10. [7] Five guys join five girls for a night of bridge. Bridge games are always played by a team of two guys against a team of two girls. The guys and girls want to make sure that every guy and girl play against each other an equal number of times. Given that at least one game is played, what is the least number of games necessary to accomplish this?
Answer: 25 Suppose that each guy plays each girl $t$ times. Since each guy plays against two girls in one game, the total number of games each guy plays is $\frac{5 t}{2}$. Then the total number of games is $\frac{25 t}{4}$, which is a multiple of 25 and therefore at least 25 . To check that 25 games is enough, we arrange the guys and girls in two circles. A good pair of guys is a pair of guys who are adjacent in the circle; a good pair of girls is defined similarly. There are 5 good pairs of guys and girls - making each good pair of guys play each good pair of girls works.