Simple

The 1968 Putnam Competition included a beautiful one-line proof that π is less than 22/7, its common Diophantine approximation:

$\displaystyle 0 \enspace \textless \int_{0}^{1}\frac{x^{4}\left ( 1 - x \right )^{4}}{1 + x^{2}} \: dx = \frac{22}{7} - \pi .$

The integral must be positive, because the integrand’s denominator is positive and its numerator is the product of two non-negative numbers. But it evaluates to 22/7 – π — and if that expression is positive, then 22/7 must be greater than π.

University of St Andrews mathematician G.M. Phillips wrote, “Who will say that mathematics is devoid of humour?”

October 2, 2023October 1, 2023 | Science & Math

Views

When [a man] puts a thing on a pedestal and calls it beautiful, he demands the same delight from others. He judges not merely for himself, but for all men, and then speaks of beauty as if it were a property of things. Thus he says that the thing is beautiful; and it is not as if he counts on others agreeing with him in his judgment of liking owing to his having found them in such agreement on a number of occasions, but he demands this agreement of them. He blames them if they judge differently, and denies them taste, which he still requires of them as something they ought to have; and to this extent it is not open to men to say: Every one has his own taste.

— Immanuel Kant, Critique of Judgment, 1790

October 1, 2023September 30, 2023 | Art

Ground Truth

In October 2005, Neil Armstrong received a letter from a social studies teacher charging that the moon landings had been faked. “[O]ver 30 years on from the pathetic TV broadcast when you fooled everyone by claiming to have walked upon the Moon,” he wrote, “I would like to point out that you, and the other astronauts, are making yourselfs a worldwide laughing stock … Perhaps you are totally unaware of all the evidence circulating the globe via the Internet. Everyone now knows the whole saga was faked, and the evidence is there for all to see.”

Armstrong replied:

Mr. Whitman,

Your letter expressing doubts based on the skeptics and conspiracy theorists mystifies me.

They would have you believe that the United States Government perpetrated a gigantic fraud on its citizenry. That the 400,000 Americans who worked on an unclassified program are all complicit in the deception, and none broke ranks and admitted their deceit.

If you believe that, why would you contact me, clearly one of those 400,000 liars?

I trust that you, as a teacher, are an educated person. You will know how to contact knowledgeable people who could not have been party to the scam.

The skeptics claim that the Apollo flights did not go to the moon. You could contact the experts from other countries who tracked the flights on radar (Jodrell Bank in England or even the Russian Academicians).

You should contact the Astronomers at Lick Observatory who bounced their laser beam off the Lunar Ranging Reflector minutes after I installed it. Or, if you don’t find them persuasive, you could contact the astronomers at the Pic du Midi observatory in France. They can tell you about all the other astronomers in other countries who are still making measurements from these same mirrors — and you can contact them.

Or you could get on the net and find the researchers in university laboratories around the world who are studying the lunar samples returned on Apollo, some of which have never been found on earth.

But you shouldn’t be asking me, because I am clearly suspect and not believable.

Neil Armstrong

(From James R. Hansen, A Reluctant Icon: Letters to Neil Armstrong, 2020.)

September 30, 2023September 29, 2023 | Science & Math · Society

Comment

On Aug. 20, 1961, Harvard physicist Percy Williams Bridgman was found dead at his home in the White Mountains of New Hampshire. After suffering for months with metastatic cancer, he had shot himself in the head. He left a two-sentence note:

“It isn’t decent for society to make a man do this thing himself. Probably this is the last day I will be able to do it myself. P.W.B.”

September 29, 2023September 28, 2023 | Death · Society

The Epsom Salts Monorail

A late odd railway: In 1922 a Los Angeles florist built a 28-mile monorail to carry hydrated magnesium sulphate from the Owlshead Mountains to a siding of the Trona Railway in San Bernardino County, California. Steel-framed locomotives crawled along a steel rail pulling carriages bearing low-slung loads like saddlebags. Its downhill speed touched 56 mph, briefly earning it the epithet “fastest monorail of the world,” but the beams warped as they dried, landslides further damaged the track, and the railway shut down in 1926, just two years after opening.

“In the late 1930s the rails were salvaged and sold for scrap, and the longitudinal timbers followed suit,” writes John Day in More Unusual Railways (1960). “In 1958 a long line of ‘A’ frames still marched across the wastes to show where the line once had run.”

