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 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 remarkable and…

Investigating Fermat’s two squares theorem — Have you ever come across a number pattern so intriguing that you can’t stop thinking about it? Here is a simple question that opens up a mysterious rabbit-hole… Which integers can be expressed as the sum of two perfect squares? Well, let’s start by testing some small integers: 0 can…

Find all primes p such that (1 + p + p² + p³ + p⁴) is a perfect square. — Don’t you love it when the problem statement fits in the subtitle!?! 😁 Find all primes p such that (1 + p + p² + p³ + p⁴) is a perfect square. This was a problem from the 2000 Greece National Mathematics Olympiad. To put this in context, the problem…

Unlikely, but mathematically possible… and it’s happened before! — In the group stage of the world cup, countries are placed in groups of 4 and play one match against each of the 3 other countries in the group. The 2 top ranked teams from each group then proceed to the knockout stage. …

Which, surprisingly, can be solved using only middle-school math! — Alice and Bob each select a real number on the interval [0,3], uniformly at random. Assuming their selections are independent, find the probability that the sum of their numbers is greater than 2. Below are two methods to solve this classic (and practical!) problem. The first uses continuous random variables…

Searching for Eulerian cycles 🚲 — At the International Mathematical Olympiad, China is the undisputed world leader and superpower. Of the last 20 IMOs, China has won 1st place 13 times, 2nd place 5 times and 3rd place twice. So it makes sense that questions from the China IMO Team Selection Tests are often more challenging…

The title makes sense… I promise! — The Three Square Geometry Problem was the subject of a 2014 Numberphile video. It was one of the first Numberphiles video I ever saw, and 8 years later it’s still a great video! (check out the link at the bottom) The problem is to find (with proof) the sum of…

Number puzzles to welcome the new year! — Some numbers can be expressed as the sum of squares of two positive integers, while others cannot. The number 2020 can, for example, because 2020 = 16² + 42². Problem 1 Can 2023 be expressed as the sum of squares of two positive integers? Problem 2 Can 2²⁰²³ be expressed as the sum of…