## Friday, March 28, 2008

### Arrange and Sum

Take the four digits: 1, 2, 3, and 4 and form a four digit number such as 2,314. You can produce 24 different four digit numbers this way (4 x 3 x 2 x 1 = 24).

Now here's the question. What is the sum of all 24 numbers?

There is a shortcut to answering this question. But feel free to figure out all 24 numbers and start adding with your favorite spreadsheet! Of course, the real question here is figure out the shortcut?

1. 66660? I did that in like 15 seconds, so that may not be right...

2. 354

3. 354??

Yeah, I thought about it a little more, and that should be right (66660). If you're adding up all the one's digits, you'll have 6 numbers with a 1, 6 with a 2, 6 with a 3, and 6 with a 4. That equals 6 X (1+2+3+4) = 6 X 10 = 60. The same will be true for all the other columns you add up, only the 6 (from 60) will carry over to the next digit.

4. You figured it out Abe:

Note that each of the columns when summing the 24 numbers consist of six 1's, six 2's, six 3's, and six 4's. Add those numbers up and you get 60. So, each column adds up to 60. Use that to add up the four columns of numbers to get
___60
__60
_60
60
-------
66,660

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