Clearance

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.)

All for One

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.)

(The error is explained here and here.)

Mirror Therapy

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].”

“The House”

“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 Stressful Game

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?

Small World

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.

The Stairs of Reconciliation

The Burg, the official headquarters of the regional government in Graz, Austria, contains a double spiral staircase, two flights of stairs spiraling in opposite directions that “reunite” at each floor, a masterpiece of architecture designed in 1499.

Bonus: Interestingly, several facades of the building bear the inscription A.E.I.O.U., a motto coined by Frederick III in 1437, when he was Duke of Styria. It’s not clear what this means, and over the ensuing centuries heraldists have offered more than 300 interpretations:

• “All the world is subject to Austria” (Alles Erdreich ist Österreich untertan or Austriae est imperare orbi universo)
• “I am loved by the elect” (from the Latin amor electis, iniustis ordinor ultor)
• “Austria is best united by the Empire” (Austria est imperio optime unita)
• “Austria will be the last (surviving) in the world” (Austria erit in orbe ultima)
• “It is Austria’s destiny to rule the whole world” (Austriae est imperare orbi universo)

At the time Styria was not yet part of Austria, so here it would refer to the House of Austria, or the Habsburg dynasty — which historically adopted the curious motto itself.

One Two Three

Discovered by Archimedes: The volume of a cone, sphere, and cylinder of the same height and radius fall in the ratio 1:2:3.

A cone plus a sphere is a cylinder.

Intentions

In 1983 Paul Desmond Taafe imported certain packages into England. He thought they contained currency, which he erroneously believed was illegal to import. The packages actually contained cannabis, which was illegal to import. Was he “knowingly concerned in [the] fraudulent evasion” of any prohibition on importing goods?

He was convicted but appealed. “If we describe his action in terms of his own beliefs (about the facts and about the law), it obviously constituted an attempt to commit (indeed, it constituted the actual commission of) that crime,” writes R.A. Duff in Criminal Attempts. But Taafe wasn’t “knowingly concerned” in evading the ban on cannabis — he didn’t know he was importing cannabis. And however guilty he may have felt for smuggling currency, that wasn’t a crime.

He was acquitted.

(Taaffe [1983] 1 WLR 627 (CA); [1984] 1 AC 539 (HL).)

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