At a dinner party for six friends, consider any two who have met before as mutual acquaintances and any two who have not met before as mutual strangers. No matter who is invited, either there are three mutual strangers or there are three mutual acquaintances. Why?

Find a path which passes through each blank square exactly once. Direction changes along the path are only allowed when facing a dark square, a square already passed through by the path, or the edges of the puzzle.

Consider N pennies which are randomly distributed in several containers. Any two random containers A and B have x and y pennies, respectively. If x is greater than or equal to y, you can remove y pennies from container A and put them into container B. This is called an operation. Show that regardless of the original distribution of pennies, a finite number of such operations can move all the pennies into one or two containers.

In the city of Gooseville, the following facts are true:

No two inhabitants have exactly the same number of hairs. No inhabitant has exactly 384,270 hairs. There are more inhabitants than there are hairs on the head of any one inhabitant.

What is the largest possible number of inhabitants of Gooseville?

What is the minimum number of stationary guards who collectively can view every point in an art gallery? Assume the art gallery is any simple n-sided polygon and each guard is a point inside the polygon who can see infinitely far in every direction.

Below is a visual example of one guard’s view not covering every point of a particular art gallery. The guard cannot see through or around walls.