## Friday, March 03, 2006

I know some folks come here just for the math, so here's another little piece of math to keep you coming back.

The other day I got an interesting e-mail from my fearless sister-in-law Ginger. (Ginger is a closet mathematician, but she just doesn't know it yet. I got her addicted to Sudoku, and now she's sending me math e-mails. It's only a matter of time before she starts doing calculus for fun.) Anyhow, here's what the message says:

1. GRAB A CALCULATOR. (YOU WON'T BE ABLE TO DO THIS ONE IN YOUR HEAD)
2. KEY IN THE FIRST THREE DIGITS OF YOUR PHONE NUMBER (NOT THE AREA  CODE)
3. MULTIPLY BY 80
5. MULTIPLY BY 250
8. SUBTRACT 250
9. DIVIDE NUMBER BY 2

I tried it on my old phone number in Illinois, 367-5384. (As an aside, that was the coolest phone number ever. It spelled DORKETH. One can be so fortunate only once in life, I suppose.)

3. 367 x 80 = 29,360
4. 29,360 + 1 = 29,361
5. 29,361 x 250 = 7,340,250
6. 7,340,250 + 5,384 = 7,345,634
7. 7,345,634 + 5,384 = 7,351,018
8. 7,351,018 – 250 = 7,350,768
9. 7,350,768/2 = 3,675,384

That is indeed a familiar number! It's the original phone number I started out with.

This trick will work with any phone number. I know you're all asking why. At least, the folks who are in this for the math are, and they're the ones who count (ha ha! a little math humor there!) so I'll now explain how this trick works.

First of all, remember from elementary school how starting from the decimal point and going to the leftt, we have the ones place, then the tens place, then the hundreds, etc.? In each of those slots, we can put any integer from 0 to 9, and this represents a number. The number is just the sum of each of those integers times the proper power of ten; for example,
436 = 6 x 1 + 3 x 10 + 4 x 100
= 6 + 30 + 400
= 436.

So, if we were to write out our phone number as an integer, we could write it as
abcdefg = g x 1 + f x 10 + e x 100 + d x 1000 + c x 10,000 + b x 100,000 + a x 1,000,000 = g + 10 f + 100 e + 1000 d + 10,000 c +100,000 b + 1,000,000 a
= g + f0 + e00 + d000 + c0,000 + b00,000 + a,000,000
= abcdefg.

So, following the above algorithm:
3. abc x 80 = (c + 10 b + 100 a) x 80 = 80 c + 800 b + 8000 a.

4. (80 c + 800 b + 8000 a) + 1 = 80 c + 800 b + 8000 a + 1.

5. (80 c + 800 b + 8000 a + 1) x 250 = 20,000 c + 200,000 b + 2,000,000 a + 250.

6. (20,000 c + 200,000 b + 2,000,000 a + 250) + defg = 20,000 c + 200,000 b + 2,000,000 a + 250 + g + 10 f + 100 e + 1000 d.

7. (20,000 c + 200,000 b + 2,000,000 a + 250 + g + 10 f + 100 e + 1000 d) + defg = 20, 000 c + 200,000 b + 2,000,000 a + 250 + 2 g + 20 f + 200 e + 2000 d
= 250 + 2 g + 20 f + 200 e + 2000 d + 20,000 c + 200,000 b + 2,000,000 a.

8. (250 + 2 g + 20 f + 200 e + 2000 d + 20,000 c + 200,000 b + 2,000,000 a) – 250 = 2 g + 20 f + 200 e + 2000 d + 20,000 c + 200,000 b + 2,000,000 a.

9. (2 g + 20 f + 200 e + 2000 d + 20,000 c + 200,000 b + 2,000,000 a)/2 = g + 10 f +100 e + 1000 d + 10,000 c + 100,000 b + 1,000,000 a
= g + f0 + e00 + d000 + c0,000 + b00,000 + a,000,000
= abcdefg (!)

I thought this was a cute trick. I wrote all the steps out there, but you can see, if you gather like terms, that after step 5, we have 2 x a,bc0,000 + 250. (The 250 is just there to throw you off the track.) After step 7, we have 2 x a,bcd,efg + 250, and after we take away the extraneous 250, we obtain 2 x a,bcd,efg. Finally, in the last step, we divide by two and get back the original number.

I have to admit that I did follow step one and pull out the old calculator when I first tried this trick. Although perhaps it would have been more fun to do it with my trusty slide rule or my new abacus that I bought in San Francisco. Until next time, math fans!