## Friday, December 31, 2010

### Beauty of abstract simplicity, again

Last night I wrote about how abstract knowledge makes complicated problems simpler. This morning I remembered my favorite example of using abstract knowledge to solve a complicated problem. Something like this appeared in the ARML test many years back. Here's the problem:
15! = 130767?368000
Now figure out what digit ? is.

Of course, looking this up on Google calculator gives you the answer, but suppose you can't use a calculator. Well, 15! = 15 x 14 x 13 x 12 x ... x 2 x 1. Do you really want to spend all your time computing that by hand?

There are a lot of concrete facts you can name about the number 15! but the only one that's relevant to this problem is that 15! is clearly divisible by 9. This, it turns out, is all you ever need to know to figure out a missing digit. We can use the method of casting out nines to solve the problem in just seconds. We have
130767?368000
Take out the 1 and the 8, a 3 and a 6, the other 3 and 6, and we're left with 7 and 7, which make 14, whose digits sum to 5. Since the number should be divisible by 9, we need ? to be a digit such that ? + 5 is divisible by 9; the only choice is ? = 4, which is indeed the answer you get on Google calculator.