[Ack]

Geoschmo is correct that you cannot use a combination equation for a round robin. This is from some website:

Example:

How many ways can we select three letters from the letters of RSTUV?

n = 5 r = 3

These are: RST, RSU, RSV, RTU, RTV, RUV, STU, STV, SUV and TUV.

From this you can see that, if RSTUV represented players, players would meet more than once (RS for example).

Permutation equations do not work for exactly the same reason.

Geoschmo,

I believe the only way you can avoid having players meet more than once is if the number of players is the square of the number of players per game. Actually another case is if the number of players is equal to the number of players per game (or 1 game). Anything other than this and you'll either having players facing each other multiple times or rounds where players do not play.

Next, the most possible games occurs if the number of players per game is equal to one. In this case the number of games is the summation of A (from A=1 to A= n-1). Where n is the total number of players. This is an important event.

So adjusting for the number of players, which should be a simple division, gives this:

Pg = players per game
n = total number of players

# games = (1/Pg) Summation (A=1 to A=n-1)

Solving a few cases:

Pg = 1, n = 1: # games = 1
Pg = 4, n = 2: # games = 3 (wrong, should be 6)
Pg = 9, n = 3: # games = 12

In summations, having variation between the odd and even entries is fairly common. So you need another equation for the even values, which I'm not going to work out tonight.

I suspect there is a more graceful solution by using some series expansions. Try looking at Taylor, Binomial, Geometric, etc. Series Expansions to find a better solution. To help you along, try determining the number of games required for 25-5 and 36-6.

One more quick thing. The number of games per round is simply n/Pg. So another option would be to use this and find a series that describes the number of rounds.