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Many problems in discrete geometry are simply stated and can be used in teaching to make certain concepts easier to understand. Searching for points with integer coordinates on a line naturally leads to applications of theorems from number theory.
Work on the notion of infinity led to paradoxes. This forced mathematicians to formalize set theory. Progressive axiomatization led to the current ZFC system, which nonetheless remains subject to various shortcomings following the work of Kurt Gödel and Paul Cohen.
The challenge: covering sets of points with as few hyperplanes as possible, particularly sets consisting of certain vertices of the hypercube.
The notion of order is part of our daily lives, whether we are putting objects away, comparing quantities or deciding what order to tackle a series of tasks in. Yet it took a very long time for these natural activities to be conceptualized mathematically.
