Category: Problem Solving and Proofs

Solve Physics and Math Problems at Brilliant.org

Problem Solving and Proofs, Resources and Freebies

I have shared quite a lot of sites in this blog about places where you can find challenging math problems for your intellectual satisfaction. In this post, I am going to introduce a website called Brilliant.org, a problem solving website which provides different levels of physics, computer science, and math problems depending on solver’s ability.

Brilliant Physics and Math Problems

Brilliant.org provides different levels of physics and math problems, from practice to Olympiad level. Recently, it has also added problems in computer science particularly on data structures and algorithm. Math problems include challenges in algebra, number theory, geometry, and combinatorics. Physics problems include challenges in mechanics, electricity and magnetism.

Just like in other challenges and games, problem solved at Brilliant.org have corresponding points. Players can exchange points for corresponding prizes. Players can also discuss problems within the website.

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SSS Congruence Theorem and Its Proof

High School Geometry, High School Mathematics, Problem Solving and Proofs

Many high textbooks consider the congruence theorems (SSS Congruence Theorem, SAS Congruence Theorem, ASA Congruence Theorem) as postulates. This is because their proofs are complicated for high school students.  However, let us note that strictly speaking, in Euclidean Geomtery (the Geometry that we learn in high school), there are only five postulates and no others. All of other postulates mentioned in textbooks aside from these five are really theorems without proofs.

In this post, we are going to prove the SSS Congruence Theorem. Recall that the theorem states that if three corresponding sides of a triangle are congruent, then the two triangles are congruent.

Before proving the SSS Congruence theorem, we need to understand several concepts that are pre-requisite to its proof. These concepts are isometries particulary reflection and translation, properties of kites, and the transitive property of congruence. If you are familiar with these concepts, you can skip them and go directly to the proof. » Read more

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10 Math Problems That Look Easy But Immensely Difficult to Solve

High School Mathematics, Problem Solving and Proofs

There are thousands of math problems that are difficult for a common person to understand even though mathematicians may find them easy to solve.  On the other hand, there are math problems that look really easy that even a middle student would understand the way they are stated, but their solution or proof is immensely difficult. Yes, such math problems exist and below are the some of the most well known. 

1. Squaring the Circle
Squaring the circle is one of the classic math problems proposed by Geometers.
Math Problem: Squaring a CircleIt was a challenge to use compass and straightedge to construct  a square with the same area as a given circle in a finite number of steps. Although the circle to square approximation was known since the time of the ancient Babylonian mathematicians, it was Anaxagoras (c. 510 – 428 BC) who was the first to be recorded in history to work on the problem.

In 1882, Ferdinand von Lindemann proved that \pi was transcendental. The consequence of this is the impossibility of squaring the circle.  » Read more

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