September 29, 2023September 28, 2023 | Technology

Caliban’s Will

A curious logic problem by Cambridge mathematician Max Newman, published in Hubert Phillips’ New Statesman puzzle column in 1933:

When Caliban’s will was opened it was found to contain the following clause:

‘I leave ten of my books to each of Low, Y.Y., and ‘Critic,’ who are to choose in a certain order:

No person who has seen me in a green tie is to choose before Low.

If Y.Y. was not in Oxford in 1920 the first chooser never lent me an umbrella.

If Y.Y. or ‘Critic’ has second choice, ‘Critic’ comes before the one who first fell in love.’

Unfortunately, Low, Y.Y., and ‘Critic’ could not remember any of the relevant facts; but the family solicitor pointed out that, assuming the problem to be properly constructed (i.e., assuming it to contain no statement superfluous to its solution) the relevant data and order could be inferred. What was the prescribed order of choosing?

	Select Click for Answer
The key is that the problem contains “no statement superfluous to its solution.” The correct solution must make use of every statement given; any solution that doesn’t do this is wrong. So, for example, Statement 1 suggests that either Y.Y. or ‘Critic’ has seen Caliban in a green tie. If this weren’t the case, then the statement would be useless and any resulting solution would be wrong. This also tells us that Low cannot choose third, because if he did then Statement 1 again would be needless. From Statement 2 we can infer that Y.Y. was not in Oxford in 1920 — if he had been, then this statement would be useless. Similarly, if no one has lent Caliban an umbrella then the second statement is not needed, so we know that someone has lent Caliban an umbrella. Very well, who lent it to him? If Low lent it, then Statement 2 informs us that Low will not choose first. Statement 1 has already told us that Low will not choose last, so if Low lent Caliban the umbrella then Low will choose second. But that can’t be — Statement 3 can be of value only if Y.Y. or ‘Critic’ chooses second. Therefore Low didn’t lend Caliban an umbrella. Is it possible that both ‘Critic’ and Y.Y. have lent Caliban an umbrella? No: In that case Statement 2 tells us that Low must choose first, and Statement 3 tells us that ‘Critic’ chooses second and Low third, so Statement 1 is not needed. So that possibility can’t be right; Caliban has received an umbrella from either ‘Critic’ or Y.Y., but not both. Similarly, if both Y.Y. and ‘Critic’ have seen Caliban in a green tie, then Statement 1 tells us that Low will choose first, but that would make Statement 2 useless. So either ‘Critic’ or Y.Y. has seen Caliban in a green tie, but not both. Okay, suppose that Y.Y. both saw Caliban in a green tie and lent him an umbrella. In that case Statement 1 would tell us that Y.Y. cannot choose first, and if that’s the case then Statement 2 is not needed. Hence if Y.Y. saw Caliban in a green tie then he can’t have lent him an umbrella, which would mean that ‘Critic’ lent Caliban an umbrella. By the same logic, if ‘Critic’ saw Caliban in a green tie, Y.Y. must have lent Caliban an umbrella. In both of those latter cases, Low must choose first (Statement 1 tells us that Low doesn’t choose last, and Statement 3 is useful only if Y.Y. or ‘Critic’ chooses second). And if that’s the case, then Statement 3 tells us that ‘Critic’ must come second and Y.Y. third. So the proper order is Low, ‘Critic’, Y.Y. (The fanciful names refer to colleagues of Phillips at New Statesman.) (Via Alex Bellos, Can You Solve My Problems?, 2017.)

Click for Answer

The key is that the problem contains “no statement superfluous to its solution.” The correct solution must make use of every statement given; any solution that doesn’t do this is wrong.

So, for example, Statement 1 suggests that either Y.Y. or ‘Critic’ has seen Caliban in a green tie. If this weren’t the case, then the statement would be useless and any resulting solution would be wrong. This also tells us that Low cannot choose third, because if he did then Statement 1 again would be needless.

From Statement 2 we can infer that Y.Y. was not in Oxford in 1920 — if he had been, then this statement would be useless. Similarly, if no one has lent Caliban an umbrella then the second statement is not needed, so we know that someone has lent Caliban an umbrella.

Very well, who lent it to him? If Low lent it, then Statement 2 informs us that Low will not choose first. Statement 1 has already told us that Low will not choose last, so if Low lent Caliban the umbrella then Low will choose second. But that can’t be — Statement 3 can be of value only if Y.Y. or ‘Critic’ chooses second. Therefore Low didn’t lend Caliban an umbrella.

