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# Always 22

By on 01/02/2016

Number magic is one of the most important “subclasses” of mathematical magic, especially useful in spontaneous situations in small groups of people as it usually requires no props, or maybe pencil and paper or a calculator, which nowadays everyone has on a cell phone. Almost everyone has at some point of his or her life met a trick of the type “think of a number – perform some calculations – the mathematician guesses your initial number or the result”. All of such tricks are based on elementary arithmetic and algebra. Here we present one of the less known of such tricks, which the author picked up years ago from a source she cannot remember any more :-).

In the beginning, the performer asks a spectator to think of a three digit number with all digits different. For the purpose of explanation, let us say that the spectator has thought of 123. Now, the spectator is asked to write all six combinations of two digit numbers with digits from the chosen number. In our case, these would be 12, 21, 13, 31, 23 and 32. (Note: As this step is often not clear to the spectator, when performing it, it is well to first make an example, and then let the spectator choose his number.)

Now, the spectator is asked to

• add all the six two digit numbers (to obtain number A) and
• to add all the three digits from the chosen number (to obtain number B).

To improve the presentation, the performer may want to talk about the arbitrarity of the chosen number, the different procedures in obtaining A and B making them impossible to be known to him (except “by magic” :-)), etc. Anyway, the spectator is finally asked to divide A by B and discard the remainder if there is any. In the most impressive way the performer can think of, he or she now announces that A/B is 22!

While the trick is relatively elaborate and thus may not be as attractive as many others, it is very suitable to use in classroom. The teacher may ask all the students to do their own calculations of A and B and compare the results: all who calculated correctly would have obtained 22. Now the teacher may use this observation to implement a inquiry lesson: Why is it so? Why was it important to start with a number with three different digits? In this way, the basics of decimal arithmetic are disovered. To help the inquiry, the teacher may direct the students’ attention to the question: Did anybody have a non-zero remainder when dividing A with B?

For those who want everything spelled out, here is the explanation: Let abc be the chosen number. Then the six two digit numbers formed from it are abbaaccabccb. When one adds them he gets

A = ab + ba + ac + ca + bc + cb = 10a + b + 10b + a + 10a + c + 10c + a + 10b + c + 10c + b = 22(a + b +c) = 22B

So A is divisible by B and the result is 22. If one would allow double digits, there would be only three two digit combinations (aaabba if a denotes the doubly appearing digit) and A would be 11B, so the result would be 11, so it is easy to adapt the trick to the beginning “think of a three digit number with not all digits equal” by finishing it of with e.g. “if A/B is less than 20, multiply the result by 4, if it is more than twenty, double it”, and thus ensure the final result to always be 44.