What is the trick?
If you want to find out how it works on your own, don't read this.....
Fun-dampener........
The trick to the game is that the gopher uses the same symbol for every multiple of 9, from 0 through 81 (it is impossible to get a higher number using only two digits). The gopher assigns the same symbol to the multiples of 9 as to other non-multiples of 9 in order to cover up the trick; the symbol picked for each game is randomized. No matter which two-digit integer the player chooses, when the subtraction is done, the resulting number will always be a multiple of 9. In fact, no matter which (nonnegative) integer the player chooses, the result will always be a multiple of 9. The former can be proven using elementary algebra.
Proof for 2 digits
Let n be a 2-digit integer. Additionally, let a be the first digit of n and b be the second digit of n. Finally, let c equal the sum of the digits of n, so c = a + b..
An equivalent form for n, by virtue of using a decimal numeral system, is n = 10a + b.
The resulting number, z, is given by z = n − c = (10a + b) − (a + b) = 9a. Hence, z is always a multiple of 9.