Core technique: turn 3‑D or constrained geometric questions into planar ones. Unfold polyhedron surfaces along shared edges to flatten a geodesic path into a straight segment; shortest path length is then Euclidean distance in the unfolding, and one must check the straight line stays within the unfolded faces.
For solids: counting cuts, colourings, or stacking requires reasoning face-by-face (a hidden/internal piece needs one cut per exposed face); shadows/silhouettes under parallel projection are bounded by silhouette edges, and a convex polyhedron shows at most half its faces to a light source.
For triangles: use the triangle inequality ∣a−b∣<c<a+b for degeneracy/validity, Heron’s formula
A=s(s−a)(s−b)(s−c),s=2a+b+c,
and A=rs for the inradius r of the largest inscribed circle.
For sphere-walk puzzles, exploit that latitude circles near a pole can have circumference 1/n for any integer n, giving infinitely many starting points besides the pole itself.