An intuitive explanation for why sample means should be normally distributed. There have been many excellent articles and videos about the Central Limit Theorem (CLT). But besides formal proofs, I was not able to find a good intuitive explanation for why it is true. So here goes… Theorem Statement The…

Sharpen your colouring pencils… or try it as a coding challenge. — Each year, about 100 of the most outstanding high school mathematics students in Australia are invited to complete the Australian Mathematical Olympiad. The competition goes over 2 days, each day students have 4 hours to attempt 4 challenging proof-style problems. This combinatorics question was the final problem on Day 1…

Our modern society is built upon mathematics. And mathematics is built upon the achievements of thousands of outstanding individuals. Explorers, creators, problem solvers… they each pushed the boundaries of knowledge and invented new methods of thinking about the world. Who was the greatest mathematician of all time? How would you…

A fun application of graph theory and combinatorics. — Does the statement seem obvious to you? How would you prove it? It probably doesn’t seem surprising for social networks like Facebook — there are millions of Facebook users and most of them only have up to several hundred friends. But wouldn’t it be possible to construct a social network…

A spider and a fly are on opposite walls of a 30 × 12 × 12 meter room. The spider is 1 meter above the floor, the fly is 1 meter below the ceiling. They are both 6 meters from adjacent walls, as shown in the diagram below. If the…

From Gauss and the birth of statistics to modern data science and machine learning — Ordinary least squares (OLS) regression is one of the first items on the menu in an introductory Statistics or Data Science course. But why do we need to square the errors? It’s a common question even high school students ask. After all, comparing the size of the errors (absolute value)…

An intuitive explanation of why summing random variables gives rise to a normal distribution. — Did you ever wonder why bell curves are everywhere? Heights, weights, neurons firing, the apparent brightness of stars… why should they all just happen to follow a similar bell curve distribution? “It reigns with serenity and complete self-effacement amidst the wildest confusion. The larger the mob, the greater the apparent…

The two squares shown both have area 10 cm² and their bases are parallel. Is it possible to find the area of the hexagon shaded in pink? — How would you go about it? I went with the algebraic method explained in Method 1 below. But there is actually a much nicer solution, which would even allow a bright primary school student to solve this problem. It is explained below as Method 2. Method 1 — Algebraic

*Simple to state. Not so simple to solve. — This article is about Problem 2 from the 2022 Indian National Mathematical Olympiad. The problem statement is beautifully concise: