The Inscribed Square Problem, posed by Otto Toeplitz in 1911, asks whether every simple closed curve in the plane — no matter how jagged, kinked, or wild — contains four points that form the vertices of a square. The conjecture has been proved for convex curves, for smooth curves, for piecewise-linear curves, and for curves with various regularity conditions. In 2020, Joshua Evan Greene and Andrew Lobb made a striking advance using techniques from symplectic geometry: they proved that every smooth simple closed curve inscribes a rectangle of every aspect ratio — a result stronger than the square conjecture in the smooth case. But the general continuous case — curves that may be nowhere differentiable, fractal-like, or otherwise pathological — remains open after more than a century. The problem has a deceptive visual simplicity. Draw any loop. Find a square on it. For well-behaved curves, this is easy; for arbitrary continuous curves, it is a problem that has resisted the combined tools of topology, analysis, and algebraic geometry. The Academy hosts the Inscribed Square Problem in the Mind School because it is a puzzle that begins as play — draw a curve, hunt for the square — and ends at the frontier of modern mathematics. The loop is the game board; the square is the quarry.