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- Diderot Mathematical Forum 2013: “Mathematics of Planet Earth”
- Pierre de Fermat and Andrew Wiles in Czech Republic stamps
- Stefan Banach (March 30, 1892 – August 8, 1945)
- Diderot Mathematical Forums
- Guessing the Numbers
- What is mathematics for Ehrhard Behrends
- What is mathematics for Krzysztof Ciesielski
- The Three Ducks Trick
- What is mathematics for Franka Brueckler

# A symmetric trick

**One of the tricks that always impress the audience who do not know the simple geometry behind it is this simple card trick. It can be done quickly, and works well in all settings**

*by Franka Miriam Brueckler*

One of the tricks that always impress the audience who do not know the simple geometry behind it is this simple card trick. It can be done quickly, and works well in all settings. One needs a pre-prepared deck of cards for it, but don’t be afraid: the preparation is easy and there is no cheating here!

First we shall describe the effect, and then the secret behind it. The performance goes like this: The performer hands a deck of cards for a spectator to shuffle. The spectator returns the deck (face down) to the performer who spreads the cards and asks the spectator to take one, look at it, and then push it back into the deck. The spectator ia asked to shuflle the cards once again. After all of this the performer is able to find the card the spectator had in his hand!

The reason is simple, and yet fair. The trick works even when the spectator wants to go through the deck beforehand to check if everything is fair. It is, and the secret is under the spectator nose, but it is quite uncommon that he notices it. This secret is simply what mathematicians call 2-fold rotational symmetry: an object posesses it it can be rotated for half a circle (i.e., about an angle of 180 degrees about an axis) and it looks the same as before; this is called a twofold symmetry because when the rotation is twice repeated, we get to the beginning position. The capital letter “Z” (in most fonts) is an example of an object with 2-fold rotational symmetry. Can you find other letters with this property? Here is another example an object with 2-fold rotational symmetry and one without:

Now check an ordinary 52 card deck. How many cards WITHOUT 2-fold symmetry can you find? The answer depends on the concrete design of the deck you use. Usually, the Jacks, Queens and King posess 2-fold symmetry, and also the diamond cards. Often also all the even numbered cards are designed in a symmetric way. Anyhow, you should find at least 15 non-symmetric cards in your deck, and probably more. Here we have 25 such cards:

Now, to perform your trick remove all the symmetric cards and keep the non-symmetric ones. Orient them in such a way that you know what is up and what is down. We use the convention above: The side with more symbols pointing “up” is the upper side; still, you may be more comfortable with another view on the “up”-directin. The only thing you have to take care about during the trick is that the spectator does not rotate any cards when shuffling. While the spectator is lookin at his chosen card, you rotate the deck (that is why you ask to pull the card out, and not just peep). When the spectator pushes the card back in, it will be “upside down” and you will easily discover it when goint through the deck. That’s how simple it is!

N: B. This trick is a great intro into the discussion of symmetry in general and its importance for arts and sciences.

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