[fh]

I'm sure there's a cute mathematical trick to calculate the missing digit, but Raphael is right, with numbers limited to 1 billion, this can be easily brute forced. You need a few gigabytes of disk space and a few days of precalculation at most.

What's more interesting to me is that the missing number is actually unique, i.e. that this trick is possible at all. Maybe it actually isn't, maybe it doesn't work in 0.1% of the cases, which is good enough for a magic trick.

[RiderOfGiraffes]

I should also have said - this trick doesn't rely on the limit of the number of digits, it works with any number of digits. Use an arbitrary precision package or sensible language, multiply numbers together for as long as you like, then tell me all but one of the digits in the answer, and 95% of the time I can tell you the one you left out.

[RiderOfGiraffes]

It works in very nearly (but not quite) 95% of cases.