Posted by: danb | October 4, 2008

## Card Trick- Guessing Pairs of Cards

When I read Anthony’s post, it reminded me of a card trick I have seen many times before, but never really sat down to think about (because I had usually just figured that the person doing the trick somehow cheated). It can be disguised in multiple ways (I’ll explain later), but each way relies on the same basic principles.  Here is how the trick works:

1. The person performing the trick starts out by removing tens and picture cards from the deck.

2. He or she fans the cards and has you pick one (let this be C1), tells you to look at it, then has you put it back in the deck.

3.  The “magician” then tells you to multiply your card’s value by 2 and add 5 to that number.  Then you multiply that number by 5.  You don’t tell the magician what this number is, but remember it in your head.  Let this number be T1.

4. You pick another card from the deck (let this be C2), and add the value of that number (let this number equal T2) to T1.

5.  The magician then tells you which two cards you had, and which was the first and which was the second.  Unfortunately the mathematical methods of this trick don’t reveal the suit.

So, let’s take a look at how this works.  After you pick the first card, you are told to multiply it by 2, add 5, then multiply it by 5.  Algebraically, we have T1=[2*(C1) + 5]* 5.  We pick the second card, so we have T2=C2, and add that to T1, giving T1+T2.  This is the number the magician is working with.  If we multiply T1 out and add T2, we get 10(C1) + 25 + C2=T1 + T2.  Now, let’s break this down and try to figure out what the magician is doing.

10(C1) is in the set 10, 20, 30…90.  The tens digit is C1.  When we add C2 (which is a single-digit because we removed tens and face cards), we get the tens digit being C1 and the ones digit being C2.  But we still add 25, and give that number to the magician.  All he does then is subtract 25 from your number and breaks down the digits to tell you your cards.

So you can see, there is very simple math behind this, and when you take the time to figure it out, you see what the magician is doing.  A magician will sometimes ask for you to simply pick two numbers 0-9 and do the same trick.  The only difference here is that 0 is in the domain, but the only effect is that if C1=0, the final number (after subtracting 25) will be a single digit.  Or, if C2=0, then the final number will be a multiple of 10.

Finally, it is not completely necessary to use the constants that I gave.  I let T1=[2*(C1) + 5]* 5.  Generalize it to this form: T1=[A*(C1) + B]* D.  The only requirements are that AD=10 and A,B,D are non-negative.  Then, at the end of the trick, instead of subtracting 25, you subtract BD.  For example, if it were T1=[1*(C1) + 4]* 10, after adding T1+T2, you would subtract 40.