If a collection of things can be ordered, consider ordering it; can’t hurt. If that ordered collection has a minimum or maximum member, it may be a good idea to examine that member. Maybe that member is so small or so large that it’ll have some weird property you can exploit. Especially if the ordering is somehow related to all the other members.
Examples
Prove that every convex polyhedron
has at least two faces with the same number of edges.
The face with the most edges will be a dominant feature of the polyhedra. It’ll be neighbors with the most other faces.
At a round-robin tournament
each player plays every other player exactly once.
During such a tournament, show that there must be some player
in the unfortunate position that every other player
either beat them, or beat someone else who beat them.
Sucks to be that player. They probably lost the most total times too …
A finite number of polygons lie in the plane.
Each pair of polygons shares at least a common point.
Show there’s a line that intersects every polygon.
The numbers along a real line are famous for being linearly ordered. I.e. for any real numbers and such that either or We can utilize this linear ordering by projecting things onto a line. In this example, for any line in the plane, our polygons will project down and become closed intervals on that line, and there will be a left-most and right-most interval …