Maps Can Always Be Colored With Only Four Colors
Maps can always be colored with only four colors in such a way that no two adjacent regions of the map share the same color. This is the Four Color Theorem, a highly controversial mathematical theorem that was proved using computer-assistance. A first in mathematics. We delve into the controversy, explain the proof concept, and present ways to make maps that are still not 4-colorable.
Coastline Paradox and Fractals
How long is the coast of Britain? This question was of utmost importance in the process of creation of the field of fractal geometry. We explain the coastline paradox, fractal geometry and dimension, and how we use fractal geometry to measure coastlines. #Fractals #Geometry #Coastlines #Fry #Lewis #Richardson #Dimension
Approximate Pi Using a Monte-Carlo Method in Matlab
We approximate pi using a Monte-Carlo method in Matlab, explain the code, provide the program for you to try for yourself, and present an error analysis. #Pi #Matlab #Tutorial
Your friends are on average more popular than you. That fact is the Friendship Paradox. We give a mathematical proof for this statement, explain the relevant graph-theoretic terms, and present an example of an application of the Friendship Paradox in real life, the early detection of flu outbreaks. #Graph #Theory #Friendship #Paradox
Game Theory and Doping
A game-theoretic discussion of doping. We discuss possible methods for solving the doping dilemma, introduce the Nash equilibrium, and establish the mathematical concept of doping games.
German Tank Problem
We explain the German Tank Problem that the Allies faced in World War 2, derive a solution by analyzing serial numbers, and provide insight with an example. #Probability #Theory #World #War #2 #German #Tank #Problem #Allies