A memorably phrased puzzle from The Graham Dial: “Consider a vertical girl whose waist is circular, not smooth, and temporarily at rest. Around the waist rotates a hula hoop of twice its diameter. Show that after one revolution of the hoop, the point originally in contact with the girl has traveled a distance equal to the perimeter of a square circumscribing the girl’s waist.”
Frightened villagers “killed” the first hydrogen balloon, launched in Paris by Jacques Charles and the Robert brothers Anne-Jean and Nicolas-Louis on Aug. 27, 1783. Allen Andrews, in Back to the Drawing Board: The Evolution of Flying Machines, quotes a contemporary account:
It is presumed that it was carried to a height of more than 20,000 feet, when it burst by the reaction of the Inflammable Gas upon the Atmospheric Air. It fell at three quarters past five near Gonesse, ten miles [actually, 15 miles] from the Field of Mars. The affrightened inhabitants ran together, appalled by the Hellish stench of sulphur, and two monks having assured them it was the skin of a Monstrous Animal, they attacked it with stones, pitchforks and flails. The Curate of the village was obliged to attend in order to sprinkle it with holy water and remove the fears of his astonished parishioners. At last they tied to the tail of a horse the first Instrument that was ever made for an Experiment in Natural Philosophy, and trained it across the field more than 6000 feet.
Perhaps forewarned, the first man to undertake a balloon flight in North America carried a pass from George Washington.
In 1943 three men came up with an ingenious plan to escape from the seemingly escape-proof Stalag Luft III prison camp in Germany. In this episode of the Futility Closet podcast we’ll learn about their clever deception, which made them briefly famous around the world.
We’ll also hear about the chaotic annual tradition of Moving Day in several North American cities and puzzle over how a severely injured hiker beats his wife back to their RV.
University of Strathclyde mathematician Adam McBride recalls that in his student days a particular teacher used to present a weekly puzzle. One of these baffled him:
Find positive integers a, b, and c, all different, such that a3 + b3 = c4.
“The previous puzzles had been relatively easy but this one had me stumped,” he wrote later. He created three columns headed a3, b3, and c4 and spent hours looking for a sum that would work. On the night before the deadline, he found one: 703 + 1053 = 354.
“This shows how sad a person I was! However, I then realised also how stupid I had been. I had totally missed the necessary insight.” What was it?
In 1948, as T.S. Eliot was departing for Stockholm to accept the Nobel Prize, a reporter asked which of his books had occasioned the honor.
Eliot said, “I believe it’s given for the entire corpus.”
The reporter said, “And when did you publish that?”
Eliot later said, “It really might make a good title for a mystery — The Entire Corpus.”
1. A puzzle from J.A.H. Hunter’s Fun With Figures, 1956:
Tom and Tim are brothers; their combined ages make up seventeen years. When Tom was as old as Tim was when Tim was twice as old as Tom was when Tom was fifteen years younger than Tim will be when Tim is twice his present age, Tom was two years younger than Tim was when Tim was three years older than Tom was when Tom was a third as old as Tim was when Tim was a year older than Tom was seven years ago. So how old is Tim?
2. Another, by Sam Loyd:
“How fast those children grow!” remarked Grandpa. “Tommy is now twice as old as Maggie was when Tommy was six years older than Maggie is now, and when Maggie is six years older than Tommy is now their combined ages will equal their mother’s age then, although she is now but forty-six.” How old is Maggie?
3. According to Wirt Howe’s New York at the Turn of the Century, 1899-1916, this question inspired an ongoing national debate when it appeared in the New York Press in 1903:
Brooklyn, October 12
Dear Tip:
Mary is 24 years old. She is twice as old as Anne was when she was as old as Anne is now. How old is Anne now? A says the answer is 16; B says 12. Which is correct?
John Mahon
Cornell mathematician Robert Connelly devised this intuition-defying card game. I shuffle a standard deck of 52 cards and deal them out in a row before you, one at a time. At some point before the last card is dealt, you must say the word “red.” If the next card I deal is red, you win $1; if it’s black you lose $1. If you play blind, your chance of winning is 1/2. Can you improve on this by devising a strategy that considers the dealt cards?
Surprisingly, the answer is no. Imagine a deck with two red cards and two black. Now there are six equally likely deals:
RRBB
RBBR
BBRR
RBRB
BRBR
BRRB
By counting, we can see that the chance of success remains 1/2 regardless of whether you call red before the first, second, third, or fourth card.
