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.
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.)
Another remarkable contribution by Lee Sallows:
(Thanks, Lee!)
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,
Millett
House of Lords
10/16/2023 UPDATE: From reader Brieuc de Grangechamps:
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.”
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
In July 1054 Chinese astronomers saw a reddish-white star appear in the eastern sky, its “rays stemming in all directions.” Yang Weide wrote:
I humbly observe that a guest star has appeared; above the star there is a feeble yellow glimmer. If one examines the divination regarding the Emperor, the interpretation is the following: The fact that the star has not overrun Bi and that its brightness must represent a person of great value. I demand that the Office of Historiography is informed of this.
It’s now believed they were witnessing SN 1054 — the supernova that gave birth to the Crab Nebula.
If Chicken McNuggets come in packs of 6, 9, and 20, what’s the largest number of McNuggets that you can’t buy?
Steve Omohundro and Peter Blicher posed this question in MIT Technology Review in May 2002, and Ken Rosato contributed a neat solution.
The answer is 43. To start, notice that we can use the 6-packs and 9-packs to piece together any multiple of 3 other than 3 itself. 43 itself is not divisible by 3, so 6-packs and 9-packs alone won’t get us there, and adding some 20-packs won’t help, since we’d have to add them to a quantity of either 23 or 3, neither of which can be assembled from packs of other sizes. So that shows that 43 itself can’t be reached.
But we still need to show that every larger number can be. Well, we can create all the larger even numbers by adding some quantity of 6-packs to either 36, 38, or 40, and each of those foundations can be assembled from the packs we have (36 = 9 + 9 + 9 + 9, 38 = 20 + 9 + 9, and 40 = 20 + 20). So that takes care of the even numbers. And adding 9 to any of these even numbers will give us any desired odd number above 43, starting with 36 + 9 = 45.
So 43 is the largest number of Chicken McNuggets that can’t be formed by combining 6-packs, 9-packs, and 20-packs.
(I think Henri Picciotto was the first to broach this arresting question, in Games magazine in 1987. Since then, McNuggets have found their way into Happy Meals in 4-piece servings, reducing the largest non–McNugget number to 11. In some countries, though, the 9-piece allotment has been increased to 10 — and in that case there is no largest such number, as no odd quantity can ever be assembled.)
In their 1981 book Facts and Fallacies, Chris Morgan and David Langford note that the biblical reference to the “four corners of the earth” would apply equally well if the world were a tetrahedron.
In a similar spirit, as American airman Joseph Portney was flying over the North Pole in 1968 he wondered, “What if the Earth were … ?” He made sketches of 12 fanciful alternate Earths and gave them to Litton’s Guidance & Control Systems graphic arts group, which created models that were featured in the company’s Pilots and Navigators Calendar of 1969. This made an international sensation, and Portney’s creations were subsequently published for use in classrooms worldwide, inviting students to ponder what life would be like on a cone or a dodecahedron.
Portney graduated from the U.S. Naval Academy and went on to work for Litton on high-altitude navigation problems — for example, designing control systems that could guide an aircraft around one of these strange worlds.