How is this trick done?

Factual Questions
1

I’ll bet this has been done on the board before, but I didn’t it find on a search.

http://digicc.com/fido/

So, how does it work?

2

The difference between a multi-digit number and any of its “anagrams” is always a multiple of nine. Proof on request.

The sum of the digits of a multiple of nine is itself always nine (or a multiple thereof). Proof on request.

So Fido adds up the numbers you type in, if the result is multiple digits then he adds 'em again, and once he’s down to a single digit he subtracts it from 9. He doesn’t let you circle 0 as he couldn’t tell if you’d circled a zero or a nine.

3

When you do the first few steps you end up with a number that, when you add the digits is divisible by 9. Say 2205 or 99 or 495. The number you type in provides the answer, it is the ndifference to the next multiple of 9 from the total of the digits.

6894
4689 = 2205

6894
4986 = 1908

9864
6894 = 2970 etc

4

Thanks, I get your first two points, but I’m fuzzy on the last step, where you “adds 'em again”. Can you provide an example?

5

Think up: 9631
Rearrange: 1963
Difference: 7668
You circle: 6
Fido adds: 7 + 6 + 8 = 21. It’s more than one digit, so Fido adds 2 + 1 = 3
Fido subtracts 3 from 9 to yield 6.