This transit map of Stuttgart’s rail network, adopted around 2000, was unique: By omitting horizontal and vertical lines and setting all diagonals at 30 degrees, the designers produced the appearance of three dimensions.
“This diagram is the only one of its type in the world,” wrote Mark Ovenden in Transit Maps of the World, “although Harry Beck did experiment briefly with a 60/120-degree variation of the London map in 1940.” Alas, it’s since been superseded.
“It is indeed a desirable thing to be well descended, but the glory belongs to our ancestors.” — Plutarch
“He who boasts of his ancestry praises the merits of another.” — Seneca
“A man who makes boast of his ancestors doth but advertise his own insignificance.” — Benjamin Franklin
“The man who has not anything to boast of but his illustrious ancestors is like a potato — the only good belonging to him is under ground.” — Sir Thomas Overbury
Sometimes in our research we come across stories that are regarded as true but that we can’t fully verify. In this week’s episode of the Futility Closet podcast we’ll share two such stories from the 1920s, about a pair of New York fruit dealers and a mythologized bank robber, and discuss the strength of the evidence behind them.
We’ll also salute a retiring cat and puzzle over a heartless spouse.
Introduced by Eberhard Faber in 1934, the Blackwing 602 premium writing pencil was stamped with the words “Half the Pressure, Twice the Speed”: Compared to an ordinary pencil, its core contained more graphite, less clay, and wax, so that it wrote like a pencil of 4B hardness but with a unique gliding feel.
It has attracted an impressive roster of creative admirers, including Walt Disney, Stephen Sondheim, Quincy Jones, Vladimir Nabokov, and John Steinbeck, who wrote, “I have found a new kind of pencil — the best I have ever had. Of course it costs three times as much too, but it is black and soft but doesn’t break off. I think I will always use these. They are called Blackwings and they really glide over the paper.”
Steinbeck would use a Blackwing pencil right down to the ferrule (pencil devotees now call this “Steinbeck stage”) and then pass them on to his son, another writer. “Writing with a Blackwing 602, more than any other pencil, feels like an event — something like a rite of passage for a pencil obsessive,” writes Caroline Weaver in The Pencil Perfect: The Untold Story of a Cultural Icon (2017). “When they are sold in my shop I always encourage the customer to sharpen it at least once and to use it for special occasions, because most of the pleasure of owning it comes from knowing what it feels like to write with it as much as it comes from the history.”
In the 1928 film Steamboat Bill, Jr., a falling facade threatens to flatten Buster Keaton, but he’s spared by the fortunate placement of an open attic window. “As he stood in the studio street waiting for a building to crash on him, he noticed that some of the electricians and extras were praying,” writes Marion Meade in Cut to the Chase, her biography of Keaton. “Afterward, he would call the stunt one of his greatest thrills.”
It’s often said that the falling wall missed Keaton by inches. Is that true? James Metz studied the problem in Mathematics Teacher in 2019. Keaton was 5 feet 5 inches tall; if that the “hinge” of the facade is 5 inches above the surface of the ground, the attic window is 12 feet above that, and the window is 3 feet high, he finds that the top of the window came only within about 1.5 feet of Keaton’s head.
“The window was tall enough to allow an ample margin of safety, so the legend about barely missing his head cannot be true,” Metz writes. “Apparently, Keaton had more headroom than was previously suspected.”
(James Metz, “The Right Place at the Right Time,” Mathematics Teacher 112:4 [January/February 2019], 247-249.)
In 1988, Florida International University mathematician T.I. Ramsamujh offered a proof that all positive integers are equal. “The proof is of course fallacious but the error is so nicely hidden that the task of locating it becomes an interesting exercise.”
Let p(n) be the proposition, ‘If the maximum of two positive integers is n then the integers are equal.’ We will first show that p(n) is true for each positive integer. Observe that p(1) is true, because if the maximum of two positive integers is 1 then both integers must be 1, and so they are equal. Now assume that p(n) is true and let u and v be positive integers with maximum n + 1. Then the maximum of u – 1 and v – 1 is n. Since p(n) is true it follows that u – 1 = v – 1. Thus u = v and so p(n + 1) is true. Hence p(n) implies p(n+ 1) for each positive integer n. By the principle of mathematical induction it now follows that p(n) is true for each positive integer n.
