I just ran across this anecdote by Jason Rosenhouse in Notices of the American Mathematical Society. In a middle-school algebra class Rosenhouse’s brother was given this problem:

There are some horses and chickens in a barn, fifty animals in all. Horses have four legs while chickens have two. If there are 130 legs in the barn, then how many horses and how many chickens are there?

The normal solution is straightforward, but Rosenhouse’s brother found an alternative that’s even easier: “You just tell the horses to stand on their hind legs. Now there are fifty animals each with two legs on the ground, accounting for one hundred legs. That means there are thirty legs in the air. Since every horse has two legs in the air, we find that there are fifteen horses, and therefore thirty-five chickens.”

(Jason Rosenhouse, “Book Review: Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles,” Notices of the American Mathematical Society, 67:9 [October 2020], 1382-1385.)

Bottema’s Theorem

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.

(Demonstration by Jay Warendorff.)

The Aventine Keyhole


The keyhole of the Priory of the Knights of Malta in Rome presents a perfectly framed view of Saint Peter’s Basilica.

It’s not clear whether this is a happy accident or a deliberate design. The property lies in the piazza Cavalieri di Malta, which was designed in 1765 by the supremely imaginative Giovanni Battista Piranesi — who imagined the Aventine Hill as a sacred ship that would sail to the heavens.

Podcast Episode 347: The Cottingley Fairies


In 1917, two young cousins carried a camera into an English dell and returned with a photo of fairies. When Arthur Conan Doyle took up the story it became a worldwide sensation. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Cottingley Fairies, a curiosity that would remain unexplained for most of the 20th century.

We’ll also remember a ferocious fire and puzzle over a troublesome gnome.

See full show notes …

Shape-Memory Alloys

At a 1963 laboratory management meeting at the U.S. Naval Ordnance Laboratory, a sample of a nickel-titanium alloy was presented that had been bent many times out of its original shape. To see its response to heat, associate technical director David S. Muzzey held the sample over his pipe lighter. To everyone’s surprise, it returned to its original shape.

The “shape memory effect” occurs because of the structures involved — when stresses deform the material, its atoms retain their relative position; no bonds are broken or reformed. So when heat is applied it can “remember” its original shape, which it retains after cooling.

The Friendship Medals


At the 1936 Olympics, Japanese pole vaulters Sueo Oe and Shuhei Nishida tied for second place, and the Japanese team were told to decide who should claim second place and who third.

After a long discussion, the team chose to favor Nishida, who had cleared 4.25 meters at his first attempt.

When they returned to Japan, Nishida and Oe had a jeweler cut each medal in half and then join the disparate halves, so that each man had a new medal, half silver and half bronze.

“The Horse of Joy”


The September 1918 issue of Popular Science Monthly describes an amusement introduced at Coney Island by Minnesota inventor Otto Fritsche: a horse made of double-walled aluminum. The patron wears a pair of shoulder straps and a gasoline engine drives the thing forward.

Aside from the fact that the contraption is supported in the rear by a pair of large wheels, that the front is supported by a human being, that a periscope protrudes from the head, and that exhaust gases puff from its ears, Otto believes it will be mistaken for a real animal.

“We would like to interview any man who has tried this form of amusement and has survived.”

Round Here


New Zealand’s Egmont National Park is pleasingly circular. Created in 1881, the reserve was specified to cover a 6-mile radius surveyed from the summit of Mount Taranaki, one of the most symmetrical volcanic cones in the world. The result is an almost perfect circle of 33,500 hectares — and it’s surrounded by pasture, which makes the boundary visually distinct.