Good Eggs

This is a puzzle by Hubert Phillips (Caliban).

'You don't like arithmetic, child?' said Humpty Dumpty. 'I don't very much.'

'But I thought you were good at sums?' said Alice.

'So I am,' said Humpty Dumpty, 'Good at sums; oh, certainly. But what has that to do with liking them? When I qualified as a Good Egg -- many, many years ago, that was -- I got a better mark in arithmetic than any of the others who qualified. Not that that's saying a lot. None of us did as well in arithmetic as in any other subject.'

'How many subjects were there?' said Alice, interested.

'Ah!' said Humpty Dumpty, 'I must think. The number of subjects was one-third of the number of marks obtainable in any one subject. And I ought to mention that in no two subjects did I get the same mark, and that is also true of the other Good Eggs who qualified.'

'But you haven't told me --' began Alice.

'I know I haven't,' said Humpty Dumpty. 'I haven't told you how many marks in all one had to obtain to qualify. Well, I'll tell you now. It was a number equal to four times the maximum obtainable in one subject. And we all just managed to qualify.'

'But how many --' said Alice.

'I'm coming to that,' said Humpty Dumpty. 'How many of us were there? Well, when I tell you that no two of us obtained the same assortment of marks -- a thing that was only just possible Ð you'll be well on the way to the answer. But to make it as easy as I can for you, I'll put it another way. The number of other Good Eggs who qualified when I did, multiplied by the number of subjects (I've told you that already), gives a product equal to half the total number of marks obtained by each Good Egg. And now you can find out all you want to know.' He composed himself for a nap.