Hex Wrench


A problem by Russian mathematician Viktor Prasolov: This hexagon has been divided into black and white triangles so that any two triangles either have a common side (in which case they’re painted different colors) or a common vertex or they have no points in common, and each side of the hexagon is a side of one of the black triangles. Prove that no such partition exists for a 10-gon.

Click for Answer

True and False

Since its launch in 1991, arXiv, Cornell’s open-access repository of electronic preprints, has cataloged more than 2 million scientific papers.

In 2010, Caltech physicist David Simmons-Duffin created snarXiv, a random generator that produces titles and abstracts of imaginary articles in theoretical high-energy physics. Then he challenged visitors to distinguish real titles from fake ones.

After 6 months and 750,000 guesses in more than 50,000 games played in 67 countries, “the results are clear,” he concluded. “Science sounds like gobbledygook.” On average, players had guessed right only 59 percent of the time. Real papers most often judged to be fake:

  • “Highlights of the Theory,” by B.Z. Kopeliovich and R. Peschanski
  • “Heterotic on Half-Flat,” by Sebastien Gurrieri, Andre Lukas, and Andrei Micu
  • “Relativistic Confinement of Neutral Fermions With a Trigonometric Tangent Potential,” by Luis B. Castro and Antonio S. de Castro
  • “Toric Kahler Metrics and AdS_5 in Ring-Like Co-ordinates,” by Bobby S. Acharya, Suresh Govindarajan, and Chethan N. Gowdigere
  • “Aspects of U_A(1) Breaking in the Nambu and Jona-Lasinio Model,” by Alexander A. Osipov, Brigitte Hiller, Veronique Bernard, and Alex H. Blin
  • “Energy’s and Amplitudes’ Positivity,” by Alberto Nicolis, Riccardo Rattazzi, and Enrico Trincherini

Try it yourself.

While we’re at it: SCIgen randomly generates research papers in computer science, complete with graphs, figures, and citations; and Mathgen generates professional-looking mathematics papers, with theorems, proofs, equations, discussion, and references.

(Via Andrew May, Fake Physics: Spoofs, Hoaxes and Fictitious Science, 2019.)

In a Word


n. a faculty or facility for forgetting; faulty memory

n. the practice of self-discipline

n. the action, process, or faculty of looking back on things past

adj. liable to vanish

“King Darius, so as not to forget the harm he had received from the Athenians, had a page come every time he sat down to table and sing three times in his ear: ‘Sire, remember the Athenians.'” — Montaigne

Cato the Elder ended each speech with the phrase Carthago delenda est, “Carthage must be destroyed.”

Tertullian observed that a slave was stationed in the chariot of a triumphant Roman general to whisper in his ear, “Remember that you are human.”

A nomenclator was “a slave with a good memory who accompanied a public figure when he went out and whispered in his ear the name of anyone important he was about to meet.” (Anthony Everitt, Cicero)

Much later, Franklin Roosevelt’s campaign manager, James Farley, would keep a file on everyone Roosevelt met so that the candidate might later ask after a spouse or child. Modern politicians maintain “Farley files” for the same purpose.

A Parabolic Calculator

mathematikum calculator

Mathematikum, the science museum in Giessen, Germany, contains this clever device for multiplying pairs of numbers. The parabola presents the curve y = x2. Now suppose we want to multiply 10 by 9. We find the points -10 and 9 on the x axis, follow them up to the curve, and hang a weighted string over the pegs that we find there. The string crosses the y axis at (0, 90), so 10 × 9 = 90.

A simulator, and an explanation of the principle, are here.

North and South


Ulysses Grant and Jefferson Davis never met, but their widows became good friends. They met at West Point in June 1893, when Varina Davis arrived to watch a cadet parade. Julia Grant presented herself and said, “I am Mrs. Grant.” “I am very glad to meet you,” Davis replied.

They ate dinner together on the piazza as curious guests looked on. “She is a very noble-looking lady,” Grant said afterward. “She looked a little older than I had expected. I have wanted to meet her for a very long time.”

They corresponded and met frequently after that. At Grant’s tomb Davis heard Julia say, “I will soon be laid beside my husband in this solemn place,” and she attended the memorial service in 1902 when these words were fulfilled, among men who had fought on both sides of the war.

