Tag Archives: Futility Closet

Existence Proofs II

Futility Closet has another example of an existence proof like their previous one taken from Peter Winkler’s 2021 Mathematical Puzzles (see my post “Existence Proofs”):

“Four bugs live on the four vertices of a regular tetrahedron. One day each bug decides to go for a little walk on the tetrahedron’s surface. After the walk, two of the bugs have returned to their homes, but the other two find that they have switched vertices. Prove that there was some moment when all four bugs lay on the same plane.”

See Existence Proofs II

Existence Proofs

Here is a seemingly simple problem from Futility Closet.

“A quickie from Peter Winkler’s Mathematical Puzzles, 2021: Can West Virginia be inscribed in a square? That is, is it possible to draw some square each of whose four sides is tangent to this shape?”

Technically we might rephrase this as, can we inscribe a flat map of West Virginia in a square, since the boundary of most states is probably not differentiable everywhere, that is, has a tangent everywhere.

But the real significance of the problem is that it is an example of an “existence proof”, which in mathematics refers to a proof that asserts the existence of a solution to a problem, but does not (or cannot) produce the solution itself.  These proofs are second in delight only to the “impossible proofs” which prove that something is impossible, such as trisecting an angle solely with ruler and compass.

Here is another classic example (whose origin I don’t recall).  Consider the temperatures of the earth around the equator.  At any given instant of time there must be at least two antipodal points that have the same temperature.  (Antipodal points are the opposite ends of a diameter through the center of the earth.)

See Existence Proofs (revised)

(Update 10/2/2021) I fixed a minor typo: “tail” should have been “head”

Bottema’s Theorem

This seemingly magical result from Futility Closet defies proof at first.  Go to the Wolfram demo by Jay Warendorff and then …

“Grab point B above and drag it to a new location. Surprisingly, M, the midpoint of RS, doesn’t move.

This works for any triangle — draw squares on two of its sides, note their common vertex, and draw a line that connects the vertices of the respective squares that lie opposite that point. Now changing the location of the common vertex does not change the location of the midpoint of the line.

It was discovered by Dutch mathematician Oene Bottema.”

As we shall see, Bottema’s Theorem has shown up in other guises as well.

See Bottema’s Theorem

A Self-Characterizing Figure

Futility Closet describes a result that is startling, amazing, and mysterious.

“This is pretty: If you choose n > 1 equally spaced points on a unit circle and connect one of them to each of the others, the product of the lengths of these chords equals n.”

The Futility Closet posting includes an interactive display using Wolfram Technology by Jay Warendorff that let’s you select different n and see the results.  It also includes a reference to a paper that proves the result; only the paper uses residue theory from complex variables, which seems a bit over-kill, though slick, for such a problem.  I found a simpler route.

See a Self-Characterizing Figure

A Tidy Theorem

This is another fairly simple puzzle from Futility Closet.

“If an equilateral triangle is inscribed in a circle, then the distance from any point on the circle to the triangle’s farthest vertex is equal to the sum of its distances to the two nearer vertices (q = p + r).

(A corollary of Ptolemy’s theorem.)”

See A Tidy Theorem