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Unsolved problems in mathematics
Geometry: Is greedy algorithm optimal for maximizing area in triangles? Malfatti circles.
Graph theory: Overfull_graph#Overfull_conjecture, Barnette's conjecture
Hadwiger conjecture (graph theory)
Keller's_conjecture about tilings of Rn with cubes.
Rectangles in the Toothpick sequence.
Geometric shape optimization Moser's worm problem Moving sofa problem
Tilings:
Prototile, Domino problem, Socolar–Taylor tile, Heesch's problem, see also [1]
"No algorithm is known for deciding whether two arbitrary polyominoes are compatible." see Polyomino.
Turing complete stuff, and computations:
Wolfram's 2-state 3-symbol Turing machine, FRACTRAN