# Cubes and Squares

MATLAB’s Loren Shure devised this lovely “proof without words” of Nicomachus’ theorem, that the sum of the first n cubes is the square of the nth triangular number:

$\displaystyle 1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(1+2+3+\cdots +n\right)^{2},$

R.J. Stroeker of Erasmus University wrote, “Every beginning student of number theory surely must have marveled at [this] miraculous fact.”

# Truth in Advertising

Another feat of self-reference — reader Hans Havermann devised these true sentences:

“The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are two out of two hundred nineteen thousand six hundred eighty-seven.”

“The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are three out of two hundred ninety-two thousand nine hundred sixteen.”

These two are in lowest terms. He has seven more.

(Thanks, Hans.)

# Ups and Downs

An Edinburgh startup called Gravitricity is hoping to create a “virtual battery” by hoisting and dropping weights in disused mine shafts. If the weights are hoisted when renewable energy is plentiful, and dropped when it’s expensive, then they can help to balance the energy grid with an efficient source of “gravity energy.”

Managing director Charlie Blair told the Guardian, “The beauty of this is that this can be done multiple times a day for many years, without any loss of performance. This makes it very competitive against other forms of energy storage — including lithium-ion batteries.”

Dropping 12,000 tonnes to a depth of 800 meters would produce enough electricity to power 63,000 homes for more than an hour. Oliver Schmidt of Imperial College London said, “I don’t expect Gravitricity to displace all lithium batteries on grids, but it certainly looks like a compelling proposition.”

(Via Tom Whitwell’s “52 Things I Learned in 2019.”)

# False Features

Just before his death in 1702, butterfly collector William Charlton delivered an unusual specimen to London entomologist James Petiver. Petiver wrote, “It exactly resembles our English Brimstone Butterfly (R. Rhamni), were it not for those black spots and apparent blue moons on the lower wings. This is the only one I have seen.” Carl Linnaeus named it Papilio ecclipsis and included it in the 12th edition of his Systema Naturae in 1767.

It wasn’t until 1793 that Danish zoologist Johan Christian Fabricius discovered that the dark patches had been painted on — it was only an ordinary brimstone butterfly after all. The curator at the British Museum “indignantly stamped the specimen to pieces” at this news, but entomologist William Jones created two new replicas to commemorate the “Charlton Brimstones.”

# Pain and Possession

A stunning finding regarding the efficacy of placebos:

In 2017 Victoria Wai-lanYeung, a psychologist at Lingnan University in Hong Kong, and her colleagues gave a placebo painkilling cream at random to half their subjects, as a gift, and then put all the subjects to a moderately painful task (immersing a hand in ice water).

Participants who’d received the cream but hadn’t used it reported lower pain levels than those who hadn’t received it — merely owning a fake painkiller had reduced their pain.

More at the link below.

(Victoria Wai-lanYeung, Andrew Geers, and Simon Man-chun Kam, “Merely Possessing a Placebo Analgesic Reduced Pain Intensity: Preliminary Findings From a Randomized Design,” Current Psychology 38:1 [2019], 194-203.)

# A Match

Do there exist a right triangle and an isosceles triangle, each with sides of rational length, that have both the same perimeter and the same area?

Yes, there’s exactly one such pair!

The right triangle has sides of lengths (377, 135, 352) and the isosceles of lengths (366, 366, 132).

# Reckoning

Mercyhurst College mathematician Charles Redmond used to get frantic phone calls at the end of each term from colleagues in other departments who’d written syllabi saying “Quiz 1 is 10% of your final grade” and now couldn’t figure out how to do the necessary calculation.

“There’s a good reason,” Redmond wrote. “They said something they didn’t mean to say, and they had only a vague notion of what they meant to say in the first place.”

He suggested that students might turn this confusion to their advantage. Suppose Quiz 1 is worth 10 or more points and your score is 9. If x is your final grade and “Quiz 1 is 10% of your final grade,” then 0.10x = 9 and suddenly your final grade is 90, an A.

“Not bad for about a week’s worth of work. Take the rest of the semester off.”

(Ed Barbeau, “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 33:2 [March 2002], 137-139.)

# Another Way

Carnegie-Mellon mathematician Po-Shen Loh has offered a new, simple proof of the quadratic formula that provides a natural, intuitive algorithm for solving general quadratic equations.

More here. “May this story encourage the reader to think afresh about old things; seeing as how new progress was made on this 4,000 year old topic, more surprises certainly await the light of discovery.”

(Thanks, Jason.)

# First Friends

In 1978, archaeologists excavating a late Paleolithic tomb in northern Israel uncovered the skeletons of an elderly human and a 5-month-old puppy. They had lain there together for 12,000 years.

“The most striking thing about these remains was the fact that whoever presided over the original burial had carefully arranged the dead person’s left hand so that it rested, in a timeless and eloquent gesture of attachment, on the puppy’s shoulder,” writes James Serpell in In the Company of Animals (1996).

“The contents of this tomb not only provide us with some of the earliest solid evidence of animal domestication, they also strongly imply that man’s primordial relationship with this particular species was an affectionate one. In other words, prehistoric man may have loved his dogs and his other domestic animals as pets long before he made use of them for any other purpose.”

(Simon J.M. Davis and François R. Valla, “Evidence for Domestication of the Dog 12,000 Years Ago in the Natufian of Israel,” Nature 276:5688 [1978], 608.)

# The Friendly Floatees

During a storm in January 1992, a container was swept overboard from a ship in the North Pacific. As it happened, it contained 28,800 children’s bath toys, and oceanographer Curtis Ebbesmeyer realized they offered the basis for a serendipitous study of surface currents. Working with his colleague James Ingraham, Ebbesmeyer began to track the toys as they drifted around the globe, accumulating reports from beachcombers, coastal workers, and local residents as they began to wash up on beaches. Using computer models, they were able to predict correctly that toys would make landfall in Washington state, Japan, and Alaska, and even become trapped in pack ice and spend years creeping across the top of the world before making an eventual reappearance in the North Atlantic. “Ultimately,” Ebbesmeyer wrote, “the toys will turn to dust, joining the scum of plastic powder which rides the global ocean.”

For some reason, media accounts of the story always carried the image of a solitary rubber duck, though the toys had also included beavers, turtles, and frogs. “Maybe it’s a kind of racism,” Ebbesmeyer speculated to journalist Donovan Hohn in 2007. “Speciesism.”