Trying to outsmart the cards doesn’t help. You might resolve to wait and see the first card: If it’s black you’ll call red immediately, and if it’s red you’ll wait until the fourth card. It’s true that this strategy gives you a 2/3 chance of winning if the first card is black — but if it’s red then it has a 2/3 chance of losing.
Similarly, it would seem that if the first two cards are black then you have a sure thing — the next card must be red. This is true, but it will happen only once in six deals; on the other five deals, calling red at the third card wins only 2/5 of the time — so this strategy has an overall success rate of (1/6 × 1) + (5/6 × 2/5) = 1/2, just like the others. The cards conspire to erase every seeming advantage.
The same principle holds for a 52-card deck, or indeed for any deck. In general, if a deck has r red cards and b black ones, then your chance of winning, by any strategy whatsoever, is r/(b + r). Seeing the cards that have already been dealt, surprisingly, is no advantage.
(Robert Connelly, “Say Red,” Pallbearers Review 9 [1974], 702.)
“It is only through fiction that facts can be made instructive or even intelligible.” — George Bernard Shaw
“People think that because a novel’s invented, it isn’t true. Exactly the reverse is the case. Biography and memoirs can never be wholly true, since they cannot include every conceivable circumstance of what happened. The novel can do that.” — Anthony Powell
“I write fiction and I’m told it’s autobiography, I write autobiography and I’m told it’s fiction, so since I’m so dim and they’re so smart, let them decide what it is or it isn’t.” — Philip Roth
In the 1870s baseball catchers played bare-faced, routinely suffering broken noses and teeth; to protect themselves they stood two dozen feet behind the batter, which prevented the pitcher from throwing his best pitches. Finally Fred Winthrop Thayer, captain of Harvard’s team, invented a “Safety-Mask for Base-Ball Players” to minimize the damage.
“It is not an unfrequent occurrence in the game of base-ball for a player to be severely injured in the face by a ball thrown against it,” he wrote in the patent application. “With my face-guard such an accident cannot happen.”
When catcher Jim Tyng first wore Thayer’s mask on April 12, 1877, it was roundly derided. Spectators yelled “Mad dog!” and “Muzzle ’em!”, and opposing players greeted Tyng with “good natured though somewhat derisive pity.” The Portland, Maine, Sunday Telegram wrote, “There is a great deal of beastly humbug in contrivances to protect men from things which do not happen. There is about as much sense in putting a lightning rod on a catcher as there is a mask.”
Catchers finally submitted when sportwriter Henry Chadwick faulted their “moral courage.” “Plucky enough to face the dangerous fire of balls from the swift pitcher,” he wrote, “they tremble before the remarks of the small boys of the crowd of spectators, and prefer to run the risk of broken cheek bones, dislocated jaws, a smashed nose or blackened eyes, than stand the chaff of the fools in the assemblage.”
Today Thayer’s Harvard mask is in the National Baseball Hall of Fame.
The sestina is an unusual form of poetry: Each of its six stanzas uses the same six line-ending words, rotated according to a set pattern:
This intriguingly insistent form has appealed to verse writers since the 12th century. “In a good sestina the poet has six words, six images, six ideas so urgently in his mind that he cannot get away from them,” wrote John Frederick Nims. “He wants to test them in all possible combinations and come to a conclusion about their relationship.”
But the pattern of permutation also intrigues mathematicians. “It is a mathematical property of any permutation of 1, 2, 3, 4, 5, 6 that when it is repeatedly combined with itself, all of the numbers will return to their original positions after six or fewer iterations,” writes Robert Tubbs in Mathematics in Twentieth-Century Literature and Art. “The question is, are there other permutations of 1, 2, 3, 4, 5, 6 that have the property that after six iterations, and not before, all of the numbers will be back in their original positions? The answer is that there are many — there are 120 such permutations. We will probably never know the aesthetic reason poets settled on the above permutation to structure the classical sestina.”
In 1986 the members of the French experimental writers’ workshop Oulipo began to apply group theory to plumb the possibilities of the form, and in 2007 Pacific University mathematician Caleb Emmons offered the ultimate hat trick: A mathematical proof about sestinas written as a sestina:
Bonus: When not doing math and poetry, Emmons runs the Journal of Universal Rejection, which promises to reject every paper it receives: “Reprobatio certa, hora incerta.”
(Caleb Emmons, “S|{e,s,t,i,n,a}|“, The Mathematical Intelligencer, December 2007.) (Thanks, Robert and Kat.)
