Paul got a very good first (think it was a top 5). Only a 2:1 for me though (although only a couple of marks off a first, well "ish").
For maths, in Part 1A, you take four 3 hour papers which are split into two sections and cover two of the modules you have studied per paper - section A is 4 easier questions worth 10 marks each (2 from each topic). Section B (in the first year) is 8 questions, four from each topic, from which you pick 3 questions from one and 2 from another and each question is out of 20. (I think in part 1B you choose 6 questions but there are far more topics to a paper).
By the old system, if you got 8, 9 or 10 you got an alpha from the section A questions and if you got 5 or 6 you got a beta (I'm not sure whether the 7 was an alpha or beta). I think the section B questions were 15, 16, 17, 18, 19 or 20 for a double alpha and 10 to 14 for a double beta (but I might be wrong).
In the new system, it's 7, 8, 9 or 10 to get a beta on the first questions and 15-20 to get an alpha or 10-14 for a beta on section B questions.
Last year you got a raw score out of 540 and an alpha and beta mark. From that, they calculate your "weighted" score by using a formula like
weighted score = raw score + 2 x beta count + 6 x alpha count (if alpha count is less than 10)
=raw score + 2 x beta count + 10 x alpha count - 36 (if alpha count is 10 or more)
or something close to that. If your score was near a grade boundary though, they look at each case individually, and if you're slightly below the boundary but have a high alpha count, then they'll look on you favourably, but if you have few alphas and lots of betas then it will disavantage you.
So yeah, it's not really that simple and it changes from year to year anyway, but that's the gist of it. For what it's worth, most of the other subjects that I know of seem to be a bit more sane with their marking and just rank order you by your score then give the top x% firsts and so on.