In a tribute to her friend published in The World in April 1897, Davis had quoted Ulysses Grant’s motto “Let us have peace.” She added, “I believe every portion of our reunited country heartily joins in the aspiration.”

(From Ishbel Ross, First Lady of the South, 1958.)



1985 saw an oddity in the chess world: Russian grandmaster Viktor Korchnoi undertook a game with Hungarian Géza Maróczy, who had been dead for 34 years. The game was arranged by amateur Swiss player Wolfgang Eisenbeiss, who enlisted medium Robert Rollans to contact the deceased grandmaster and communicate his moves. (Rollans did not play chess and was not paid.)

Closely watched in Germany, the game took nearly eight years to unfold, hampered by Korchnoi’s schedule, Rollans’ illness, and Maróczy’s unhurried pace. Korchnoi, who won after 47 moves, remarked that his opponent had shown weakness in the opening but made up for it with a strong endgame.

After an analysis in 2007, neuropsychiatrist and amateur player Vernon Neppe declared that Maróczy had played at master level and that his moves could not have been found by a computer. Further, when asked to confirm his identity, the deceased grandmaster had dictated 38 pages of text to Rollans, complaining, “I am astonished when somebody does not believe me to be here personally.” Historian and chess expert Laszlo Sebestyen determined that 87.9 percent of Maróczy’s assertions there (about his playing, tournament wins, and personal life) had been accurate.

But in a 2021 critique, Edwin C. May and Sonali Bhatt Marwaha point out that Maróczy had typically taken 10 days to make each move, during which time Rollans might easily have consulted outside assistance. And the medium had had ample time to prepare Maróczy’s 1986 communication confirming his identity. Ultimately the answers lie with Rollans, who, ironically, passed quickly out of reach — he died just 19 days after Maróczy’s resignation.


Letter to the Times, June 23, 2000:

Sir, The shortest ambiguous sentence I have come across is a road sign found everywhere in New York. It consists of three words: ‘Fine for Parking.’

But I would not like to argue the point with a New York traffic cop.

Yours faithfully,

House of Lords

10/16/2023 UPDATE: From reader Brieuc de Grangechamps:

schrödinger's dumpster

Time Tables

polish-american system

Elizabeth Peabody hated rote learning. So the 19th-century American educator adopted the “Polish system,” a graphical means to help students recall reams of historical facts. Invented in the 1820s by Antoni Jażwiński and popularized by Józef Bem, the system relies on a series of 10×10 grids. The location of an event gives its date, the symbol that records it shows its type, and its color indicates the nations involved. A student who wanted to indicate that a revolution took place in America in 1776 would choose the grid for the 18th century, find the square for the 76th year (row 7, column 6), and paint a square using the color designating the British colonies in North America. The square indicates insubordination; a triangle would mean a revolt, an X a conspiracy.

The system is largely forgotten today, but it was immensely popular in Europe and North America in the early 18th century. In the 1830s it was approved for use throughout the French educational system, and Peabody toured the United States offering a book of her own, blank charts and a special set of paints. “Instructors who are themselves not well educated in history, may yet dare to undertake to teach chronology with the help of this manual,” she wrote, “and it must be obvious that highly-accomplished teachers can unfold and develop the subject to an indefinite extent.”

“The results of Peabody’s appropriation of the Polish System are both handsome and surprising,” write Daniel Rosenberg and Anthony Grafton in Cartographies of Time (2012). “Surviving copies of the charts in libraries look nothing like one another. Each bears the imprint of an individual student’s imagination.”

“English as She Is Pronounced”

The wind was rough
And cold and blough,
She kept her hands within her mough.

It chill’d her through,
Her nose grough blough
And still the squall the faster flough.

And yet although
There was nough snough,
The weather was a cruel fough.

It made her cough —
Pray do not scough! —
She coughed until her hat blew ough.

Ah, you may laugh,
You silly caugh!
I’d like to beat you with my staugh.

Her hat she caught,
And saught and faught
To put it on and tie it taught.

Try as she might
To fix it tight
Again it flew off like a kight,

Away up high
Into the skigh.
The poor girl sat her down to crigh.

She cried till eight
P.M., so leight!
Then home she went at a greight reight.

— J.H. Walton