This problem, by V. Tchepizhni, won fifth prize in the Bohemian Centenary Tourney of 1962. It’s four puzzles in one:
(a) The diagrammed position.
(b) Turn the board 90 degrees clockwise.
(c) Turn the board 180 degrees.
(d) Turn the board 270 degrees clockwise.
Each of the resulting positions presents a helpmate in two: Black moves first, then White, then Black, then White, the two sides cooperating to reach a position in which Black is checkmated. What are the solutions?
Dakar is closer to Toronto than to Mogadishu.
(Thanks, Johnny.)
Crockett Johnson, author of the 1955 children’s book Harold and the Purple Crayon, was trained as an engineer and produced more than 100 paintings based on diagrams used in the proofs of classical theorems. This one, Polar Line of a Point and a Circle (Apollonius), appears to have been inspired by a figure in Nathan A. Court’s 1966 College Geometry. The two circles are orthogonal: They cut one another at right angles. And as the square of the line connecting their centers equals the sum of the squares of their radii, these three segments form a right triangle.
Johnson was inspired to this work by his admiration of classical Greek architecture. Sitting in a restaurant in Syracuse in 1973, he managed to construct a heptagon using seven toothpicks and the edges of a menu and a wine list, a construction that had eluded the Greeks. (He found later that Archibald Finlay had illustrated similar constructions in 1959.)
(Stephanie Cawthorne and Judy Green, “Harold and the Purple Heptagon,” Math Horizons 17:1 [2009], 5-9.)
A curious detail from Ernest Shackleton’s 1919 memoir South — he and his companions have just crossed 800 miles of the icy Southern Ocean and traversed unexplored South Georgia Island to get help for their friends on Elephant Island:
When I look back at those days I have no doubt that Providence guided us, not only across those snowfields, but across the storm-white sea that separated Elephant Island from our landing-place on South Georgia. I know that during that long and racking march of thirty-six hours over the unnamed mountains and glaciers of South Georgia it seemed to me often that we were four, not three. I said nothing to my companions on the point, but afterwards Worsley said to me, ‘Boss, I had a curious feeling on the march that there was another person with us.’ Crean confessed to the same idea. One feels ‘the dearth of human words, the roughness of mortal speech’ in trying to describe things intangible, but a record of our journeys would be incomplete without a reference to a subject very near to our hearts.
T.S. Eliot picked up the image in The Waste Land:
Who is the third who walks always beside you?
When I count, there are only you and I together
But when I look ahead up the white road
There is always another one walking beside you
Gliding wrapt in a brown mantle, hooded
I do not know whether a man or a woman
— But who is that on the other side of you?
Shackleton’s description encouraged other survivors of extreme hardship to share similar experiences — it appears to be most common among mountain climbers, solo sailors, and polar explorers. It’s called the Third Man factor.
In 1989, William Faulkner’s niece founded an annual contest to parody her uncle’s distinctive writing style. It ran for 16 years, with the winners published in United Airlines’ Hemispheres magazine. Here’s the winner from 2001, “The (Auto) Pound and the Jury (or Quentin Gets His First Parking Ticket),” by Louisiana Philharmonic clarinetist Allan Kolsky:
For the fifth time in as many minutes, the bright shapes slowly passed us through the somnolent dust, each moving left to right, each in its ordered place. As we (once again) passed beneath the grim and merciless statue of the confederate soldier (that still unravish’d sentinel of quietude, his implacable marble hand forever shading the inscrutable carven eyes) our hearts sank a little deeper, not because we now realized that our quest was futile, but because it always had been, because we now seemed doomed forever to circle this postage stamp of land like slow planets orbiting some inescapable star.
‘Well well well,’ said Ratliff, ‘I reckon thats the fifth time weve been around this square and I still aint seen no parkin space. Why dont you just pull up in front of that fire hydrant — its only for a minute, anyhow.’
And now the musty smell of old leather—the thick, bound books containing what Father once called the sum total of mans ignorance: ceteris paribus and tempus fugit and caveat emptor too, and Oliver Wendell Holmes with Saint Francis himself, who never had a parking ticket and first thing lets kill all the lawyers and i father i have committed grand theft auto and he this looks more like a parking ticket to me and i but are they not the same and he you would take a perfectly common automotive error, an inevitable consequence of operating a motor vehicle and you would make it monstrous and i but it IS monstrous and he its only fifteen dollars, its not exactly the end of the world and i but i have still failed and he arbitrary lines delimiting segments of tarmac, the sum total of mans folly reduced to lines drawn ceteris paribus on some cosmic concrete chalkboard and i but did you ever get one and he of course and i how many times and he you want me to Count—NoCount would ever satisfy you and i but dont you believe in sin and he sin quentin was a term coined by those without courage to describe the actions of those who did indeed possess it and i but then our lives are just and he our lives are just so many tiny clumsy sandcastles before the godless oceans angry tide.
I took the ashtray from the table and I placed it on the floor. Then I realized that I had forgotten the gasoline and so I had to open the cabinet and take the can and remove the cap. The gasoline stung my nostrils as I poured it into the ashtray. I replaced the cap and I put the can back in its cabinet. I placed the parking ticket in the ashtray and I soaked it well with the gasoline. Then I remembered that I needed a match, but my hand had already found the matchbook in my pocket, and so I didnt have to open the cabinet any more.
Alas, the contest has been suspended since 2005, but some of the winners are archived here.
Dan Pearcy’s Twitter feed highlights a pretty link between Pascal’s triangle and the tangent angle formulae:
In 2010, as the Colombian government was preparing to rescue 16 soldiers held by armed FARC guerrillas, it looked in vain for a way to alert the soldiers without tipping off their captors. Finally Colonel Jose Espejo arranged to have local radio stations broadcast a pop song that contained a message in Morse code, which the soldiers had learned in basic training but that the guerrillas likely wouldn’t recognize.
The lyrics run, “In the middle of the night / Thinking about what I love the most / I feel the need to sing … About how much I miss them.” And hidden at three points in the song (1:30, 2:30, 3:30), in Morse code, is the message “19 people rescued. You are next. Don’t lose hope.”
“The hostages were listening to our own stations, so we made sure the song was played,” Espejo told The Verge. “The code message said, ‘you’re next’ because the hostages thought if they ran away, they would die in the jungle. We let them know that our troops were nearby.”
It worked. “We know of hostages who heard the message,” Espejo said, “and were able to escape and provide information that led to the release of more hostages.”
“Reynolds said that ‘[Samuel] Johnson always practised on every occasion the rule of speaking his best, whether the person to whom he addressed himself was or was not capable of comprehending him. ‘If,’ says he, ‘I am understood, my labour is not lost. If it is above their comprehension, there is some gratification, though it is the admiration of ignorance;’ and he said those were the most sincere admirers; and quoted Baxter, who made a rule never to preach a sermon without saying something which he knew was beyond the comprehension of his audience, in order to inspire their admiration.'” — James Boswell, Life of Samuel Johnson, 1791
But:
“To write well, express yourself like the common people, but think like a wise man.” – Aristotle
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