Dork Scratchings

Can you always reorient a four legged chair to make it stable on an uneven surface? If we can assume the feet form a perfect square, then: Yes! Without loss of generality let's suppose that the chair starts off in the position shown top left, with its front right foot in the air and the others on the ground.  If we keep the two left feet on the ground while pushing down on the other two, the above ground foot will eventually touch the floor and the remaining foot will sink below it. Now imagine instead rotating the chair by 90$^\circ$, keeping the same 3 feet touching the ground at all times, and ending with the two back feet in the positions previously occupied by the two left feet.  The two back feet and the front left foot occupy three corners of the square described by the feet at the end of the previous paragraph, so the fourth must be below ground.  Since the front right foot starts off in the air and ends up below ground we can infer from t...

In More Mathematical Puzzles and Diversions , Martin Gardner makes a passing reference to the Ham Sandwich Theorem.  It goes like this Any 3 shapes in 3 dimensional space can be simultaneously bisected by a single plane So imagine you have two roughly cut pieces of bread and a slice of ham, then you can always cut the sandwich in half such that each half has exactly half of each piece of bread and half of the ham, no matter how roughly strewn the pieces are. According to Gardner the generalized version has been proved by Tukey and Stone: any n shapes in $\mathbb{R}^n$ can be simultaneously halved by a single $n-1$ dimensional hyperplane.  But I thought I'd have a go at proving it myself in the 3D case, just for kicks. First observe that there are at least enough degrees of freedom to make it not impossible .  A plane (other than one going through the origin) can be described by the equation $k_xx+k_yy+k_zz = 1$ for some $\boldsymbol{k} \neq \boldsymbol{0}$ so the...