This is not a Twisty Puzzle, but is an entertaining 'puzzle' aside:

Try this

Perhaps the mathematical bods amongst us could explain to a simpleton like me how this works?

Richard

Weird, someone posted this to rec.puzzles the other day. If you go to that newsgroup, there is a discussion about it, the subject is something like: "7-up" puzzle
Hope that helps.

-d

Here's my half-assed go at an explanation.

First off, whatever your number is, after you subtract the scrambled version from the original (or vice versa), one crucial condition has to be met for the problem to work: the subtraction must result in a non-zero number. They don't really tell you this in the problem, but the game won't work if you choose a number and a scramble of it that subtract to 0.

Next, no matter how you scramble it up, when you subtract the smaller from the larger, the digits in the result will always add up to a multiple of 9. Knowing that, the rest is easy. To detect which number from 1-9 was removed from a group of digits, simply calculate the difference between the sum of the digits and the next highest multiple of 9.

Why does the sum of the digits in the difference always add up to 9? I dunno!

Sandy

The sum of the digits of a multiple of 9 always adds up to a multiple of 9, because of a strange quirk in algebra that mathematicians call 'casting out nines.' The condensed version of this law, is that whenever you start subtracting a bunch of nines from any two-or-more digit number, you'll eventually get the sum of the digits of your original number.

So, the trick works like this: When you do the first bit of subtraction, you get a multiple of nine. This means that the digits have to add up to a multiple of nine. (Why? Here's a hint: Every time you subtract nine from a multiple of nine, you still have a multiple of nine...)

When you remove a non-zero digit, you end up with another number. You enter in this number. It first checks to see if the digits add up a multiple of nine. If so, your number has to be nine, because of casting out nines and the fact that you can't remove a zero. Otherwise, it finds out the smallest multiple of nine that is greater than the sum it generated, and uses simple subtraction to figure out your number.

If you didn't follow a word of that, don't worry... it's not that important.