# the Math behind the Magic

Here is a little trick I like to call calculator magic. You will need a calculator, a 7-digit phone number and an unwitting bystander. Here goes:

1. Key in the first three digits of your phone number
2. Multiply by 80
4. Multiply by 250
7. Subtract 250
8. Divide the number by 2
9. Surprise! It is your phone number!

Before we analyze this little problem, it is probably best for you to try it with your own phone number.

Done? All right, I will now show you the math behind the magic with the help of one of my favourite friends, Algebra.

While there are numerous variations on this party trick, what they all have in common are a couple of things. First, they treat the 7 digit number as a complete unit, without the break. That is, they view   123-4567 as 1 234 567. The second thing they do is view the first 3 numbers as one subsection of that unit and the last 4 digits as a second subsection.

Another key piece to this trick is to do something to the first subsection so that it can be easily combined with the second section in numerical fashion.  What we do in this case is to multiply 123 by  10 000 so that we get 1 230 000. As you can see, if we add the second subsection (4567) to this manipulated first section, we get 1 234 567.

Hence, in general, if we call the first piece “a” and the second piece “b”, we can write your number as seen below.

(a × 10 000) + (b)

(123 × 10 000) + (4567) = 1 234 567

Now let us walk through the above trick in 2 ways, one with a specific number, say 123-4567 and one with our general number we defined above.

Step 1. 123 = 123

Step 2. 123 × 80 = 9840

Step 3. (9840) + 1 = 9841

Step 4. (9841) × 250 = 2460250

Step 5. 2460250 + 4567 = 2464817

Step 6. 2464817 + 4567 = 2469384

Step 7. 2469384 – 250 = 2469134

Step 8. 2469134 / 2 = 1234567

Step 1. a

Step 2. a × 80

Step 3. (a × 80) + 1

Step 4. ((a × 80) + 1) × 250

Step 5. ((a × 80) + 1) × 250 + b

Step 6. ((a × 80) + 1) × 250 + b + b

Step 7. ((a × 80) + 1) × 250 + b + b – 250

Step 8. [((a × 80) + 1) × 250 + b + b – 250] / 2

As you can see in the left box, the trick worked as intended, we arrived at our phone number. In the right box at the bottom, we have a very messy formula. The reason for this is that on each line in the left box we were able to calculate the number since we were working with a specific example. Since we were working in general on the right side however, we need to wait until the end to simplify. So time to simplify and see what we get.

[((a × 80) + 1) × 250 + b + b – 250] / 2

= [((a × 80) + 1) × 250 + 2b – 250] / 2                        By combining the 2 b’s

= [(80a + 1) × 250 + 2b – 250] / 2                   By combining the 80 a’s

= [20 000a + 250 + 2b – 250] / 2                                By multiplying the 250

= [20 000a + 2b + 250 – 250] / 2                                By rearranging

= [20 000a + 2b + 0] / 2                                              By adding and subtracting

= 10 000a + b                                                  By dividing by 2

= (a × 10 000) + (b)                                         By rearranging

Presto, the trick is up! All those fancy additions, subtractions, multiplications and divisions caused us to end up with our phone number automatically. By investigating the trick in general using Algebra we can see how no skill was required on the magician’s part, it was all in the Math.

For the avid reader I have made an example of my own below. For the really brave, try to create your own trick! Remember, you should end up with (a × 10 000) + (b) when all the fancy math is over.

1. Key in the first three digits of your phone number
2. Multiply by 16