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In the web game Ancient Greek Geometry, there are challenges to construct regular polygons and circle packings using ruler and compass constructions. The game measures the number of line and circles used to make a given shape. I was wondering if there are known minima for the number of moves needed for the shapes given in the challenges? In the comments on the blog about the game, many users show that there are fewer numbers of moves needed for some of the challenges than the game requires. I'm looking both for records and proofs that certain figures take at least $x$ moves, so upper and lower bounds. For example, I know how to make a pentagon (in the origin circle) in 12 moves, but someone claims to be able to do it in 11. Can someone prove that it can't be done in 10?

Remark: There is a related question of how many lines and circles are needed in classical ruler & compass constructions of these figures? This is slightly different, in that in a ruler & compass construction, any circle may be drawn centered at a point of intersection of previous lines and circles, as long as its radius is the distance between two of these points, whereas in the web game, the radius must be the distance from the center to a preexisting point. One can construct the same figures with the web game, but they could possibly take more moves than a ruler & compass.

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I'd like to see a list of those best known solutions. – user21820 Feb 8 '15 at 9:21

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