Here is the solution to Adam McBride’s problem.
What one needs to explain here is why the result is independent of the 3 digit number you start with.
A general 3 digit whole positive number has the form , where are integers between 0 and 9. So let us start with some such number and without loss of generality we can assume that (because is not allowed and if , we anyway are going to turn the number around and subtract the smaller from the bigger one).
Turning this around, we have
It is not yet clear what the form of is, because is negative as by assumption. So let us do the oldest trick in the Book and add and subtract 1:
So , with , , and . Now we can turn around. This gives
Finally, all we have to do is add and :
and all the and have cancelled ( got rid of long ago, in the computation of ).
By the way note that is necessarily divisible by 9 and 11 because .
Postscript. Some of you may have recognised the title to this post as a reference to Sellar & Yeatman’s 1066 and All That, the classic account of school history as remembered years later. It’s also the title of a book by the remarkably versatile applied mathematician David Acheson — see this article on plus.maths.org to find out more. And speaking of Books… (DP)