Show that the average degree of a vertex in the triangulation is strictly less than 6.

A string of length L is cut into arbitrarily many pieces. Some or all pieces are made to circles and other pieces are made to rectangles. What is the maximum possible area enclosed by these shapes?

If a graph with six vertices is embedded in S2, what is the maximum number of edges it can have. Prove your answer and illustrate the maximal case.

In an​ orchestra, 23 people can play stringed​ instruments, 21 can play​ brass, and 15 can play percussion.​ Further, 9 of the performers can play both strings and​ brass, whereas 8 can play both strings and percussion. If no one can play all three types of​ instruments, what are the maximum and minimum numbers of people in the​ orchestra? ​(Hint. Consider expressing the numbers in some of the regions of your diagram in terms of a single​ unknown, say​ x.)