Open problems and research questions, each a git repository: many come from curated collections and credit them. Routes are branches anyone can start or continue; results count once the checks pass.
Find how many half-plane floodlights placed at vertices always suffice to light a polygon with n vertices; the answer lies between 9n/14 and 2n/3.
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Find the smallest point set on which every planar graph with n vertices can be drawn with straight edges: do O(n) points suffice?
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Prove that the crossing points of every simple arrangement of great circles can be 3-coloured, or find one that needs four colours.
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Bound how close two sums of k square roots of integers up to n can be, and decide whether such sums can be compared in polynomial time.
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Find an algorithm for linear programming whose number of arithmetic steps is polynomial in the numbers of variables and constraints alone.
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Sort all n² pairwise sums of two sets of n numbers in o(n² log n) time, faster than general-purpose sorting allows.
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Find the maximum number of k-sets and halving lines of n points in the plane; the known bounds are barely superlinear and O(n^4/3).
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Determine whether finding the longest tour through n points in the plane, with Euclidean distances, is NP-hard or solvable in polynomial time.
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Decide whether every polyhedron can be cut open, with cuts anywhere on its surface, and flattened into one non-overlapping piece.
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Decide whether 2–3 and 3–2 flips connect all tetrahedralizations of every point set in general position in three dimensions.
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Compute the Euclidean minimum spanning tree of n points in three or more dimensions in near-linear time; the best known is about n^4/3 in 3D.
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Show that every polycube can be cut along cube edges and unfolded flat in one non-overlapping piece, or find one that cannot.
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Prove that every convex polyhedron can be cut along its edges and unfolded flat in one piece without overlap, or find one that cannot.
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Count the polyominoes with n cells, known exactly up to n = 70, and pin down Klarner's growth constant, now between 4.0025 and 4.5252.
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3SUM now has a truly subquadratic algorithm; find one for a 3SUM-hard geometric problem, such as deciding whether n points have three on a line.
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