## mardi, décembre 12, 2006

### La marmotte magique: la solution

The Magic Gopher asks you to first pick a two-digit number. He should ask you to pick a two-digit whole number, because a decimal like 2.5 (that's what I picked the first time through) doesn't work in the trick.
No matter what two-digit whole number you decide on, this number can be represented algebraically as 10 × a + b, or 10a + b. In this form, the number 99 is written as 10(9) + 9, and the number 10 is written 10(1) + 0. You can see that no matter what number you pick, it can be written in this form. You can also see that the variables a and b above are the digits of your number. So, the number 65, for example, is written as 10(6) + 5. The a represents the first digit, 6, and the b represents the second digit in your number, 5.
Next, the Magic Gopher asks you to find the sum of the digits in your number. Because the digits are a and b, we can represent this step algebraically as a + b.
Finally, the Magic Gopher asks you to subtract the sum you found in the second step from your original number. In other words, he's (she's?) asking you to subtract (a + b) [the sum of the digits in the number] from 10a + b [your original number].
This is represented as 10a + b − (a + b), which can be rewritten as
10a + b − a − b.
After we subtract and add, we get 9a. So, no matter what two-digit number you pick, your result, after following all the steps, will be 9 times the first digit in your original number. This means that your result will only ever be a multiple of 9. The first nine multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81. This list is enough because your first digit cannot be less than 1 and it cannot be greater than 9.
So, suppose I pick 65. I add the two digits of the number (6 + 5 = 11), and I subtract this from my original number (65 − 11 = 54). Nine times the first digit in my original number is 9 × 6, or 54.
Next, the Magic Gopher shows you a chart with different symbols next to the numbers 0 to 95. Notice that the symbol next to each multiple of 9 is the exact same. This is how the Magic Gopher can "read your mind." He knows you will pick a multiple of 9, so he sticks the same symbol next to each one.
The clever part of the magic trick is that your symbol changes each time you go back. Helps to throw you off the scent. But, although the symbol varies, it is always the same symbol next to the multiples of 9.