You and I have to travel from Startville to Endville, but we have only one bicycle between us. So we decide to leapfrog: We’ll leave Startville at the same time, you walking and I riding. I’ll ride for 1 mile, and then I’ll leave the bicycle at the side of the road and continue on foot. When you reach the bike you’ll ride it for 1 mile, passing me at some point, then leave the bike and continue walking. And so on — we’ll continue in this way until we’ve both reached the destination.
Will this save any time? You say yes: Each of us is riding for part of the distance, and riding is faster than walking, so using the bike must increase our average speed.
I say no: One or the other of us is always walking; ultimately every inch of the distance between Startville and Endville is traversed by someone on foot. So the total time is unchanged — leapfrogging with the bike is no better than walking the whole distance on foot.
You are. My argument would be sound if each of us simply stood by the bike after dismounting until the pedestrian caught up. But instead we’re investing that extra time in walking, which accounts for the faster progress.
Suppose the total trip is 2 miles, and each of us can walk at 4 mph and ride at 12 mph. I ride 1 mile in 5 minutes and leave the bike for you, walking the remaining 1 mile in 15 minutes. You take 15 minutes to reach the bike and then ride to Endville in 5 minutes. We both arrive at the destination in 20 minutes, where it would have taken 30 minutes if we’d walked the whole way.
Personal ads from the New York Herald in the 1860s:
IF THE LADY WHO, FROM AN OMNIBUS, SMILED on a gentleman with a bunch of bananas in his hand, as he crossed Wall street, corner of Broadway, will address X., box 6,735 Post office, she will confer a favor. (March 21, 1866)
ON WEDNESDAY AFTERNOON A LADY WITH black silk quilted hat walked nearly side by side with a gentleman in a drab overcoat from Tenth to Fourteenth street, in Broadway. Both were annoyed by the wind and dust. Her smile has haunted him ever since. Will she send her address to Carl, Union square Post office? (March 8, 1861)
BOOTH’S THEATRE, THURSDAY EVENING, 11TH. Will the lady who met the gent’s gaze through an opera glass and smiled please address, in confidence, Harry Wilton, Herald office? (March 13, 1869)
A YEAR AGO LAST SEPTEMBER OR OCTOBER TWO ladies with a child were travelling on the Hudson River cars, one of whom offered a seat to a middle aged gentleman, with light whiskers or goatee, slightly gray, who kindly pointed out to her the red leaved trees, and said he had a number of them on his place, and made himself otherwise agreeable; and when she was leaving him (ten miles this side of where he stopped) gave her a parting embrace, which she has never been able to forget. If the gentleman has any recollection of the circumstance he will greatly oblige by addressing a note to Lena Bigelow, Madison square Post office, giving some description of the lady, also name of the paper he gave her. (Jan. 25, 1862)
Here’s a related curiosity. If a circle of diameter L is placed at random on a pattern of circles of unit diameter, which are arranged hexagonally with centers C apart, then the probability that the placed circle will fall entirely inside one of the fixed circles is
If we put k = C/(1 – L), we get
And a frequency estimate of P will give us an estimate of π.
Remarkably, in 1933 A.L. Clarke actually tried this. In Scripta Mathematica, N.T. Gridgeman writes:
His circle was a ball-bearing, and his scissel a steel plate. Contacts between the falling ball and the plate were electrically transformed into earphone clicks, which virtually eliminated doubtful hits. With student help, a thousand man-hours went into the accumulation of N = 250,000. The k was about 8/5, and the final ‘estimate’ of π was 3.143, to which was appended a physical error of ±0.005.
“This is more or less the zenith of accuracy and precision,” Gridgeman writes. “It could not be bettered by any reasonable increase in N — even if the physical error could be reduced, hundreds of millions of falls would be needed to establish a third decimal place with confidence.”
(N.T. Gridgeman, “Geometric Probability and the Number π,” Scripta Mathematica 25:3 [November 1960], 183-195.)
When my brother and I built and flew the first man-carrying flying machine, we thought that we were introducing into the world an invention which would make further wars practically impossible. That we were not alone in this thought is evidenced by the fact that the French Peace Society presented us with medals on account of our invention. We thought governments would realize the impossibility of winning by surprise attacks, and that no country would enter into war with another of equal size when it knew that it would have to win by simply wearing out its enemy.
In 1855 Pedro Carolino decided to write a Portuguese-English phrasebook despite the fact that he didn’t actually speak English. The result is one of the all-time masterpieces of unintentional comedy, a language guide full of phrases like “The ears are too length” and “He has spit in my coat.” In this episode of the Futility Closet podcast we’ll sample Carolino’s phrasebook, which Mark Twain called “supreme and unapproachable.”
We’ll also hear Hamlet’s “to be or not to be” rendered in jargon and puzzle over why a man places an ad before robbing a bank.
Sources for our feature on Pedro Carolino’s disastrous phrasebook:
(This edition, like many, incorrectly names José da Fonseca as a coauthor. Fonseca was the author of the Portuguese-French phrasebook that Carolino used for the first half of his task. By all accounts that book is perfectly competent, and Fonseca knew nothing of Carolino’s project; Carolino added Fonseca’s name to the byline to lend some credibility to his own book.)
