## 2 Mar 2011

### Number Trick

Here is a math trick that might get you thinking...

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

Try it out first!

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Have you tried it out yet? If you're too lazy to figure out what the result is, what happens is that the final answer becomes the phone number you have. And you don't have to use a phone number, just any 7-digit number would do.

So let's say I use 1234567. At step 2 I key in 123.
Step 3: 123 * 80 = 9840.
Step 4: 9840 + 1 = 9841.
Step 5: 9841 * 250 = 2460250.
Step 6 and 7: 2460250 + 4567 + 4567 = 2469384.
Step 8: 2469384 - 250 = 2469134.
Step 9: 2469134 / 2 = 1234567. (!)

But what happens exactly when doing this trick? Usually algebra helps in dissecting such number tricks. I'll attempt to see what happens when you do the sequence, but this time using unknown variables to make it clearer to see how each digits is manipulated.

Let's call our 7-digit number ABCDEFG. At step 2, the three digits used are ABC, but since A, B and C are just digits, the actual numerical value of the 3-digit number is 100A + 10B + C.

Step 3:
`80(100A + 10B + C) = 8000A + 800B + 80C`

Step 4: adding 1 just gives
`8000A + 800B + 80C + 1`

Step 5:
```250(8000A + 800B + 80C + 1)
= 2000000A + 200000B + 20000C + 250```

Step 6 + 7: The other 4 digits are DEFG, but the numerical value is 1000D + 100E + 10F + G. So adding these 4 digits twice gives...
```2000000A + 200000B + 20000C + 250
+ 2(1000D + 100E + 10F + G)
= 2000000A + 200000B + 20000C + 2000D
+ 200E + 20F + 2G + 250```

Step 8: Subtracting by 250 gives this
`2000000A + 200000B + 20000C + 2000D + 200E + 20F + 2G`

Step 9:
```(2000000A + 200000B + 20000C + 2000D + 200E + 20F + 2G) / 2
= 1000000A + 100000B + 10000C + 1000D
+ 100E + 10F + G```

If you haven't discovered yet, 1000000A + 100000B + 10000C + 1000D + 100E + 10F + G as a 7-digit number would look like ABCDEFG. So all these steps are just a very very long-winded way of getting your digits in the right place value. The addition of 1 and subtraction of 250 don't affect anything since they cancel off each other.

And that's what number tricks tend to be like. It lists a really long sequence of steps which when observed carefully is actually just a long way of doing a simple operation. However, it's not easy to see at first glance since it's all just words.

Well, I guess that's two uses of algebra in real life there: 1) breaking down a number "magic" trick and 2) making your own tricks! Although both aren't really considered that productive...