### A Magical Incident due to the Paypal's Policy

The great online company paypal allows payment processing in one form or another from 190 countries. Lebanon is not one of them. Due to this mischief I was contacted by the Lebanese amateur puzzlist and magician Akram Najjar who wanted but could not purchase one of my applets.

Akram is the author of three puzzle ebooks (two are being sold at amazon.com, see his website *The Hidden Paw's Puzzle Site*. The third one came as an attachment to his email; I link to it with his permission. (The book *Mathematical Card Tricks* is in MSWord docx format.) As many other authors, Akram had to face the "his/her dilemma" in a way not offensive to either of the sexes. His charming solution betrays a practical aspect of the book: "To avoid saying 'he/she' all the time I will use the term ** victim** (VIC) for anyone who participates with the magician." The tricks included in the book are intended to be run by an audience with the audience participation.

The first one has immediately attracted my attention because of the recent posts and tweets of the New York city math teacher Patrick Honner. Patrick posted a well known problem:

Imagine yourself sitting in front of a cup of coffee and a cup of cream. Suppose you take a spoonful of cream, pour it into the coffee, and stir it up. Now once that’s thoroughly mixed, you take a spoonful of the mixture and pour it back into the cream. Then you mix that up.

After all of this, is there more more coffee in the cream, more cream in the coffee, or equal amounts in both?

Akram converted that problem to a magical card trick:

Sit across a table from VIC. Let VIC shuffle the deck and give you approximately half of it. You tell VIC to turn VIC’s pile FACE UP and you keep your pile FACE DOWN, with VIC’s full knowledge. Insist that throughout the exchange, the original direction of the pile should be maintained: VIC’s FACE UP and your FACE UP.

Ask VIC to call out a number between 1 and 10. Say the number was 6.

VIC should then give you 6 cards that are FACE UP.

Before adding VIC’s cards to yours, you also give VIC 6 cards of yours, FACE DOWN.

Each of you should place the cards as received. You place VIC’s cards as you received them: FACE UP. VIC also places your cards on top of VIC’s pile FACE DOWN.

Ask VIC to shuffle the deck without turning it over. In fact, VIC should never turn it over, nor, for the moment, should you.

Let VIC call out another number between 1 and 10. Now VIC gives you that number of cards (which may contain some of your FACE DOWN cards). Before you place them on top of yours, you give VIC the same number of cards (which may contain some of VIC’s FACE UP cards). You both place your “gifts” on top of your piles without changing their orientation. You both shuffle your piles, always keeping the

original orientation of the pile fixed.Repeat step 6 as many times as VIC wishes.

When VIC decides to stop, ask VIC to count the number of FACE DOWN cards (your cards) that VIC has in VIC’s deck.

While VIC is doing that, turn your deck upside down. This is a deception which you should carry out openly, ie, not under the table. Later on, you can say, I did it in front of you, but of course VIC would not have noticed.

Result: VIC’s FACE UP cards with you are now FACE DOWN. MAGIC: the number of FACE DOWN cards is the same in both decks? And how can it be after so much random exchange and

shuffling?

Explanation: no matter how many times you exchange cards and which cards are exchanged, at the end each of you will have the same number of cards in your piles as you started out with. This is because you always exchanged the same number of cards. So, it follows logically, that if I have X cards from VIC, VIC would have X cards from me. VIC’s cards in my pile would be FACE UP. My cards in VIC’s pile would be FACE DOWN. Therefore, by turning my deck upside down, the FACE UP cards will become FACE DOWN, hence matching those in VIC’s pile. MAGIC.

I believe that the discrete variant makes the solution more transparent. For example, think of coffee and cream not as liquids but as collections of molecules. Since the number of molecules in the two glasses remains the same even after repeated iterations, cream molecules in the "water" glass come at the expenses of the water molecules in the "cream" glass and, therefore, the two quantities are equal.

The discrete variant is reminiscent of a problem of matching dots of two colors arranged in two rows and its generalization. There is also a related card trick:

The deck of cards is held face up and fanned through with the player removing pairs of cards of the same color whenever they occur adjacent in the deck. Of course, the removal of adjacent pairs may create other adjacent pairs which will also have to be removed. The game ends when there are no more adjacent same-colored pairs to remove. Winning the game means having no cards left at the end. This can be turned into a magic trick as follows:

The magician calls upon two spectators to each take half the deck of cards and shuffle them independently. They then merge their stacks (alternatively dealing the cards into a single stack) and play the solitaire game. To the amazement of everyone, they find they have won.