The following problems require reasoning for their solution. To prove that an answer is correct requires an argument (often containing subsidiary arguments) whose premises are contained in the statement of the problem—and whose final conclusion is the answer to it. If the answer is correct, it is possible to construct a valid argument proving it. In working these problems, readers are urged to concern themselves not merely with discovering the answers but also with formulating arguments to prove that those answers are correct.

Imagine a room with four walls, with a nail placed in the center of each wall, as well as in the ceiling and floor, six nails in all. The nails are connected to each other by strings, each nail connected to every other nail by a separate string. These strings are of two colors, red or blue, and of no other color. All these strings obviously make many triangles, because any three nails may be considered the apexes of a triangle. Can the colors of the strings be distributed so that no one triangle has all three sides (strings) of the same color? If so, how? And if not, why not?