benlehman

So, in the interests of full disclosure, I should say that I'm writing this from a local internet cafe whilst people of all ages tango around me. Just, you know, in case it should bias you.

I started working on a game design this afternoon, and ending up spending nearly 6 hours investigating the basics of combinatorics, a pretty cool little branch of mathematics that has a little to do with everything. In public interest, I thought I would post about it. Those who are mathematically interested might find the problems enjoyable. You can also spoil them very easily by looking in any probability textbook, but what fun is that? :-)

Here are the problems, in general assortment:

1) Say you have a number of different types of objects, some of which are distinguishable and some are indistuingishable. Given a general number of object types m, with number of objects in each type n1, n2, etc., how many possible orderings are there? (I have solved this one, although I started with the case of only (n1, n2).)

1a) What particular set (n1, n2, n3...) produces, with growing n1, the point buy sequence (1, 3, 6, 10...) so loved by FGS GMs? What games was/is this sequence used for?

2) Say you have the same initial set-up, but only a limited number of "slots," k, in which the objects can be placed? How many combinations are possible then? (not by me solved except in trivial cases)

3) Say you have the object types m, with number of objects n1, n2, etc., and number of slots, k, but the order of the objects does not matter. How many combinations are possible then? (also not yet solved by me.)

Have fun, if it's your sort of thing... I just like playing with coins, really...