Two Puzzles

This just caught my eye: In the centenary issue of the Mathematical Gazette, in 1996, Sir Bryan Thwaites offered monetary prizes for the proofs of two conjectures.

The first is what’s known as the Collatz conjecture, which Sir Bryan had been puzzling over since 1951. He had come to believe it was unprovable, so he offered a prize of £1,000.

The second conjecture was “even simpler as far as numerical skill is concerned”:

Take any set of N rational numbers. Form another set by taking the positive differences of successive members of the first set, the last such difference being formed from the last and first members of the original set. Iterate. Then in due course the set so formed will consist entirely of zeros if and only if N is a power of two.

Sir Bryan felt that this conjecture was likely provable, so he offered £100 for its proof or disproof.

“Both are easily understood and handled by the average ten-year-old,” he wrote, “and so there will be some who will attribute my continuing interest in such apparently elementary mathematics to well-advanced senility.” But that was 20 years ago, and I believe both conjectures remain unproven — the first is a famously unsolved problem, and I can’t find any record that the second has ever been worked out (though evidently the case of N = 4 had been known since the 1930s).

(Bryan Thwaites, “Two Conjectures or How to Win £1100,” Mathematical Gazette 80:487 [March 1996], 35-36.)

12/03/2018 UPDATE: It has been proven. (Thanks, Yotam.)

| Science & Math

History

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“It is conceivable that Alexander the Great — for all the military successes of his youth, for all the excellence of the army he trained, for all the desire he felt in himself to change the world — might have stopped at the Hellespont, and never crossed it, and not out of fear, not out of indecisiveness, not out of weakness of will, but from heavy legs.” — Kafka

| History · Literature · Quotations

Footwork

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Each day after work, Smith takes the train to the suburb where he lives, and his chauffeur meets him at the station and drives him home. One day Smith finishes work early and arrives at the suburb one hour earlier than usual. He starts walking home, and the chauffeur meets him on the road and drives him the rest of the way. This gets Smith home 10 minutes earlier than usual. How long did he walk? (Disregard the time spent in stopping and picking up Smith, and assume that the chauffeur normally arrives at the station just as the train does.)

Click for Answer
| Puzzles

Solace

A thinly clad man, who was trudging afoot through a wintry and shelterless region, met another wrapped in a big black cloak. The cloak hung heavily on its wearer, and seemed to drag him back, but at least it kept off the cold.

‘That’s a fine warm cloak you’ve got,’ said the first man through his chattering teeth.

‘Oh,’ said the other, ‘it’s none of my choosing, I promise you. It’s only my old happiness dyed black and made over into a sorrow; but in this weather a man must wear what he’s got.’

‘To think of some people’s luck!’ muttered the first man, as the other passed on. ‘Now I never had enough happiness to make a sorrow out of.’

— Edith Wharton, The Valley of Childish Things, and Other Emblems, 1896

| Literature

Rinehart

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On June 11, 1900, someone in Harvard Yard called out “Oh, R-i-i-i-n-e-HART!” There must have been something in the air, because hundreds of students repeated the cry, and for the next 40 years it took on a strange life of its own. Journalist (and alumnus) George Frazier mentioned it in his 1932 song “Harvard Blues,” recorded in 1941 by Count Basie. John Barrymore mentioned it in his 1939 film The Great Man Votes. Thomas Pynchon describes it in his novel Against the Day. Today it’s documented in slang dictionaries and has entered the realm of legend: A Harvard man menaced by Arabs in Africa supposedly cried “Rinehart!” and was rescued by a fellow alumnus from the nearby French Foreign Legion.

