Some generalizations.

The General Pigeon Hole Principle asserts that if we must put Nk+1 or more pigeons into N pigeon holes, then some hole must contain at least k+1 pigeons. Here is an example:

Problem 4. Twenty-five crates of apples are delivered to a store. The apples are of 3 different sorts, and all of the apples in each crate are of the same sort.Show, that among these crates there are at least nine containing the same sort of apples.

Hint. N=3, k=8.

A similar but still distinct application of the pigeon hole principle is the following scheme:

If the sum of n or more integers is equal to S, then at least one of these integers must be less than or equal to S/n, and, also, at least one of them must be not less than S/n.

Problem 5. On a certain planet in the solar system Tau Cetus, more than half of the surface of the planet is dry land. Show that the Tau Cetus can dig a tunnel,straight trough the center of their planet, beginning and ending on dry land. (Assume that their technology is sufficiently developed.)

Hint. Consider the big circles passing through the north and the south poles. At least one of them will have the property that more than half of the length of the circle will be covered by dry soil. Why? How this implies that there is at least one diameter of the planet, which intersects with the circle at ``dry'' points?