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From the Fun Fact files, here is a Fun Fact at the Medium level:

Red-Black Pairs Card Trick

Here's a terrific mathematical card trick that will impress your friends. When you do this trick, the effect of the card trick will look like this:

You have a deck of cards, and you ask for a volunteer who knows how to do a riffle shuffle. You then cut the deck and then give the volunteer the halves of the deck and ask him to do one riffle shuffle and return the deck to you. Now say "There's no way I could know anything about the deck right now, right? Well, I was born with the amazing ability to feel the redness and blackness of cards with my fingertips. However, my talent is not that refined. I can only feel red and black cards in pairs." As you say this, put the deck of cards behind your back (so that you cannot see them) and then, at regular intervals, you fish around in the deck and pull out pairs of cards and show them to the audience. These pairs will all have exactly one black and one red card!

Presentation Suggestions:
Before performing the trick, order the deck alternating colors, all the way through, red-black-red-black-... etc. (When you flash the deck before their eyes, they really won't notice this pattern if you do it quickly.)

After this, there is really only one thing you need to remember to ensure that the trick works: you must cut the deck (not the spectator), and you must do it in such a way that the bottom of each half of the deck is a different color. Then, no matter how the spectator riffle shuffles the deck, the cards will always drop in red-black or black-red pairs. See below for explanation.

Then, all you have to do after the deck is returned and you put it behind your back is to pull out the top 2 cards. It will be either red-black or black-red! Then pull out the next 2 cards, which again will be red-black or black-red. You can continue in this fashion to the end of the deck, if you like!

Of course, you should make it look as if you are trying really hard to find the cards (even though what you are really doing is very easy). Spectators will wonder if you are pulling one card off the top and one card off the bottom; but you can pull the deck out and show them that this is not the case.

The Math Behind the Fact:
The reason the trick works at the point of the riffle shuffle is both simple and stunning: if you cut the deck so that the cards at the bottom of each half are different colors, then the first card that gets "dropped" in the shuffle will be a different color then the second card that gets dropped, no matter which half of the deck they come from. As an example, if the first card that gets dropped is black, then after that both halves will have red cards at the bottom, so no matter which card falls next it will be red! After this, both halves again have different colored cards at bottom and we are back to the situation at start. So all the cards will fall off in either red-black or black-red pairs. This amazing fact is a special case of something known as the Gilbreath principle.

The message of this trick is that one shuffle is not enough to randomize a deck of cards-- you really can know something about the deck after one shuffle... but only if you stack the deck in a particular way first!

There's more mathematical magic in the Fun Fact files.

How to Cite this Page:
Su, Francis E., et al. "Red-Black Pairs Card Trick." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Subjects:    other
Level:    Medium
Fun Fact suggested by:   Francis Su
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