[ Huh. My statistics page says that someone came to me yesterday looking for the “mathematics behind rap music”. I don’t doubt there is such mathematics, but I’ve never written anything approaching it. I admit that despite the long intertwining of mathematics and music, and my own childhood of playing a three-quarter size violin in a way that must be characterized as “technically playing”, I don’t know anything nontrivial about the mathematics of any music. So, whoever was searching for that, I’m sorry to have disappointed you. ]
Now, let me try my first guess at saying whether it’s easier to tip the cube over by pushing along the middle of the edge or by pushing at the corner. I laid out the ground rules, and particularly, the symbols used for the size of the box (it’s of length ) and how far the center of mass (the dead center of the box) is from the edges and the corners last time around. Here’s my first thought about what has to be done to tip the box over: we have to make the box pivot on some point — along one edge, if we’re pushing on the edge; along one corner, if we’re pushing on the corner — and so make it start to roll. If we can raise the center of mass above the pivot then we can drop the box back down with some other face to the floor, which has to count as tipping the box over. If we don’t raise the center of mass we aren’t tipping the box at all, we’re just shoving it.