Go here..

http://digicc.com/fido/

I know this is just a sequence of math manipulations to get the answer but what are they?

 

Well, the way I can figure it out is because when you take a number and subtract out a mixture of itself, the result of the numbers will add up to 9.

For example, take 9721 and subtract out 7912 = 1809 = 1+8+0+9 = 18
9721-1279 = 8442 = 8+4+4+2 = 18
9721-2179 = 7542 = 7+5+4+2 = 18 etc.

So if you put in 109 for 1809, it will guess 8 because 9-1=8 and 9-9=0 and so on. That's the best I can explain it. Hopefully a real mathematician can make it more clear.

 

ZGiRL has the essential points:

1. When you have two numbers with the same digits and subtract one from the other, the difference is always divisible by nine.

2. When you add the digits of a number divisible by nine, the sum is divisible by nine.

All the website is doing is adding up the digits you type in; if the answer is a two-digit number, they add those digits till they get a single digit number. Because all of the digits must add to nine, subtracting this final sum from nine gives the missing digit.

Note that if the final sum is a nine, the missing digit is a nine, because the rules say you may not circle a zero.

(If you want to see why the first two statements are true, let me know; but be prepared for some number theory that may be difficult to follow.)