Lee,
I am surprised that scheduling golf threesomes has not been raised here before. Anyway, the answer to your question is yes. With four rounds of threesomes it's possible to arrange that everyone plays everyone else exactly once.
(G F A) (B D C) (E H I)
(F C H) (I G D) (A E B)
(B I F) (D A H) (E C G)
(F D E) (H B G) (C I A)
The example above is the first in a whole series of threesome schedules. If the number of players is three more than an a multiple of 6 then the all-plays-all-exactly-once property is always possible. The next example is 15 players, which is famously known in mathematics as Kirkman's Schoolgirl problem (search the internet for more info on this, and you will find out how the schedule below can be constructed).
(K O H) (L N J) (E A G) (F I C) (D M B)
(B H N) (A K I) (F L D) (O G C) (M J E)
(M K L) (O D J) (G N I) (E F H) (B C A)
(C M N) (H J G) (L O A) (I D E) (K B F)
(L B G) (N E K) (O I M) (H C D) (J A F)
(I L H) (A D N) (C J K) (G F M) (B E O)
(M H A) (E C L) (J I B) (D G K) (N F O)
Below is the third in the series for 21 players.
(G F T) (B I J) (A U D) (O S L) (Q H K) (P R M) (N E C)
(K I U) (T M H) (S C B) (J R L) (A Q E) (F P N) (D O G)
(H I R) (L B D) (T O P) (U G Q) (K A C) (S J N) (E F M)
(A N M) (E L P) (S T D) (G C R) (U J H) (I F Q) (B K O)
(I P A) (L U T) (C O M) (H B F) (Q R S) (J D E) (N K G)
(E H G) (I D C) (O N R) (M S U) (T A J) (K F L) (Q B P)
(M L I) (Q C T) (F U O) (D N H) (R A B) (P J G) (K E S)
(C L H) (E U R) (N T I) (D P K) (F A S) (O Q J) (M G B)
(C P U) (B T E) (H O A) (G S I) (J M K) (R D F) (L Q N)
(R K T) (D M Q) (C J F) (L G A) (P H S) (I E O) (U N B)
Hope that helps,
Ian.
Mike,
My comments above relate to the situation where everyone plays everyone else the same number of times. However there are other things you could do. How about dividing your 10 players into 5 teams of 2 players each. If the players are A to J, then make the teams up like this
(A B) (C D) (E F) (G H) (I J)
and then play the schedule below.
(B D F H I) (A C E G J)
(B C E G I) (A D F H J)
(A D E G I) (B C F H J)
(A D E H J) (B C F G I)
(B D E H I) (A C F G J)
(B D F G J) (A C E H I)
(A C F H I) (B D E G J)
(A D F G I) (B C E H J)
In the 8 rounds each player will play with all players from opposing teams exactly 4 times, and never play with their own team mate.
Ian.