Most sides a cube's shadow can have
Question
A cube is lit so that it casts a shadow on a flat surface. Over all orientations of the cube, what is the greatest number of sides the shadow outline can have?
Short answer
6. The shadow is the projection of the cube, which is a parallelogram-sum of the three edge directions, so its outline is a hexagon at most. Looking along a main diagonal, the three faces at one corner project to a regular hexagon.
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