Hello
Sorry to be slow in answering, but I try not to look at my emails too often. Anyway, I'm amazed that you asked such a question to a nice economist such as Hamish. What do economists know about sums?
Anyway, 'formula' isn't the right word for what you're looking for. Some mathematical problems, such as finding the possible integer values for the sides of a right-angled triangle, do result in formulae that give all the possible answers, but when you're dealing with a number-letter substitution puzzle such as this one, the only reliable method is basically an algorithm that is really just well-informed trial and error.
Looking at it as ABC + DEF = GHI, where all the letters have distinct non-zero values, you solve it by making a table of all possible values for C, with all possible values of C beneath each one. That gives you the corresponding I values, and then you let the thing branch out further with the other letters (all on a huge sheet of paper).
Actually, there are 336 solutions, but since the values of A/D, B/E and C/F can be swapped around, any one solution leads to seven more, which means that only 42 of the solutions are independent. But that's still a lot of them, and any bright eight-year-old (or indeed a journalist or economist) ought to stumble across one pretty quickly.
It's a better puzzle if you make it XXX + XXX = XXXX using all the digits including zero (but not allowing any number to begin with zero). Even that has 96 solutions (or 12 independent ones).
My favourite one of these letter-number puzzles was one I composed for the Independent years ago which was the multiplication sum J x AITKEN = SLEAZE which has a unique solution. And that's very difficult.
Luv
Bill
PS: Lots of luv to Hamish too.
