Perhaps I am misinterpreting, but it seems to me that some of your testsuite solutions are incorrect. For example, for N=5 I seem to be getting 19 instead of 27 cubes entirely within the sphere. I double-check using this code: x=rand(1e6,3); r=sum((x-.5).^2,2)
You are totally right. I've generated answers using my solution, which was wrong. My fault is I haven't examined it any other way. Considering N=5, there could fit inside the sphere only a 3x3x3 cube, but its space diagonal is 3*sqrt(3) which is bigger than 5, the diameter of sphere. So, there cannot be more than 27-8=19 cubes inside. Thanks, for pointing that out, Alfonso!
Fixed & rescored now. It leads now to smaller-size answers. Consider the fact that for some N (for example 6) there are some quasi-cubes containing a very small amount of peel from those ultra-thin-peel-sweet-juicy-ideal-oranges ;-)
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