Tag Archives: Futility Closet

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

Moving Up

Here is a simple problem from an old Futility Closet posting.

“My wife and I walk up an ascending escalator. I climb 20 steps and reach the top in 60 seconds. My wife climbs 16 steps and reaches the top in 72 seconds. If the escalator broke tomorrow, how many steps would we have to climb?”

Answer.

See Moving Up for solutions.

Square Deal

Here is a simple Futility Closet problem from 2014.

“This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?”

Answer.

See the Square Deal for solutions.

Pointing Fingers

Here is a nice logic puzzle from 2014 Futility Closet.

“Only one of these statements is true. Which is it?

_________A. All of the below
_________B. None of the below
_________C. One of the above
_________D. All of the above
_________E. None of the above
_________F. None of the above”

Answer.

See Pointing Fingers for solutions.

Modern Art

This is a problem from a while back (2015) at Futility Closet.

“Which part of this square has the greater area, the black part or the gray part?”

Answer.

See Modern Art. for solutions.

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

Threewise Problem

Here is another simple problem from Futility Closet.

“Draw an arbitrary triangle [ABC] and build an equilateral triangle on each of its sides, as shown. Now show that [straight lines] AP = BQ = CR.”

See Threewise Problem

Magic Pythagorean Circle

This statement showed up recently at Futility Closet and I found it to be another one of those magical results that seemed so surprising. I don’t recall ever seeing this before.

“The radius of a circle inscribed in a 3-4-5 triangle is 1.
(In fact, the inradius of any Pythagorean triangle is an integer.)”

(A Pythagorean triangle is a right triangle whose sides form a Pythagorean triple.) Futility Closet left these remarkable statements unproven, so naturally I felt I had to provide a proof.

See Magic Pythagorean Circle

Straight and Narrow Problem

The following interesting behavior was found at the Futility Closet website:

“A pleasing fact from David Wells’ Archimedes Mathematics Education Newsletter: Draw two parallel lines. Fix a point A on one line and move a second point B along the other line. If an equilateral triangle is constructed with these two points as two of its vertices, then as the second point moves, the third vertex C of the triangle will trace out a straight line. Thanks to reader Matthew Scroggs for the tip and the GIF.”

This is rather amazing and cries out for a proof. It also raises the question of how anyone noticed this behavior in the first place. I proved the result with calculus, but I wonder if there is a slicker way that makes it more obvious. See the Straight and Narrow Problem.

(Update 3/25/2019) Continue reading →

Meditations on Mathematics

A compendium of puzzles, problems, math inquiries, and math commentary.

Tag Archives: Futility Closet

Bottema’s Theorem

A Self-Characterizing Figure

Moving Up

Square Deal

Pointing Fingers

Modern Art

A Tidy Theorem

Threewise Problem

Magic Pythagorean Circle

Straight and Narrow Problem