from 1 to 10 inclusive are placed in order in a fan that is presented

upside-down to the subject. The subject then is allowed to perform, as frequently as

desired, and in any order, the following two operations:

(i) the deck is cut, and the top part transferred to the bottom;

(ii) a consecutive pair of cards is grasped, turned over and put back in the same position

(with the card on the left going on the right, and vice versa, and a card showing its back

now showing its spot, and vice versa).

At the end, the dealer conceals the deck under the table, does something, and then shows the

deck, now having the odd cards facing in one direction (up or down) and the even in the other.

What did the dealer do, and why does it work? Other tricks of greater complexity will be shown,

and participants are welcome to bring along some of their own.

Ed Barbeau