Category Archives: Discrete Mathematics

Why is a² + b² ≥ 2ab ?

This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again). How can one … Continue reading

Posted in Calculus, Discrete Mathematics, Geometry, Inequalities, Linear Algebra, Probability | 1 Comment

The discrete Gauss-Bonnet theorem

This is a slight extension of my previous note on discrete Gauss-Bonnet theorem. As mentioned in that note, this is a generalization of the well-known fact that the sum of the exterior angles of a polygon is always , which … Continue reading

Posted in Calculus, Combinatorics, Discrete Mathematics, Geometry, Topology | Leave a comment

“Useless” Circle Properties: The Green Flash In Mathematics

Since F.4 I’ve been thinking long and hard about real-life applications of circle properties. Why do we study them in secondary school anyway? Someone told me they’re used in designing cylindrical structures, but I couldn’t find a satisfactory book or website that … Continue reading

Posted in Applied mathematics, Discrete Mathematics, Geometry | Tagged , , , , , | 2 Comments

Water puzzles

Here is a typical water puzzle: You have two cups. Their capacities are 5 units and 3 units respectively. You may get water, pour water way or to another cup, but there are no marks on the cups. If you … Continue reading

Posted in Discrete Mathematics | 2 Comments

Stokes’ theorem on a simplicial complex

This is a note on a Stokes’ theorem on a simplicial complex. Originally I wanted to establish some formulas on a graph, it turns out that it’s better to work on a simplicial complex. After discussing with Raymond, we arrived … Continue reading

Posted in Analysis, Discrete Mathematics | 2 Comments