The Magic Gopher Math Game is an interactive online game presented by the British Council Learn English website and was created by Cambridge English Online, LTD. The Magic Gopher is presented as a math-magic trick that, once decoded, you can use to amaze your friends. To completely understand the math behind how it works, familiarity with modular math is helpful, but the basic premise is easy to understand and just as easy to do yourself.
In the Magic Gopher game, the gopher asks you to choose any two-digit number. You are then asked to add the two numbers of your two-digit number until it is reduced to a single-digit number. You are then instructed to subtract that result from your original number. The gopher then shows you a chart of symbols and tells you to find your number on the chart and remember the symbol next to it. He then guesses what your symbol is -- and he's always right.
Casting out Nines
The Gopher Game is based on the mathematical operation for checking multiplication and addition called casting out nines, which is detailed in the next section. The way this works in the Gopher Game is that for any two-digit number, if you add the two digits until they are reduced to a single digit, then subtract that single digit from the original two-digit number, it will always be a multiple of nine. For example, if your number is 32, summed it is 5, subtracted from 32 it is 27, which is divisible by nine. It works every time. Try it.
The only reason this method works is because of the base-10 numbering system we use. Here's an example. If you add 13,759 and 549, the answer is 14,308. To reduce this number to a single digit, throw out the 8 and the 1, as they sum to 9, and add the 4 and the 3, which equals 7. To check your work, add the digits in 13,759 together, but cast out the 9 and 1, 3 and 5 because together they total 9. What's left is the 7, which is the answer. Next take 549, cast out the 9 and the 5 and 4 because they sum nine, and the result of these two operations is 7, which matches the result of reducing the original answer, 14,308, to one digit. This means the answer to the original addition problem is correct.
The Gopher's Trick
First, you must understand that the gopher is not guessing the number you chose, he is guessing the symbol that represents your number. The gopher is always able to guess your symbol, because he knows that your number must be a multiple of nine -- 9, 18, 27, 36, 45, 54, 63, 72 or 81. When you are presented with the symbol chart, if you look at the multiples of nine, you will see that they are all represented by the same symbol -- the gopher can't lose.