The truth is more prosaic. Rinehart is John Bryce Gordon Rinehart, class of 1900. A contemporary article in the Harvard Crimson explained:

Rinehart, who is an earnest student, has been in great demand as a tutor to other men in his courses. As he lives at the top of Grays hall his friends have sought to find out whether he was in or not by directing plaintive cries of ‘Rinehart, O Rinehart’ at his windows. This made the studiously inclined who swell in the neighboring dormitories very tired and they determined to quell Rinehart, so promptly at dark for the past three nights the college yard has resounded with the cries of ‘Rinehart, O Rinehart.’ First one end of the yard and then other would send up the plaintive cry, and then all the buildings would swell as if in chorus with the same old plaint. Last night the college police tried to stop the racket, but the boys by a little teamwork kept them running from one dormitory to the other. One man with a megaphone was particularly offensive, but despite the police vigil of three hours the megaphonist was still summoning Rinehart in tearful tones.

Rinehart himself, “stocky, gray, and genial” at 61, finally confirmed this at the university’s tercentenary celebration in 1936, his first time back to campus:

It was in the Spring of 1900. Examinations were over and the atmosphere was tense, as it usually is between examinations and commencement.

My classmates always looked upon me as a grind. They were continually calling for me to go out on a spree, but I have never touched a drop in my life.

That Spring evening, in 1900, they came and called up to my room — I was living in Gray’s 49, on the top floor — for me to join them. The late Frank Simonds, living in Matthews, who was a friend of mine, heard the call and just for a joke stuck his head out of his window and repeated the call.

The cry was taken up. Among those who joined in were John Price Jones and Charles Underwood, who is director of the Manly School here in Cambridge. Within a few minutes the yard was a bedlam.

Why it caught on, though, still seems to be a mystery.

| Oddities

Podcast Episode 226: The Great Match Race

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America’s first national sports spectacle took place in 1823, when the North and South sent their best horses for a single dramatic race that came to symbolize the regional tensions of a changing nation. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Great Match Race, which laid the foundations of modern American thoroughbred racing.

We’ll also ponder a parasite’s contribution to culture and puzzle over a misinformed criminal.

See full show notes …

| Entertainment · History · Podcast

Also-Rans

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Bylines appearing in L&N Employees’ Magazine, a house organ of the Louisiana and Nashville Railroad, in the 1940s:

  • R.R. South
  • Steele Raylor
  • Dick C. Lyon
  • Lou Nash
  • L.M. Lynes
  • C. Ross Tye
  • Lincoln Penn
  • Cole Carr
  • M.T. Hopper
  • Rowan House
  • Rowland Stock
  • C.A. Boose

Thinking these fishy, writer Robert Rennick inquired of the railroad’s public relations department and learned that editor Julian James had barred any writer from receiving two bylines in a single issue. So they’d adopted these pseudonyms.

“The assumption was that no reader would ever imagine that these were real names. Yet, W.R. Heffren, writing as C. Ross Tye, once received a letter from a lady genealogist stating that she was researching the Tye family and would he kindly send her his family line to see if it could be related to hers.”

(Robert Rennick, “Fictitious Names,” Word Ways 37:1 [February 2004], 3.)

| Language

Room of the Giants

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Image: Wikimedia Commons

Painter Giulio Romano decorated the Palazzo del Te outside Mantua with a series of illusionistic spaces and special effects, culminating in a bewildering room in which giants that have rebelled against Zeus are crushed for their transgression — Giulio “paints the walls away,” leaving the viewer in a crumbling city into which Zeus flings lightning from the heavens. Poet Gregorio Comanini praised Giulio’s fantastic imagination:

In Mantua, in a room in the Palazzo del Te, Giulio Romano has painted giants struck by lightning at Flegra. They are crushed under the rubble of rock and mountain, in positions so strange and horrible that anyone who saw such a spectacle in reality would surely be horrified and feel great distress. None the less, since this is an imitation and a painting, anyone would welcome a chance to see it and would be highly pleased with it, as can be attested to by the frequency with which visitors flock to view it.

Giorgio Vasari wrote, “Let no one think ever to see any work of the brush more horrifying, or more realistic, than this.”

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| Art