Is it possible that both ‘Critic’ and Y.Y. have lent Caliban an umbrella? No: In that case Statement 2 tells us that Low must choose first, and Statement 3 tells us that ‘Critic’ chooses second and Low third, so Statement 1 is not needed. So that possibility can’t be right; Caliban has received an umbrella from either ‘Critic’ or Y.Y., but not both. Similarly, if both Y.Y. and ‘Critic’ have seen Caliban in a green tie, then Statement 1 tells us that Low will choose first, but that would make Statement 2 useless. So either ‘Critic’ or Y.Y. has seen Caliban in a green tie, but not both.

Okay, suppose that Y.Y. both saw Caliban in a green tie and lent him an umbrella. In that case Statement 1 would tell us that Y.Y. cannot choose first, and if that’s the case then Statement 2 is not needed. Hence if Y.Y. saw Caliban in a green tie then he can’t have lent him an umbrella, which would mean that ‘Critic’ lent Caliban an umbrella. By the same logic, if ‘Critic’ saw Caliban in a green tie, Y.Y. must have lent Caliban an umbrella.

In both of those latter cases, Low must choose first (Statement 1 tells us that Low doesn’t choose last, and Statement 3 is useful only if Y.Y. or ‘Critic’ chooses second). And if that’s the case, then Statement 3 tells us that ‘Critic’ must come second and Y.Y. third. So the proper order is Low, ‘Critic’, Y.Y.

(The fanciful names refer to colleagues of Phillips at New Statesman.)

(Via Alex Bellos, Can You Solve My Problems?, 2017.)

September 28, 2023September 27, 2023 | Puzzles

Dancing the Slaves

After the morning meal came a joyless ceremony called ‘dancing the slaves.’ ‘Those men who were in irons,’ says Dr. Thomas Trotter, surgeon of the Brookes in 1783, ‘were ordered to stand up and make what motions they could, leaving a passage for such as were out of irons to dance around the deck.’ Dancing was prescribed as a therapeutic measure, a specific against suicidal melancholy, and also against scurvy — although in the latter case it was a useless torture for men with swollen limbs. While sailors paraded the deck, each with a cat-o’-nine-tails in his right hand, the men slaves ‘jumped in their irons’ until their ankles were bleeding flesh. One sailor told Parliament, ‘I was employed to dance the men, while another person danced the women.’ Music was provided by a slave thumping on a broken drum or an upturned kettle, or by an African banjo, if there was one aboard, or perhaps by a sailor with a bagpipe or a fiddle. Slaving captains sometimes advertised for ‘a person that can play on the Bagpipes, for a Guinea ship.’ The slaves were also told to sing. Said Dr. Claxton after his voyage in the Young Hero, ‘They sing, but not for their amusement. The captain ordered them to sing, and they sang songs of sorrow. Their sickness, fear of being beaten, their hunger, and the memory of their country, &c., are the usual subjects.’

— Daniel P. Mannix and Malcolm Cowley, Black Cargoes: A History of the Atlantic Slave Trade, 1518-1865, 1962

September 27, 2023September 26, 2023 | History

Unquote

“Everything at a distance turns into poetry: distant mountains, distant people, distant events: all become Romantic.” — Novalis

September 27, 2023September 26, 2023 | Quotations

Planned Forgiveness

My neighbor has been stealing my newspaper. And when I confront him, he apologizes with a sarcastic, condescending air, as if to say that he’s surprised I can read at all. I find it impossible to forgive him, but then I learn that he’s about to lose his job. He’s an aging executive with a large family to support, and I’m sure that this misfortune will soften his scorn and make him more sincerely apologetic. I decide to forgive him when all this happens.

This seems odd — if I’m sure that he’ll lose his job and express real contrition for stealing the paper, why do I have wait for this to happen? Why can’t I forgive him now?

Another twist: I learn that I (and only I) can save his job. This would amount to doing him a large favor, so I feel justified in withholding my help until I’ve forgiven him. But is this fair? Can I refuse to help him until I get a sincere apology, knowing that this will happen only after he loses his job?

Xanthippe is angry that Socrates is late, but she knows that he’ll apologize when she starts making dinner. Knowing this, can’t she skip the dinner and just forgive him? “In other words,” asks Tennessee State University philosopher James Montmarquet, “knowing that he would apologize, may she not still forgive him — having elected, for quite good reasons, not to allow conditions apt for his apology even to take place?”

(James Montmarquet, “Planned Forgiveness,” American Philosophical Quarterly 44:3 [July 2007], 285-296.)

