Part 1/7:

The Enigmatic Inscribed Square Problem

The inscribed square problem, first posed by Otto Toeplitz in 1911, presents a tantalizing mathematical mystery: Does every closed continuous curve necessarily have an inscribed square? This question, which belongs to the realm of topology, is still unsolved and invites mathematicians to explore its intricate and engaging concepts.

Understanding Closed Loops

To comprehend the problem, it’s essential to grasp what a closed continuous curve entails. It refers to a shape—a "squiggle"—that can be drawn on paper without lifting the pen, ending at the starting point. The challenge here is to determine if, within any such curve, it’s possible to find four points that form the vertices of a square.