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Message: [QUOTE="Joshua Cranmer, post: 3614141"] The graph I gave is embeddable in 3 dimensions, but not 2. Pick an arbitrary point A on the y-axis, with B and C in the x-axis. F would be on the plane determined by the y and z axes, and D and E at the intersections of the spheres. If you tweak the numbers in the right way, you can push the graph into 2 dimensions. So let me give you a simpler one in 2 dimensions: A is at (0, 10) B and C at (+/- 1, 0) D is at (0, 9) Your algorithm gives A->B->D->C->A, a length of 2*[sqrt(11)+sqrt(10)] (~13), while the optimal is A->D->B->C->A, a length of 2+sqrt(11)+sqrt(10) (~8.5). Would you like me to scan this page in and send it to you? Here's some rough ASCII art: A / \ / D \ /_( )_\ B-------C [/QUOTE]