Famous Solved Math Problems: Four Color Theorem, Sphere Packing, and Aperiodic Tiling

We continue exploring problems solved by mathematics — today a bit of combinatorics and packing problems.

Solved Math Problems in Combinatorics and Geometry

11. Coloring any planar map with four colors

Problem: prove that any map on a plane can be colored so that adjacent regions have different colors, using no more than four colors.

Solution: a proof was constructed after decades of searching. The key insight: if a map requiring five colors existed, it would necessarily contain a “forbidden” configuration — one that provably cannot arise. The conclusion is rigorous: four colors are always sufficient.

12. Maximum-density packing of equal spheres in 3D

Problem: arrange spheres of equal radius in space so that the fraction of occupied volume is maximized, with no overlaps.

Solution: it has been proven that the classical “pyramidal” packing — the same arrangement seen with oranges at a grocery store — is the densest possible. No other configuration fills space with a greater fraction. This problem is also known as Kepler’s Conjecture.

Circle Packing and Game Theory as Solved Mathematical Problems

13. Maximum-density packing of circles on a plane

Problem: arrange identical circles on a plane as densely as possible without overlapping.

Solution: the optimal arrangement is the honeycomb pattern — each circle surrounded by six neighbors, with centers forming a hexagonal lattice. No denser arrangement on a plane is possible; this has been rigorously proven.

14. Optimal strategies in games

Problem: for games like tic-tac-toe, determine whether perfect play leads to a win or a draw.

Solution: the game is represented as a move tree and systematically traversed, with each position labeled as winning, losing, or drawing. For tic-tac-toe, the strict conclusion is: perfect play from both sides always results in a draw.

Aperiodic Tiling — A Famous Solved Problem in Real Life Geometry

15. Tiling the plane with a single tile, aperiodically

Problem: find a tile shape that covers the plane without gaps or overlaps, but in such a way that the pattern never becomes periodic — no translation maps the tiling onto itself.

Solution: specific tile shapes have been found for which it is proven that tiling is possible, yet every resulting pattern is inevitably non-periodic. The result is an “ornament without wallpaper”: local motifs repeat, but there is no global repetition.

To be continued…