One of my favorite board games is Hex🔶🔷. The rules are incredibly simple. Hex is a two-player game played on a rhombus-shaped board covered with hexagonal tiles. The standard board is an \( 11 \times 11 \) grid with \( 121 \) hexagons, though Hex can be played on boards of any size. Hex was…

Here, I’ll discuss a famous fixed-point theorem from topology. Definition and Statement of the Theorem A fixed point is literally a point that moves to itself when moved by a continuous mapping. Formally, let \( X \) be a topological space, and \( f:X→X \) a continuous map. A point \( x \) that satisfies…

This is a well-known joke, so many of you might already be familiar with it. Nonetheless, it’s quite amusing as a joke. 😆 The Assertion: “Everyone is Bald” As indicated in the title, we will prove the assertion that “everyone is bald” using mathematical induction (often simply called induction). Understanding Mathematical Induction Let \( P(n)…

Definition Let \( K \) be a subset of the \( n \)-dimensional Euclidean space \( \mathbb{R}^n \). When any line segment \( \overline{xy} \) that connects two any points \( x \) and \( y \) in \( K \) is also contained within \( K \), we say that \( K \) is…

Laplacian The Laplacian on \(\mathbb{R}^3 \), \[ \Delta = \left( \frac{\partial^2}{\partial x^2},\ \frac{\partial^2}{\partial y^2},\ \frac{\partial^2}{\partial z^2 } \right), \] is a second-order differential operator. When this operator is expressed in spherical polar coordinates in \( \mathbb{R}^3 \), where \[ x = r \sin \theta \cos \varphi,\ y = r \sin \theta \sin\varphi,\ z = r…