Revealing the Magic Gopher's Secrets, and Learning How to Debunk Psychics

I recently received this puzzle in my e-mail, in which a psychic gopher magically predicts the number you are thinking of, after putting you through some apparently innocuous math on the way.

Of course, the math isn't innocuous, that's what makes the trick work. If the gopher could really read your mind, he wouldn't bother with the math, he'd just reveal your number right away. That is a good question to keep in mind any time one is confronted with a supposed psychic feat: if the psychic could really do what he claims he can do, is there a more direct path to the answer than the one he is taking. If there is, then the answer to how he does it is embedded somewhere in the process he goes through, be it asking you questions, waving a wand, chanting, whatever.

In this case, the math he makes you go through will always result in a multiple of nine, and if you look at the grid of numbers and symbols, all multiples of nine (0, 9, 18, 27, 36, etc.) have the same symbol. The symbol will change on each play, but it will always be the same for all multiples of 9.

Here's the math behind it. Any 2-digit number you choose can be represented as 10X + Y. For example, for 56, X = 5 and Y = 6. Now the gopher asks you to add the digits together, and then subtract that total from your original number. So you take (X + Y) and subtract it from (10X + Y):

10X + Y - (X + Y)

which is equivalent to:

10X + Y - X - Y

or the more visually appealing:

10X + Y
- X - Y

Since the Y's cancel out, that leaves 10X - X, or 9X. Thus, no matter what X and Y you choose, you will always have 9X when it comes time to pick the symbol from the list, and since the Gopher knows this, he need only predict the symbol next to all the multiples of nine to make it appear that he is being psychic.

I too am psychic, and I predict that no matter how many times you play the gopher's game, you will never finish with 62. Or 11. Or 50. Unless, that is, you need to work on your math.

Math like this is very often at the heart of psychic tricks. Watch for it, and ask the right questions. For me, understanding the math behind the tricks is far more interesting than believing it is some sort of magic.