I am sorry that I can't offer a complete solution here, as the list of limitations is something I can't address, and as you will see later there are other issues as well. But I can offer some guidance on how I would approach this problem. The first thing to note is that it is possible to have every couple meet with every other couple exactly once in a 5 month period. So it seems natural to look for a 15 round schedule where every pair of couples meet once in the first 5 months, meet once in months 6 to 10, and meet once in the last 5 months. So this goes a long way towards meeting the "same pair two months in a row" criterion, as this can only happen in between each block of 5. Of course this will be a month short, but I think I would attempt to do something special in the last month based on the couples' performance over months 1 to 15.
It is not actually possible to do as you ask: "to go to every pair's home over the two year period". If there are 16 rounds, and every couple hosts 4 times, then they can only visit a maximum of 12 out of the 15 other homes.
If the social mixing is to be optimal, then you should try to avoid having the same 3, or the same 4 couples come together more than once. Or putting that another way, on the 3 occasions that couples A and B meet, the other 6 couples present should all be different. It is possible to achieve this, and my
schedule here in reply #1 shows you how to do it - just take any 3 of the 7 blocks.
As I have been experimenting with this problem, it's become apparent that my 3 blocks of 5 approach is not compatible with the idea of each couple having a long break between months where they host. So I can only offer the 15 month schedule below where quite often a couple hosts in 3 or 4 consecutive months. You could choose to reorder the 15 rounds to improve the breaks between hosting, but in the process you would make the breaks between seeing the same couple much worse. So there is definitely a trade off here - you can not have everything that you ask for above.
So here is the 15 round (the 15 rows) schedule for 16 couples A to P. The square brackets give the hosts, so in round 1 from the 1st row, the four couples (I M O P) come together at M's home.
Group 1 Group 2 Group 3 Group 4 Hosts
(A E G H) (B C D F) (I M O P) (J K L N) [E B M L]
(O L F H) (I J C E) (G D N P) (A B K M) [F C D M]
(N C M H) (O A J D) (F K E P) (G I B L) [M O F L]
(L A C P) (M F G J) (D I K H) (E N O B) [L M D E]
(C G K O) (B H J P) (D E L M) (F A N I) [O B L I]
(J N A H) (K L E G) (B F I P) (C D M O) [J K P D]
(C G B H) (D E F A) (K O J P) (L M N I) [C D P N]
(I E O H) (J C L F) (A M G P) (B K D N) [O J P K]
(E B M J) (D H L P) (F G N O) (A C I K) [J P N K]
(N C E P) (O A B L) (F K M H) (G I J D) [N B K J]
(G D F H) (A B C E) (O L N P) (I J K M) [H A O I]
(N K E H) (O I B D) (F C M P) (G A J L) [H I C G]
(M B L H) (N G I C) (E J D P) (F O A K) [H G E A]
(K G B P) (L E F I) (C O J H) (D M N A) [G E H A]
(B F J N) (A H I P) (C D K L) (E G M O) [F A C G]
If you use this, then a couple will never visit another couple's home more than once.
The table below gives another way of looking at the schedule above, one row gives the movements for a single couple, so for example couple D starts by going to the home of couple B, then in month 2 the asterisk indicates that they host, while in month 3 they go to the home of couple O.
Couple/Round
1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
-----------------------------
A | E M O L I J D P K B * G * * *
B | * M L E * P C K J * A I H G F
C | B * M L O D * J K N A * G H *
D | B * O * L * * K P J H I E A C
E | * C F * L K D O J N A H * * G
F | B * * M I P D J N K H C A E *
G | E D L M O K C P N J H * * * *
H | E F M D B J C O P K * * * * A
I | M C L D * P N O K J * * G E A
J | L C O M B * P * * * I G E H F
K | L M F D O * P * * * I H A G C
L | * F * * * K N J P B O G H E C
M | * * * * L D N P J K I C H A G
N | L D M E I J * K * * O H G A F
O | M F * E * D P * N B * I A H G
P | M D F L B * * * * N O C E G A
Finally note that couples [B F I N] are the ones who host only 3 times in the 15 round schedule, so they should be the hosts for whatever happens in month 16.
Hope that is of some help.
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