Trick-Taking Card Games

Consider one round of a four-player trick-taking card game with no trumps and no breakages (for instance, if hearts did not need to be broken in Hearts). (That is, players must follow suit of the first card played in the trick unless they can’t.) How many of the 52! orderings of cards represent a legal order in which the cards could be played in the round?

To provide examples, any ordering that starts

6♣,K♣,Q♢,10♣,A♣,J♣,3♣,K♢,…

is illegal, as [6♣,K♣,Q♢,10♣] constitutes the first trick, and thus the player who played the K♣ starts the next trick, and thus the player after them has broken the rules as they clearly have at least one club (the J♣ they played this trick) that they did not play last trick (where they played a Q♢), but

2♣,2♢,2♡,2♠,3♣,3♢,3♡,3♠,4♣,4♢,4♡,4♠,…,A♣,A♢,A♡,A♠

is, as it turns out in this case the four parties were each dealt all the cards of one suit, and thus subsequent players will never be able to follow suit of the first player.

Here are some ideas for extensions: have a trump suit, have a breakage rule, have both, solve this for Napoleon.

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