OEIS A000241: Crossing numbers of the complete graphs K_n

An introduction to A000241: the minimum number of edge crossings required to draw the complete graph on n vertices. We trace the problem from Turan's brick factory through Harary-Hill, explain why exact values are known only for small n (up to 27 and 30) and bounded otherwise, and contrast the standard and rectilinear crossing numbers. We also survey algorithmic approaches, NP-completeness, and connections to geometry and computation, with examples like K5=1, K6=3, K7=9, and K8=18 (straight-line) vs 19 (allowing curves).

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