Now let x and y be any two positive integers. Take n to be the maximum of x and y. Since p(n) is true it follows that x = y.
“We have thus shown that any two positive integers are equal. Where is the error?”
(T.I. Ramsamujh, “72.14 A Paradox: (1) All Positive Integers Are Equal,” Mathematical Gazette 72:460 [June 1988], 113.)
When a limb is paralyzed and then amputated, the patient may perceive a “phantom limb” in its place that is itself paralyzed — the brain has “learned” that the limb is paralyzed and has not received any feedback to the contrary.
University of California neuroscientist V.S. Ramachandran found a simple solution: The patient holds the intact limb next to a mirror, looks at the reflected image, and makes symmetric movements with both the good and the phantom limb. In the reflected image, the brain is now able to “see” the phantom limb moving. The impression of paralysis lifts, and the patient can now move the phantom limb out of painful positions.
A 2018 review called the technique “a valid, simple, and inexpensive treatment for [phantom-limb pain].”
“Two years ago,” she said, “when I was so sick, I realized that I was dreaming the same dream night after night. I was walking in the country. In the distance, I could see a white house, low and long, that was surrounded by a grove of linden trees. To the left of the house, a meadow bordered by poplars pleasantly interrupted the symmetry of the scene, and the tips of the poplars, which you could see from far off, were swaying above the linden.
“In my dream, I was drawn to this house, and I walked toward it. A white wooden gate closed the entrance. I opened it and walked along a gracefully curving path. The path was lined by trees, and under the trees I found spring flowers — primroses and periwinkles and anemones that faded the moment I picked them. As I came to the end of this path, I was only a few steps from the house. In front of the house, there was a great green expanse, clipped like the English lawns. It was bare, except for a single bed of violet flowers encircling it.
“The house was built of white stone and it had a slate roof. The door — a light oak door with carved panels — was at the top of a flight of steps. I wanted to visit the house, but no one answered when I called. I was terribly disappointed, and I ran and I shouted — and finally I woke up.
“That was my dream, and for months it kept coming back with such precision and fidelity that finally I thought: surely I must have seen this park and this house as a child. When I would wake up, however, I could never recapture it in waking memory. The search for it became such an obsession that one summer — I’d learned to drive a little car — I decided to spend my vacation driving through France in search of my dream house.
“I’m not going to tell about my travels. I explored Normandy, Touraine, Poitou, and found nothing, which didn’t surprise me. In October, I came back to Paris, and all through the winter I continued to dream about the white house. Last spring, I resumed my old habit of making excursions in the outskirts of Paris. One day, I was crossing a valley near L’Isle-Adam. Suddenly I felt an agreeable shock — that strange feeling one has when after a long absence one recognizes people or places one has loved.
“Although I had never been in that particular area before, I was perfectly familiar with the landscape lying to my right. The crests of some poplars dominated a stand of linden trees. Through the foliage, which was still sparse, I could glimpse a house. Then I knew that I had found my dream château. I was quite aware that a hundred yards ahead, a narrow road would be cutting across the highway. The road was there. I followed it. It led me to a white gate.
“There began the path I had so often followed. Under the trees, I admired the carpet of soft colors woven by the periwinkles, the primroses, and the anemones. When I came to the end of the row of linden, I saw the green lawn and the little flight of steps, at the top of which was the light oak door. I got out of my car, ran quickly up the steps, and rang the bell.
“I was terribly afraid that no one would answer, but almost immediately a servant appeared. It was a man, with a sad face, very old. He was wearing a black jacket. He seemed very much surprised to see me, and he looked at me closely without saying a word.
“‘I’m going to ask you a rather odd favor,’ I said. ‘I don’t know the owners of this house, but I would be very happy if they would permit me to visit it.’
