So I managed to get part 1 of the day, but it took 2 seconds to run on the real input, which is a bad sign.
I can't see any kind of optimisation that means I can skip checks and know how many combinations are in those skipped checks (aside from 0.) I can bail out of branches of combinations if the info so far won't fit, but that still leads me to visiting every valid combination which in one of the examples is 500k. (And probably way more in the input, since if I can't complete the example near instantly the input is not happening.)
Right now I take the string, then replace the first instance of a ? with the two possible options. Check it matches the check digits so far then use recursion on those two strings.
I can try to optimise the matching, but I don't think that solves the real problem of visiting every combination.
I don't think (or hope) it's just bad code but this is my code so far (python.)
edit:
spoiler
a cache was the solution!
One of the nice things about Gmail at the time, was that you could access your emails when not home. If you were at a friend's or on holiday at a net café, all you needed was to know your email and password.
That sounds silly, but at the time the majority of ISP mailboxes were pop only. Or those Webmails you could get were attached to what you would now think of comically small mailboxes. Full history Webmail added a convenience we didn't get before.