This came to me whilst trying to balance a Rubik’s cube on the circular rim of a coffee cup. There are three ways of orienting a cube on top of a cup, which I’ll refer to as face-down, edge-down (resting on the rim in four places) and point-down (resting on the rim in three places. For these latter two cases the cube extends below the level of the rim of the cup. Depending upon the particular cube and the particular cup, the cube might extend deeper into the cup when it is edge-down or when it is point-down.

Assuming that the edge-down and point-down orientations are possible for the given cube and cup, how can you tell, very quickly and without using any measuring devices, only looking at and manipulating the cube and the cup, which of those two orientations extends deeper into the cup?