I can guess your age!

     A popular magic act or parlor trick is to guess another person's age, someone whom you have just met for the first time, for example. I will present here a unique method for doing this sort of game that I found in a math magazine some years ago.

     Learn it, then amaze your friends with your magical skills.

     The procedure begins with three questions to ask the person:

  1. What is the remainder when your age is divided by 3?

  2. What is the remainder when your age is divided by 5?

  3. What is the remainder when your age is divided by 7?

     As you hear the answers, you secretly call the first answer x, the second one y, and the third one z. (Of course, you are going to have to assume the person can do arithmetic correctly!) Then, mentally or out of view of the other person, you substitute those answers into this algebraic expression:

70x + 21y + 15z

     The value that comes out will be the person's age, or (as you will see below) lets you easily adjust it to the correct age.


     Here are some examples to explain this. First, if the person is 36 years old, the remainders are x = 0, y = 1, and z = 1. Substituting into the expression gives

		  70(0)  +  21(1)  +  15(1)
		=  0	 +   21    +   15
		=  36.
     Next, for a 37-year-old person the remainders, in order, are 1, 2, and 2. This time substitution yields 70 + 42 + 30 = 142. But you can clearly see that the individual is not over 100 years (no gray hairs, no wrinkles, etc.). So you subtract 105, obtaining 37.

     Finally, if the person is 38, the remainders are 2, 3, and 3, producing a value of 248 in the expression. This time merely subtracting 105 is not sufficient -- 143 is about the same as the second example. But twice 105 (or 210) does the trick, giving the age of 38.

     It get easier with practice; that's always the way it is with magic. One last note: 105 is just the product of the three divisors, 3 × 5 × 7. So it's not hard to remember.

     Have fun...

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