SDP Mind of Mystery Competition Winners

I announced a competition on December 23rd: play the Flash Mind Reader, and then e-mail us with explanations. 

Our readers being unusually shrewd, we like to think, it took only a few days for three winning submissions to arrive.  Mike Buus expressed a hope that we would keep the answer to ourselves, but SDP is far too chatty for that.  Besides, I like to think that nature has mystery enough in it without using fraudulent means to create new ones.  This is not to say there was anything wrong with the mind reader.  Its fun, and to the mathematically primitive (like myself), it is at first pleasantly mystifying.  But it is based on trickery.

In the game you choose a number between 10 and 99, and add the two digits together and then subtract from the original number (for example, with 99, 9 + 9 = 18; and 99-18 = 81.  Then you find the last number on a chart and are told to "concentrate on the symbol, and when you have it clearly in your mind" click on the crystal ball.  Invariably (if you did the math right) the symbol in your mind will appear on the crystal ball.  As I said, for many folk, it feels like magic the first time through.  But without the benefit of much math, I noticed that although there were a lot of different symbols, the mathematical formula produced only a handful of answers.  For example, 99 yields 81, but so does 98 and the next seven numbers down.  89-80 all produce 72.   The machine makes sure that all the possible  resulting answers have the same symbol, and that is the one it produces. 

Aside from Mr. Buus, Joel Arends, and Gene Johansen submitted solutions.  Mr. Johansen has the most thorough explanation of the math. 

Any two digit number can be expressed in this form:
10x + y

Then, subtracting the sum of the two digits gives:

10x + y -(x + y)
= 10x + y - x - y
= 9x

So, following the instructions will always give 
you a number divisible by 9.

Look at all the symbols next to numbers divisible by 9 
(except for 90 and 99,
which can't be hit).  
They are all the same, and will match the symbol that
shows up in the crystal ball.

The symbols are scrambled each time so it's harder 
to see the pattern.

And if that didn't make any sense then it's done with Voodoo.

Mr. Johansen shrewdly points out that the symbols are changed with each play, which makes it look much more mysterious.  Well done.