As long as we’re at it, here’s Monty Python’s “Dirty Hungarian Phrasebook” sketch:
Hamlet’s “to be or not to be” soliloquy rendered in jargon, from Arthur Quiller-Couch’s On the Art of Writing (1916):
To be, or the contrary? Whether the former or the latter be preferable would seem to admit of some difference of opinion; the answer in the present case being of an affirmative or of a negative character according as to whether one elects on the one hand to mentally suffer the disfavour of fortune, albeit in an extreme degree, or on the other to boldly envisage adverse conditions in the prospect of eventually bringing them to a conclusion. The condition of sleep is similar to, if not indistinguishable from, that of death; and with the addition of finality the former might be considered identical with the latter: so that in this connection it might be argued with regard to sleep that, could the addition be effected, a termination would be put to the endurance of a multiplicity of inconveniences, not to mention a number of downright evils incidental to our fallen humanity, and thus a consummation achieved of a most gratifying nature.
This week’s lateral thinking puzzle was contributed by listener Lawrence Miller, who sent this corroborating link (warning — this spoils the puzzle).
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Many thanks to Doug Ross for the music in this episode.
A puzzle from J.A.H. Hunter’s Fun With Figures (1956):
A man paddling a canoe upstream sees a glove in the water as he passes under a bridge. Fifteen minutes later, he turns around and paddles downstream. He passes under the bridge and travels another mile before reaching the rock from which he started, which the glove is just passing. If he paddled at the same speed the whole time and lost no time in turning around, what is the speed of the current?
Say the canoeist paddled at x miles per hour through the water and the current moved at y miles per hour relative to the land. Fifteen minutes of paddling took the canoe x/4 miles from the glove, and the glove took 1/y hours to travel the 1 mile between the bridge and the rock. So after the canoeist turned around, the glove took (1/y – 1/4) or (4 – y)/4y hours to reach the rock. During that time the man moved x(4 – y)/4y miles through the water. We know that he was x/4 miles from the glove at the start of that interval, so x/4 = x(4 – y)/4y and hence 1 = (4 – y)/y and y = 2. The speed of the current is 2 miles per hour.
A simpler solution, sent in by reader Peter McLeod: “You need to switch reference frames. Take the reference frame of the river, i.e. that which is moving downstream at the speed of the current. In that frame, the canoeist passes the glove, spends fifteen minutes paddling away from it, then turns and paddles back towards it. How long does he take to reach the glove again? It must be fifteen minutes, since we know his paddling speed is constant. Hence, the canoeist passes the glove half an hour after he left it. We know that in that time, the glove travelled a mile, so the current is going at 2 mph!”
In 1940, just before the release of her film They Knew What They Wanted, Carole Lombard’s press agent, Russell Birdwell, approached the filmmakers with a novel publicity scheme. Lombard would be scheduled to fly to New York for the opening, but they would arrange for the plane to “go down” en route and remain missing for 12 hours.
“And in those twelve hours, fellas, we’re going to be on every goddam front page in the United States of America,” Birdwell said. “Not only Carole Lombard’s name, but the name of the picture and the name of the theatre it’s going to open at and how would you like to foot the bill for that kind of advertising?” He planned that eventually Lombard and the pilot could wander out of the woods saying that the plane’s engines and radio had died.
In his 1967 memoir Hollywood, director Garson Kanin remembers that in the meeting Lombard began to slap her thigh, yelling, “I’ll die! I’ll die! Isn’t that something? I’ll die!”
Birdwell’s plan was considered seriously but finally canceled due to the cost. Two years later, Lombard died in a plane crash in Nevada. “I could hear Carole’s voice and the sound of her hand slapping her thigh, her voice yelling delightedly, ‘I’ll die! I’ll die!'” Kanin wrote. “I remembered Russell Birdwell’s notion of the fake crash for publicity. I stood there hoping against hope that perhaps this was a postponed version of his scheme. … Carole Lombard … could not be dead at thirty-five. But she was.”
One of democracy’s ideals is egalitarianism: Each person gets one vote, and all votes are equally consequential, so that all people have equal power over the world. For that reason we consider it improper for one person to vote twice in the same election. But then shouldn’t we also consider it improper for dual citizens to cast votes in two different places?
It’s true that dual citizens vote in different elections, but they’re still exercising twice as much power over the world as other voters. And it’s true that power is already unequally distributed among the world’s voters, but this is no reason to shrink from the ideal.
The fact that a dual citizen has the legal right to cast two ballots doesn’t mean that this accords with democratic principles. Suppose that Texas passed a law saying that any native-born Texan can vote in Texas, regardless of where he currently lives. Then a Texan living in Pennsylvania could cast ballots in both states, whereas a native Pennsylvanian could vote only once. This might be legal, but we would object morally to the unfairness of such a law.
Given the unequal influence of the world’s nations, one idealistic way to equalize power among all voters would be to give everyone a right to vote in every election, everywhere. This would give each of us an equal amount of power over the world. “That vision remains pretty visionary, we concede,” write Robert E. Goodin and Ana Tanasoca. “Still, visions matter.”
(Robert E. Goodin and Ana Tanasoca, “Double Voting,” Australasian Journal of Philosophy, vol. 92, no. 4, 743-758.)
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