How does the 'mind reading' algorithm on websites such as the one on mysticalball.com work?

Question

(Think of a two digit number, add the digits, then subtract that number from your original and look at the symbol.) This certain website has more than one possible resaulting symbol, yet it is always right!

Answer 1

Write down the symbols of several of the other numbers-symbol combinations first then click the silver ball. You will see that the symbols changed. Next on the new page do the math that you were told to do on the first page for several different 2 digit Numbers. You will see that on the page of symbols it is impossible to do the math that they ask you to do without coming up with the same symbol as the symbol that will later be in the Chrystal ball. The math leads to, as the above answerers have said, a multiple of 9.

It is just that the human mind doesn't really remember what all of the symbols were, so it doesn't recognize that every time you reset the page it has been restacked to give a different symbol, thus giving the illusion of always being right. Also, people seldom will try doing the same number twice, but if they did they would quickly see through the trick because if you picked the same number twice in a row you would realize that the same number lead to a different symbol.

By the way good job to those above answerers for the math involved I prefered to just explain the trickery.

Answer 2

I think Pascal is thinking of something slightly different, but pretty much everything is there.
First think of a two digit number [ab]. This number (as Pacal said) can be written as 10a+b. The sum of the digits is naturally a+b. Thus the two digit number [ab] minus the sum of it's digits (a+b) is 10a+b-(a+b)=9a. and thus a multiple of 9.

As Pascal showed, this can also be done by not subtracting the sum of the digits but the reverse of the number (ie [ba] instead of [ab]).

Answer 3

Consider a two digit number ab: it can be written algebraically as the sum 10a+b. Its reverse is 10b+a. 10a+b-(10b+a) = (b-10b)+(10a-a) = 9a-9b = 9(a-b) -- therefore the answer you get is always a multiple of nine. Now look at the solution key, do you notice anything about the multiples of nine on that list? Yes, they are all the same symbol.