There are 12 couples in a card group. Once a month, 4 card games are scheduled at different locations. Each game consists of a host couple and 2 guest couples (3 couples total).
I am looking for an annual schedule that:
1. Assigns hosting duty to all couples equitably,
2. No couple hosts 2 consecutive months,
3. Pairs the host and guest couples so that no 2 couples play with each other 2 consecutive months, and
4. Allows as few repeat pairings as possible.
I'm not great at math so if you could, please keep your answer as simple as possible. Thanks.
This schedule is not perfect, as each couple only gets to play with $6$ other couples (twice with each of them), but it does meet the other requirements.
Give each couple a hat with a number ranging from $1$ to $12$. Couples with hat numbers $1, 4, 7, 10$ are hosts, while hats $2$ and $3$ join $1$, $5$ and $6$ join $4$ et cetera.
After the game, we do the following reassignment of hats:
$$\begin{array}{c c} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\ 2 & 6 & 10 & 5 & 3 & 7 & 8 & 12 & 1 & 11 & 9 & 4\end{array}$$
i.e. whoever has hat $1$ gives it to the couple with hat $2$, the couple with hat $4$ passes it to the holders of hat $5$ and so on.
Now, again, for the second game, couples with hat numbers $1, 4, 7, 10$ are hosts, while hats $2$ and $3$ join $1$, $5$ and $6$ join $4$ et cetera.
This reassignment is a $12-cycle$, which means each couple will get their original hat again after exactly $12$ months. The matchups at that point (after one year) will be identical again, but you asked for an annual schedule.
As illustration, here are the matchups for one year:
$ \tiny\left(\begin{array}{c| |cc|c||cc|c||cc|c||cc} \mathrm{Adler} & \mathrm{Bertoli} & \mathrm{Caesar} & \mathrm{Davidsen} & \mathrm{Erdos} & \mathrm{Franklin} & \mathrm{Garibaldi} & \mathrm{Holmes} & \mathrm{Indra} & \mathrm{Jones} & \mathrm{Khan} & \mathrm{Lopez}\\ \mathrm{Bertoli} & \mathrm{Franklin} & \mathrm{Jones} & \mathrm{Erdos} & \mathrm{Caesar} & \mathrm{Garibaldi} & \mathrm{Holmes} & \mathrm{Lopez} & \mathrm{Adler} & \mathrm{Khan} & \mathrm{Indra} & \mathrm{Davidsen}\\ \mathrm{Franklin} & \mathrm{Garibaldi} & \mathrm{Khan} & \mathrm{Caesar} & \mathrm{Jones} & \mathrm{Holmes} & \mathrm{Lopez} & \mathrm{Davidsen} & \mathrm{Bertoli} & \mathrm{Indra} & \mathrm{Adler} & \mathrm{Erdos}\\ \mathrm{Garibaldi} & \mathrm{Holmes} & \mathrm{Indra} & \mathrm{Jones} & \mathrm{Khan} & \mathrm{Lopez} & \mathrm{Davidsen} & \mathrm{Erdos} & \mathrm{Franklin} & \mathrm{Adler} & \mathrm{Bertoli} & \mathrm{Caesar}\\ \mathrm{Holmes} & \mathrm{Lopez} & \mathrm{Adler} & \mathrm{Khan} & \mathrm{Indra} & \mathrm{Davidsen} & \mathrm{Erdos} & \mathrm{Caesar} & \mathrm{Garibaldi} & \mathrm{Bertoli} & \mathrm{Franklin} & \mathrm{Jones}\\ \mathrm{Lopez} & \mathrm{Davidsen} & \mathrm{Bertoli} & \mathrm{Indra} & \mathrm{Adler} & \mathrm{Erdos} & \mathrm{Caesar} & \mathrm{Jones} & \mathrm{Holmes} & \mathrm{Franklin} & \mathrm{Garibaldi} & \mathrm{Khan}\\ \mathrm{Davidsen} & \mathrm{Erdos} & \mathrm{Franklin} & \mathrm{Adler} & \mathrm{Bertoli} & \mathrm{Caesar} & \mathrm{Jones} & \mathrm{Khan} & \mathrm{Lopez} & \mathrm{Garibaldi} & \mathrm{Holmes} & \mathrm{Indra}\\ \mathrm{Erdos} & \mathrm{Caesar} & \mathrm{Garibaldi} & \mathrm{Bertoli} & \mathrm{Franklin} & \mathrm{Jones} & \mathrm{Khan} & \mathrm{Indra} & \mathrm{Davidsen} & \mathrm{Holmes} & \mathrm{Lopez} & \mathrm{Adler}\\ \mathrm{Caesar} & \mathrm{Jones} & \mathrm{Holmes} & \mathrm{Franklin} & \mathrm{Garibaldi} & \mathrm{Khan} & \mathrm{Indra} & \mathrm{Adler} & \mathrm{Erdos} & \mathrm{Lopez} & \mathrm{Davidsen} & \mathrm{Bertoli}\\ \mathrm{Jones} & \mathrm{Khan} & \mathrm{Lopez} & \mathrm{Garibaldi} & \mathrm{Holmes} & \mathrm{Indra} & \mathrm{Adler} & \mathrm{Bertoli} & \mathrm{Caesar} & \mathrm{Davidsen} & \mathrm{Erdos} & \mathrm{Franklin}\\ \mathrm{Khan} & \mathrm{Indra} & \mathrm{Davidsen} & \mathrm{Holmes} & \mathrm{Lopez} & \mathrm{Adler} & \mathrm{Bertoli} & \mathrm{Franklin} & \mathrm{Jones} & \mathrm{Erdos} & \mathrm{Caesar} & \mathrm{Garibaldi}\\ \mathrm{Indra} & \mathrm{Adler} & \mathrm{Erdos} & \mathrm{Lopez} & \mathrm{Davidsen} & \mathrm{Bertoli} & \mathrm{Franklin} & \mathrm{Garibaldi} & \mathrm{Khan} & \mathrm{Caesar} & \mathrm{Jones} & \mathrm{Holmes} \end{array}\right)$