Magic Gopher "Reading your mind" puzzle solved!

My daughter Deborah introduced me to this puzzle - The Magic Gopher last evening. We were truly puzzled by the fact that it could correctly guess the symbol of our chosen numbers correctly no matter how many of us were playing the game. My younger daughter Rachel suggested that we should not even whisper our choices of numbers, while I suggested to punch the gopher, hoping that he would at least get it wrong in a daze.

Anyway, later that evening, we solved the puzzle. Here's how we did it.

1. Deborah noticed that the symbols associated with the numbers are different between one game and the next.
2. I looked at the numbers, selected those symbols with the lowest number (i.e. the rare symbols), and tried to work backwards to the numbers, but could not get any.
3. I then tried computing all the numbers in the 70 series, and found that 70 to 79 all resulted in 63. I tried 50-59 and found that they all worked out to 45. Then it dawned upon me that all numbers from 10 to 99 will always work out to be a multiple of 9.
4. I then looked at the gopher's chart and found that multiples of 9 are always assigned the same symbol in each game.
5. So, the gopher's game is up :D

Have a nice day :)