For the next Olympic Winter Games the organizers wish to exp

For the next Olympic Winter Games, the organizers wish to expand the number of teams competing in curling. They wish to have 14 teams enter, divided into two pools of seven teams each. Right now, they’re thinking of requiring that in preliminary play each team will play seven games against distinct opponents. Five of the opponents will come from their own pool and two of the opponents will come from the other pool. They’re having trouble setting up such a schedule, so they’ve come to you. By using an appropriate graph-theoretic model, either argue that they cannot use their current plan or devise a way for them to do so.

Solution

Consider one pool of 7. the condition: Five of the opponents will come from their own pool is enough to see that the plan is not possible.

Lets the competitor be named from 1 to 7.

Assume with loss of generality that the opponents of 7 are 1, 2, 3, 4, and 5. Notice that 6 has not choices for it\'s opponents. We are sure they are 1, 2, 3, 4, and 5.

Hence your are now left with the same problem but for a poll of 5 where you need de find 3 opponents for each.

I let you continue for the pool of 5 with the same argument and them for a pool of 3 and so on until you find a problem ;)