# The Pigeon Hole Principle: 7 gorgeous proofs 0 (0)

From Mathologer. Let’s say there are more pigeons than pigeon holes. Then, if all the pigeons are in the holes, at least one of the holes must house at least two of the pigeons. Completely obvious. However, this unassuming pigeon hole principle strikes all over mathematics and yields some really surprising, deep and beautiful results.…

# The ultimate algorithm 0 (0)

From Mathologer. There must be millions of people who have heard of the Tower of Hanoi puzzle and the simple algorithm that generates the simplest solution. But what happens when you are playing the game not with three pegs, as in the original puzzle, but with 4, 5, 6 etc. pegs? Hardly anybody seems to…

# How many ways to make change for a googol dollars? (infinite generating functions) 0 (0)

From Mathologer. Okay, as it says in the title of this video, today’s mission is to figure out how many ways there are to make change for one googol, that is 10^100 dollars. The very strange patterns in the answer will surprise, as will the explanation for this phenomenon, promise. A very nice Mathematica file…

# The ARCTIC CIRCLE THEOREM or Why do physicists play dominoes? 0 (0)

From Mathologer. I only stumbled across the amazing arctic circle theorem a couple of months ago while preparing the video on Euler’s pentagonal theorem. A perfect topic for a Christmas video. Before I forget, the winner of the lucky draw announced in my last video is Zachary Kaplan. He wins a copy of my book…

# 700 years of secrets of the Sum of Sums (paradoxical harmonic series) 0 (0)

From Mathologer. Today’s video is about the harmonic series 1+1/2+1/3+… . Apart from all the usual bits (done right and animated ðŸ™‚ I’ve included a lot of the amazing properties of this prototypical infinite series that hardly anybody knows about. Enjoy, and if you are teaching this stuff, I hope you’ll find something interesting to…

# The hardest “What comes next?” (Euler’s pentagonal formula) 0 (0)

From Mathologer. Looks like I just cannot do short videos anymore. Another long one ðŸ™‚ In fact, a new record in terms of the slideshow: 547 slides! This video is about one or my all-time favourite theorems in math(s): Euler’s amazing pentagonal number theorem, it’s unexpected connection to a prime number detector, the crazy infinite…

# How did Ramanujan solve the STRAND puzzle? 0 (0)

From Mathologer. Today’s video is about making sense of an infinite fraction that pops up in an anecdote about the Indian mathematical genius Srinivasa Ramanujan. 00:00 Intro 04:31 Chapter 1: Getting a feel for the puzzle 08:27 Chapter 2: Algebra autopilot 12:37 Chapter 3: Infinite fraction 17:51 Chapter 4: Root 2 21:19 Chapter 5: Euclidean…

# What does this prove? Some of the most gorgeous visual “shrink” proofs ever invented 0 (0)

From Mathologer. Bit of a mystery Mathologer today with the title of the video not giving away much. Anyway it all starts with the quest for equilateral triangles in square grids and by the end of it we find ourselves once more in the realms of irrationality. This video contains some extra gorgeous visual proofs…

# What is the best way to lace your shoes? Dream proof. 0 (0)

From Mathologer. A blast from the past. A video about my fun quest to pin down the best ways of lacing mathematical shoes from almost 20 years ago. Lots of pretty and accessible math. Includes a proof that came to me in a dream (and that actually worked)! 0:00 Intro 1:31 What’s a mathematical lacing?…

# Euler’s crazy pi formula generator 0 (0)

From Mathologer. Today we derive them all, the most famous infinite pi formulas: The Leibniz-Madhava formula for pi, John Wallis’s infinite product formula, Lord Brouncker’s infinite fraction formula, Euler’s Basel formula and it’s infinitely many cousins. And we do this starting with one of Euler’s crazy strokes of genius, his infinite product formula for the…

# Why did they prove this amazing theorem in 200 different ways? Quadratic Reciprocity MASTERCLASS 0 (0)

From Mathologer. The longest Mathologer video ever, just shy of an hour (eventually it’s going to happen ðŸ™‚ One video I’ve been meaning to make for a long, long time. A Mathologerization of the Law of Quadratic Reciprocity. This is another one of my MASTERCLASS videos. The slide show consists of 550 slides and the…

# Why was this visual proof missed for 400 years? (Fermat’s two square theorem) 5 (1)

From Mathologer. Today’s video is about a new really wonderfully simple and visual proof of Fermat’s famous two square theorem: An odd prime can be written as the sum of two integer squares iff it is of the form 4k+1. This proof is a visual incarnation of Zagier’s (in)famous one-sentence proof. 0:00 Intro 2:20 Chapter…

# Fermat’s Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+… 0 (0)

From Mathologer. NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) Leibniz’s formula pi/4 = 1-1/3+1/5-1/7+… is one of the most iconic pi formulas. It is also one of the most surprising when you first encounter it. Why? Well, usually when we see pi…

# Secret of row 10: a new visual key to ancient Pascalian puzzles 0 (0)

From Mathologer. NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) Today’s video is about a recent chance discovery (2002) that provides a new beautiful visual key to some hidden self-similar patterns in Pascal’s triangle and some naturally occurring patterns on snail shells. Featuring,…

# Power sum MASTER CLASS: How to sum quadrillions of powers … by hand! (Euler-Maclaurin formula) 0 (0)

From Mathologer. The longest Mathologer video ever! 50 minutes, will this work? Let’s see before I get really serious about that Kurosawa length Galois theory video ðŸ™‚ Today’s video is another self-contained story of mathematical discovery covering millennia of math, starting from pretty much nothing and finishing with a mathematical mega weapon that usually only…

# 500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can’t handle? 0 (0)

From Mathologer. Why is it that, unlike with the quadratic formula, nobody teaches the cubic formula? After all, they do lots of polynomial torturing in schools and the discovery of the cubic formula is considered to be one of the milestones in the history of mathematics. It’s all a bit of a mystery and our…

# 2000 years unsolved: Why is doubling cubes and squaring circles impossible? 0 (0)

From Mathologer. Today’s video is about the resolution of four problems that remained open for over 2000 years from when they were first puzzled over in ancient Greece: Is it possible, just using an ideal mathematical ruler and an ideal mathematical compass, to double cubes, trisect angles, construct regular heptagons, or to square circles? 00:00…

# Why don’t they teach this simple visual solution? (Lill’s method) 0 (0)

From Mathologer. Today’s video is about Lill’s method, an unexpectedly simple and highly visual way of finding solutions of polynomial equations (using turtles and lasers). After introducing the method I focus on a couple of stunning applications: pretty ways to solve quadratic equations with ruler and compass and cubic equations with origami, Horner’s form, synthetic…

# The Secret of Parabolic Ghosts 0 (0)

From Mathologer. NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) Today we’ll perform some real mathematical magic—we’ll conjure up some real-life ghosts. The main ingredient to this sorcery are some properties of x squared that they don’t teach you in school. Featuring the…