Quandaries and Queries


Dear Q&Q,

I'm a father and a grandfather and have come up with a game for my offspring to play while we're on the road. When we see a license plate, the object is to be the first one to add all the numbers on it, and come up with THE one digit number that sums them up.

For example: ABC-787 = 7+8+7 = 22 = 2+2 = 4

Another example is 2932 = 2+9+3+2 = 16 = 1+6 = 7

Up 'til now, I've called it just plain "Numerology", but I'm sure that there's a math term for what we're doing, and I'd sure appreciate it if you could tell me what it is!



Hi Gerry,

The procedure is called "casting out 9's". Here are three other ways to do it that will make the name made sense and maybe speed your calculations.

  1. Add the digits, divide by 9 and report the remainder.

    7+8+7 = 22. Then 22 divided by 9 is 2 with a remainder of 4.
    (Cast out 2 nines and the remainder is 4.)

    2+9+3+2 = 16. Then 16 divided by 9 is 1 with a remainder of 7.
    (Cast out 1 nine and the remainder is 7.)

  2. Once you see this you can do it faster by casting out the nines as you go.

    7+8 = 15, cast out a nine and have 6 remaining. 6+7 = 13, cast out a nine and be left with 4.

    The second problem is even easier. From 2+9+3+2 cast out the 9 and have 2+3+2 which is 7.

  3. You can also work with negatives. 8 is one less than 9 and 7 is two less than 9 so the first problem is

    7+8+7 = 7-2-1 = 4

Chris and Penny


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