September 26, 2023 | Oddities

“An Elevation Puzzle”

This does my head in — it’s a puzzle from the October 1958 issue of Eureka, the journal of the Cambridge University Mathematical Society:

“Below are shown the front elevation and plan of a mathematical figure. What is the side elevation?”

https://archive.org/details/eureka-21/page/7/mode/2up?view=theater — Image: Eureka

The terms (I believe) refer to multiview orthographic projection, the illustration technique used in architectural drawings: The front elevation is the view looking squarely at the “front” of the object, and the plan view looks down from above. What is the side view?

	Select Click for Answer
Here’s the answer given in the journal: Image: Eureka No explanation is given. I had imagined something like a curved disk, but that wouldn’t give a circular plan view. The nearest similar puzzle I know is the “architect’s puzzle,” in which you’re asked to imagine a single object that can cast three different shadows: a circle, a square, and a triangle. That’s possible, but it’s a distinctly different task, and it’s hard to see how to adapt that object into the one we’re seeking here. 09/27/2023 UPDATE: I’ve received a ton of mail about this puzzle — a million thanks to everyone who’s contributed; I’m constantly impressed by your intelligence, imagination, and resourcefulness. The consensus is that the solution was published upside down! And the puzzle as presented is a bit ambiguous. Reader Catalin Voinescu explains: Note there are no additional lines to the semicircle that is the front elevation: that’s all there is to see from that direction. The plan view could be filled or not, and we can’t tell. So the figure must be the intersection of the bottom half of a horizontal cylinder surface (like a half-round gutter) and either a vertical cylinder surface, like a gutter downpipe, or a solid vertical cylinder. If you think of what a T joint of two cylinders of the same diameter looks like from the side, the joints appear as straight lines that bisect the angle between the cylinders. (They must be straight lines bisecting the angle for reasons of symmetry.) So the given answer is one of two possibilities. When the plan view is hollow, the figure is a closed curve in three dimensions: draw an ellipse with a long diameter $\displaystyle \sqrt{2}$ times the short diameter, and bend the paper at 90 degrees along the short diameter. The top view is a circle, the side view a semicircle, the other side view is the angle made by the paper. However, if the plan view was solid, the figure would be a curved surface: what you need to cut out of the half-round gutter to get the vertical cylinder to go through it. Viewed from the side, it would look like a triangle — like the given solution, but with a line at the bottom. That second solution looks like this: Here’s an interactive plot of the first solution, and here’s the second. Click and drag these figures to examine them. In both cases the side elevation is an inverted V, so the published solution is upside down (and I’m not crazy!). My thanks to Drake Thomas and to Catalin for their work on all this, and thanks again to everyone else who wrote in about it. I’ve actually gone through the subsequent issues of the journal to see whether an erratum notice was ever published, and I don’t find one. I wonder how much mail this generated in 1958!

Click for Answer

Here’s the answer given in the journal:

https://archive.org/details/eureka-21/page/29/mode/2up?view=theater — Image: Eureka

No explanation is given. I had imagined something like a curved disk, but that wouldn’t give a circular plan view.

The nearest similar puzzle I know is the “architect’s puzzle,” in which you’re asked to imagine a single object that can cast three different shadows: a circle, a square, and a triangle. That’s possible, but it’s a distinctly different task, and it’s hard to see how to adapt that object into the one we’re seeking here.

09/27/2023 UPDATE: I’ve received a ton of mail about this puzzle — a million thanks to everyone who’s contributed; I’m constantly impressed by your intelligence, imagination, and resourcefulness. The consensus is that the solution was published upside down! And the puzzle as presented is a bit ambiguous. Reader Catalin Voinescu explains:

Note there are no additional lines to the semicircle that is the front elevation: that’s all there is to see from that direction. The plan view could be filled or not, and we can’t tell. So the figure must be the intersection of the bottom half of a horizontal cylinder surface (like a half-round gutter) and either a vertical cylinder surface, like a gutter downpipe, or a solid vertical cylinder. If you think of what a T joint of two cylinders of the same diameter looks like from the side, the joints appear as straight lines that bisect the angle between the cylinders. (They must be straight lines bisecting the angle for reasons of symmetry.)

So the given answer is one of two possibilities.

When the plan view is hollow, the figure is a closed curve in three dimensions: draw an ellipse with a long diameter $\displaystyle \sqrt{2}$ times the short diameter, and bend the paper at 90 degrees along the short diameter. The top view is a circle, the side view a semicircle, the other side view is the angle made by the paper.

However, if the plan view was solid, the figure would be a curved surface: what you need to cut out of the half-round gutter to get the vertical cylinder to go through it. