My favourite trigonometric identity
I enjoyed proving trigonometric identities when I was in high school. The following identity that appeared on an old mathematics contest (Descartes contest) is perhaps my favourite one as it …
Break it down
Life is overwhelming but a good general principle is to pick one small thing you can do. Finish it. Then repeat the process. Each iteration will give you more confidence …
More Food for Thought
Enjoy the privileges of having, but also enjoy the freedom of not having.
Food for Thought
You might go further in life if you rely on other people, but you will live a more meaningful life if you are able to do it yourself.
A Problem from "Problem Solving through Problems"
One of my favourite mathematical books is "Problem Solving Through Problems" by Loren C. Larson. It contains a wealth of techniques and problems, many of which appeared in previous Putman …
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A Superficial Proof of Fermat's Last Theorem
I mentioned my interest in learning about Fermat's Last Theorem in an earlier post. As I wrote in that post, the theorem is surprisingly simple to state but the proof required …
Marriage Lemma
The Marriage Lemma is a neat result that has the following interesting formulation: Consider a group of men and women in a party. Suppose in any group of n men, there …
Mathematical Induction
The Principle of Mathematical Induction is arguably one of the most powerful techniques in mathematics. Formally, the principle asserts the following: Principle of Mathematical Induction: Consider a set of statements …
The Pigeonhole Principle
The pigeonhole principle is based on a very simple idea but is very powerful in mathematical problem solving. Here is the principle: Pigeonhole Principle: If there are objects to …
Telescoping series
One of my favourite tricks in mathematical problem solving is the idea of a telescoping series. Consider the following problem: Find the sum of This seems to lead to …