Mathematics is often celebrated for its precision and certainty, yet some of its most fascinating corners are shrouded in mystery. This section is dedicated to the unsolved problems that continue to puzzle and inspire mathematicians around the world. These are not mere curiosities—they are deep, fundamental questions that cut across fields such as number theory, topology, algebra, analysis, and geometry. Some, like the Riemann Hypothesis or the P vs NP problem, are famous even outside academic circles; others are lesser-known gems with the potential to reshape mathematical understanding. Here, we explore these enduring enigmas—not just their statements, but their history, significance, and the progress made toward solving them. Whether you’re a curious student or a seasoned researcher, this space invites you to stand at the edge of what is known—and look beyond.
- Unsolved Problems in Geometry
- The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?
- The disk covering problem about finding the smallest real number
such that
disks of radius
- Carathéodory conjecture: any convex, closed, and twice-differentiable surface in three-dimensional Euclidean space admits at least two umbilical points.
- The filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length.
- Osserman conjecture: that every Osserman manifold is either flat or locally isometric to a rank-one symmetric space.
- Yau’s conjecture on the first eigenvalue: The first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of
is
- Bellman’s lost-in-a-forest problem – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation.
- Ehrhart’s volume conjecture: a convex body
in
- Falconer’s conjecture: sets of Hausdorff dimension greater than
in
must have a distance set of nonzero Lebesgue measure.
- Inscribed square problem, also known as Toeplitz’ conjecture and the square peg problem – does every Jordan curve have an inscribed square?
- The Kakeya conjecture – do
- Lebesgue’s universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one.
- Moser’s worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?
- The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
- Is there a non-convex polyhedron without self-intersections with more than seven faces, all of which share an edge with each other?
- The Thomson problem – what is the minimum energy configuration of
- Convex uniform 5-polytopes – find and classify the complete set of these shapes.
- Unsolved Problems in Number Theory
- The Gauss circle problem: how far can the number of integer points in a circle centered at the origin be from the area of the circle?
