Problem 6. In the kingdom, there live counts, dukes, and marquises. One day, each count dueled with three dukes and several marquises. Each duke dueled with two counts and six marquises. Each marquise dueled with three dukes and two counts. It is known that all counts dueled with an equal number of marquises. How many marquises did each count duel with?
Answer: With 6 marquises.

Solution. Let there be $x$ counts, $y$ dukes, and $z$ marquises in the kingdom. Each count fought with three dukes, so there were $3 x$ duels between counts and dukes. But each duke fought with two counts, meaning there were $2 y$ such duels. Therefore, $3 x=2 y$.

Each duke fought with six marquises. Thus, there were $6 y$ duels between dukes and marquises. Each marquis fought with three dukes, so there were $3 z$ such duels. Therefore, $6 y=3 z$, from which $z=2 y=3 x$.

Each marquis fought with two counts, so there were $2 z=6 x$ duels between counts and marquises. Therefore, each count fought with six marquises.