“‘The château is for rent, madame,’ he said, with what struck me as regret, ‘and I am here to show it.’
“‘To rent?’ I said. ‘What luck! It’s too much to have hoped for. But how is it that the owners of such a beautiful house aren’t living in it?’
“‘The owners did live in it, madame. They moved out when it became haunted.’
“‘Haunted?’ I said. ‘That will scarcely stop me. I didn’t know people in France, even in the country, still believed in ghosts.’
“‘I wouldn’t believe in them, madame,’ he said seriously, ‘if I myself had not met, in the park at night, the phantom that drove my employers away.’
“‘What a story!’ I said, trying to smile.
“‘A story, madame,’ the old man said, with an air of reproach, ‘that you least of all should laugh at, since the phantom was you.'”
— Andre Maurois, 1931
A puzzle by Ezra Brown and James Tanton:
Three gnomes sit in a circle. An evil villain puts a hat on each gnome’s head. Each hat is either rouge or puce, the color chosen by the toss of a coin. Each gnome can see the color of his companions’ hats but not of his own.
At the villain’s signal, all three gnomes must speak at once, each either guessing the color of his own hat or saying “Pass.” If at least one of them guesses correctly and none guesses incorrectly, all three gnomes will live. But if any of them guesses incorrectly, or if all three pass, they’ll all die.
They may not communicate in any way during the game, but they can confer beforehand. How can they arrange a 75 percent chance that they’ll survive?
The Sherlock Holmes story “The Adventure of the Priory School” concerns the Duke of Holdernesse and the kidnapping of his son, Lord Saltire. The family name of the duke is never given in the story, but Holmes mentions in “The Adventure of the Blanched Soldier” that his real title is Duke of Greyminster.
Greyminster calls to mind Lord Greystoke, the title of Tarzan’s father, John Clayton, in Edgar Rice Burroughs’ Tarzan adventures. Greystoke is marooned on the west coast of Africa while investigating a political intrigue there, and his wife gives birth to a son who becomes lord of the apes.
Is there a link here? Did Burroughs discreetly alter Clayton’s title from Greyminster to Greystoke in telling his story?
Writing in the Baker Street Journal in 1960, Princeton English professor H.W. Starr points out that John Clayton is also the name of the driver of the cab in which Stapleton trails Sir Henry and Dr. Mortimer in The Hound of the Baskervilles.
That’s an interesting coincidence. Can Tarzan’s father and the cab driver be the same man? No, the timing doesn’t work: Tarzan’s father was marooned in 1888 and never returned to England, and the John Clayton of Baskervilles appears at 221B Baker Street in 1889.
But the cabbie could be Tarzan’s grandfather. In the Tarzan stories, when Tarzan’s father disappears off Africa, the title passes to his younger brother (the cabbie’s other son, in our hypothesis), and we are told that that son bore a son of his own, William Cecil Clayton. William Cecil Clayton was 19 or 20 in 1909, similar to the age of Lord Saltire, the kidnapped son of the Duke of Holdernesse in “Priory.” Starr writes, “This seems too close a correspondence of facts to be mere coincidence.”
If all this is true, and assuming that Burroughs changed “Greyminster” to “Greystoke,” then John Clayton the cab driver is really the fifth Duke of Greyminster, father of two sons: John Clayton (“Lord Greystoke” in Burroughs’ rendering) and the “sixth Duke of Holdernesse” (in Doyle’s rendering). And the “Lord Saltire” who is kidnapped in “The Adventure of the Priory School” is the latter’s son, William Cecil Clayton, seventh Duke and the first cousin of John Clayton III, who is the eighth Duke of Greyminster and Tarzan of the Apes.
(All this might also help to explain Holmes’ whereabouts during his supposed death in the period 1891-1894 — possibly the British government had sent him to Africa to investigate Greystoke’s disappearance.)
(H.W. Starr, “A Case of Identity, or The Adventures of the Seven Claytons,” Baker Street Journal, New Series X, i, January 1960.)
Philip José Farmer hatched an even more elaborate theory in 1972.
