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What is maximum size of solid cube that can rest on top of a semi-sphere of radius 1m?
A question like this has to be non-trivial, so I'll suppose the criterion is stability of the cube in the "upright", centered position and that the cube is in nonslip rolling contact with the hemisphere. It then becomes necessary that the upright state is the lowest-energy one and thus any deviation from upright raises the cube's CM. The intuitive answer is that the CM should be 1 hemisphere radius above the contact point. I don't yet have a proof worked out but it certainly works when plotted, so it's a spare time project. There must be other fans who can contribute.
Circle Kingz 2008 Semi Final - Top Mavericks(UK/Russia) vs Jinjo Crew(Korea)