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. Another…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
5 days agoMember-onlyHow Much Does Medium Pay for 1000 Views? (2023)A breakdown of my earnings from 100 Medium articles — I have now written 100 articles on Medium! 97 of these articles are monetized under the Medium Partner Program. In total these 97 articles have: accumulated a total of 261,129 views earned a total of $3904 Doing a little arithmetic give the following average: I have earned an average of…Medium6 min readMedium6 min read
Published in Cantor’s Paradise·Mar 24Member-onlyGPT-4 is Amazing but Still Struggles at High School Math CompetitionsIts greatest strength is its biggest flaw — GPT-4 is the latest version of the GPT architecture developed by OpenAI, released on 14 March 2023. We are told on the landing page that GPT-4 “surpasses ChatGPT in its advanced reasoning capabilities”, and presented with some impressive results for the USA Bar Exam and Biology Olympiad.Mathematics7 min readMathematics7 min read
Mar 23Mathematics — A Young Man’s Game?2818Areeba MerriamThanks Areeba, this is a topic that interests me too.Thanks Areeba, this is a topic that interests me too. There is definitely a balance between the free creativity of youth (what you called "fluid intelligence") and the wisdom that only comes with age. Here is a data visualisation I did last year on the age of famous mathematicians when they made their major contributions: https://medium.com/p/d6dadac4f6eb1 min read1 min read
Published in Nice Math Problems·Mar 22Member-onlyA Quick Algebra Problem from the 2022 Oxford Maths Admission TestHow many solutions does this equation have? — This was the first question on the 2022 Mathematics Admissions Test, an entry test for students applying for a Mathematics or Computer Science Degree at Oxford University, Imperial College London and/or University of Warwick. The question is multiple choice:Mathematics3 min readMathematics3 min read
Published in Nice Math Problems·Mar 17Member-only2023 Australian Mathematics Olympiad — Problem 1A nice counting puzzle that anyone can try — The Problem Mia is playing the following game. She writes the numbers 1,2,3,…, n in some order on the sides of a regular n-sided polygon. Then, on each vertex of the polygon, she writes the sum of the numbers on the two sides that meet at that vertex. Mia wins if the…Mathematics7 min readMathematics7 min read
Published in Nice Math Problems·Mar 14Member-onlyYou’ll Love This Japanese Geometry ProblemTrust me. It’s lovely. — A regular hexagon is inscribed in a rectangle, creating a triangle with area 20 cm² and quadrilateral with area 23 cm², as shown below. Find the area of the hexagon. This problem appeared in the Preliminary Round of the 2023 Japan Mathematical Olympiad. Where could we start? The key with…Mathematics3 min readMathematics3 min read
Published in Nice Math Problems·Mar 9Member-onlyHow Many Roots?A clever complex equation from the 2022 IIT JEE — Every year, hundreds of thousands of aspiring Indian students sit the Indian Institute of Technology Joint Entry Examinations (IIT JEE). This complex number question appeared in the 2022 IIT JEE (Advanced). Let z̄ denote the complex conjugate of a complex number 𝑧 and let 𝑖 = √(−1). In the set…Mathematics4 min readMathematics4 min read
