How Does the Crystal Ball Know What Number I Chose?

Date: 01/05/2010 at 02:44:25
From: Cheryl
Subject: algerbra

Hi Dr. Math,

I'm Cheryl.  I'm 12 years old this year.  I'm primary 6.

Can I ask you a question?

Go to this website:

The Flash Mind Reader

  http://www.exstatica.net/flash/psychic.swf 

How do you solve this?

I don't know how did they do that.

Date: 01/05/2010 at 10:18:23
From: Doctor Ian
Subject: Re: algerbra

Hi Cheryl,

The first thing to notice is that if you click without even thinking
of a number, the crystal ball comes up with something.  That's an
important clue!

Now that we know that our thoughts have no effect on the crystal ball,
let's pick a number -- say, 10.  You add its two digits to get 1, 
subtract from 10 to get 9, and the symbol next to 9 shows up.

Keep clicking the crystal ball.  Different symbols will show up ...
but more importantly, that symbol will always appear next to the 9!

What does this tell us?  Well, it suggests that only certain numbers
can be obtained through the operation they describe.

Here's the short answer to why it works is:  They put the same symbol
next to all the outputs that it's possible to get; so no matter what
number you think of -- or whether you even think of a number at all 
-- the crystal ball shows you the symbol next to the output for that 
number.

To explore this a little more, think about representing the two-digit
number 'ab' as

  10*a + b

For example, we could represent 74 by letting a = 7 and b = 4:

  10*7 + 4 = 74

So a stands for the tens digit, and b stands for the ones place.  

(To guarantee that we get only two-digit numbers, restrict a to the 
positive integers 1, 2, ..., 8, 9; and b to the integers 0, 1, 2, ... 
8, 9.)

Now let's take the operation they describe, and perform it on our 
variables.  Add up the digits, and subtract that sum from your number:

   (10a + b) - (a + b)
  = 10a + b - a - b
  = 10a - a
  = 9a

Interesting!  What you get has to be a multiple of 9.  And all the
products of 9 and a single non-zero digit are in the list:

  9 18 22 27 28 36 40 45 48 54 57 63 72 81 87 91 92
  *  *     *     *     *     *     *  *  *

The other numbers are thrown in just to confuse you.

So a more complete answer is:  No matter what number you pick, your
result is a multiple of 9; and all those multiples always have the
same symbol, which is the one the crystal ball shows you.

Does that make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/