Published in Math Simplified·PinnedMember-onlyGraph Theory 101: Why all Non-Planar Graphs Contain K₅ or K₃,₃An intuitive explanation of Kuratowski’s Theorem and Wagner’s Theorem, with lots of diagrams! — A planar graph is one that can be drawn in a plane without any edges crossing. For example, the complete graph K₄ is planar, as shown by the “planar embedding” below. One application of planar graphs would be designing a rail network without any intersections. …Mathematics7 min readMathematics7 min read
Published in Cantor’s Paradise·PinnedMember-onlyLeast Squares vs Least Absolute Errors —A 250-Year-Old DebateFrom 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. …Math10 min readMath10 min read
Published in Cantor’s Paradise·PinnedMember-onlyThe Central Limit Theorem — Why Is It So?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…Statistics7 min readStatistics7 min read
Published in Nice Math Problems·Jan 13Member-onlyGiven f(f(x)) = x² − x + 1, can you find f(0)?Surprisingly, the answer is yes… — I came across this problem in a YouTube video by the Math Booster channel. There is a link to the video below, in which the presenter gives a perfectly good algebraic solution. But I found myself searching for a more intuitive way to think about the problem. Here it is… Visualising f(f(x)) …Mathematics4 min readMathematics4 min read
Published in Nice Math Problems·Jan 3Member-onlyA Nice Algebra Problem from the 2023 China Math OlympiadThe Chinese IMO team eat sequences like this for breakfast!* — When it comes to Olympiad Mathematics, China rules. The Chinese team has won the IMO (International Mathematical Olympiad) for the last 4 years, and has not finished outside the top 3 in the last 25 years. China runs its National Olympiad earlier than most countries. All countries run it early…Math10 min readMath10 min read
Published in Nice Math Problems·Dec 9, 2022Member-onlyBouncing Light Rays in the Iran Mathematical OlympiadAn enlightening geometry problem ;) — A rectangle has sides that are perfect mirrors. A light ray enters from the top left corner of the rectangle, and after being reflected several times, hits the bottom right corner. Prove that the light ray must have passed through the center of the rectangle. If you actually draw out…Mathematics5 min readMathematics5 min read
Dec 2, 2022Member-onlySimulating the World Cup Finals with the Elo Rating System50,000 simulations — want to see how many your team won? — There are few events that rival the FIFA World Cup for focusing the world’s attention. …Math6 min readMath6 min read
Published in Cantor’s Paradise·Nov 29, 2022Member-onlyA Curious Connection Between Squares and PrimesInvestigating 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…Mathematics11 min readMathematics11 min read
Published in Nice Math Problems·Nov 23, 2022Member-onlyA Nice Prime Number Problem from the 2000 Greek OlympiadFind 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…Mathematics5 min readMathematics5 min read
Nov 22, 2022Member-onlyWorld Cup 2022: Winning 2 out of 3 Group Matches but Failing to Advance?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. …Mathematics4 min readMathematics4 min read
