Cutting up a circle to make a square We know that there is no paper-and-scissors solution to Tarski's circle-squaring problem (my six-year-old daughter told me this while eating lunch one day) but what are the closest approximations, if we don't allow overlapping?
More precisely: For N pieces that together will fit inside a circle of unit area and a square of unit area without overlapping, what is the maximum area that can be covered?
N=1 seems obvious: (90.9454%)

A possible winner for N=3: (95%)

It seems likely that with, say, N=10 we could get very close indeed but I've never seen any example, and I doubt that my N=3 example above is even the optimum. (Edit: It's not!) And I've no idea what the solution for N=2 would look like.
This page discusses some curved shapes that can be cut up into squares. There's a nice simple proof here that there's no paper-and-scissors solution for the circle and the square.

I believe I have found the optimal method for n=6 and n≥10.
Above you state that an inscribed dodecagon yields coverage of 95.49%, which you demonstrate is not the best answer. You were on the right track, but a better approach is to overlay the circle with a dodecagon of equal area, and then dissect that into its classic 6 pieces. These can be rearranged into a square of equal area with ~99.108% coverage, which unless the linked computer program rounds incorrectly, is better than the best answer achieved by that program so far. Pictures and discussion of best answers for other numbers of pieces can be found here: http://imgur.com/gallery/xHLAL.
What I've found suggests that possibly the best you can get with n=10 is 99.347%, which as you predicted, is very close indeed.
