Math CentralQuandaries & Queries


Question from Valentino, a student:

What is the explanation for the following math trick: you think of any four digit number add those digits take that sum and subtract it from the first number then you say three of those new numbers in any order and the other person guesses the last digit.
How does he do it?

Any number minus the sum of its digits gives a multiple of 9. And the sum of the digits of a multiple of 9 is again a multiple of 9. So the magician will pick a number so that the four numbers add up to 9.

Most of the times, there is only one way to go, and the magician is sure to get it right. However, if the three digits already add up to 9, the magician cannot be sure whether the fourth digit is a 0 or a 9, so he will have to take a guess and hope he's right.

For instance, If I pick 2001, I get 2001-3 = 1998 and I can say 1, 9, 8 to the magician. He's supposed to guess the last 9. But If I had picked 1101, I'd have gotten 1101-3 = 1098 and I could have said 1, 9, 8 to the magician. That time he would be supposed to guess 0 rather than 9.

You can search our database with the keywords casting out nines to find other similar tricks.


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