Can one of you geniuses help me with a real-world problem?

Eight of us are soon going to Scotland to play golf for a week or so. How many rounds do we need to play to insure that each player plays in the same group with every other player exactly the same number of times during the trip? [note: a "group" on the golf course is 2,3, or 4 players].

My brother is considering going (which would make 9 of us). Same question. A general formula would help for the following year's trip, as the group seems to be getting bigger each year.

If one of you will solve this for me, I can spend more time practicing my chip shots before we go!

Originally posted by camilli Can one of you geniuses help me with a real-world problem?

Eight of us are soon going to Scotland to play golf for a week or so. How many rounds do we need to play to insure that each player plays in the same group with every other player exactly the same number of times during the trip? [note: a "group" on the golf course is 2,3, or 4 players].

My ...[text shortened]... e of you will solve this for me, I can spend more time practicing my chip shots before we go!

Let:

N = number of players in total
G = size of each group

Then there are N!/(G!*(N-G)!) or "N choose G" games to be played in total. Each player will participate in (N-1) choose (G-1) of those games. So, if 8 people play in foursomes, you will need to play 8 choose 4 or 70 games total, with each player playing 35 of those games. If your brother plays, you will need to play 9 choose 4 or 126 games total, with each player playing 56 of those games.

N = number of players in total
G = size of each group

Then there are N!/(G!*(N-G)!) or "N choose G" games to be played in total. Each player will participate in (N-1) choose (G-1) of those games. So, if 8 people play in foursomes, you will need to play 8 choose 4 or 70 games total, with each player playing 35 of those games. If your brother plays ...[text shortened]... tal, with each player playing 56 of those games.

I hope you bring your traveler's cheques.

I've just worked out by hand that with 7 total rounds per person, arranged as two foresomes for each round (since there are 8 golfers), it can be arranged so that each player plays with every one of the others exactly 3 times over the seven rounds. So I now have this one, but not the one with 9 players

So I don't understand your general formula, since it gives the wrong answer for 8 golfers

n players, group size is g, r is the number of rounds. If we keep to a simple case where g divides into n exactly:

In any particular group each player effectively plays (g - 1) individual matches. So r(g - 1) matches in total. If you want each player to play each other exactly the same number of times, then we need r(g - 1) to be a multiple of (n - 1).

So, r = LCM(n - 1, g - 1)/(g - 1)

(This doesn't prove a solution exists, it just gives a way of eliminating some that definitely won't).

So with 8 players in 2 foursomes, we need 3r to be a multiple of 7. The minimum value of r that will do that is 7 - which matches your solution.

With 9 players we'd have to play threesomes. 2r is a multiple of 8. Which suggests you can do it in four rounds, with everyone playing each other once (and a little hand-calculation shows this is possible). Or you can have 8 rounds, 12 rounds, etc.

If you have a group size that doesn't divide the total number - I might have a go at that later!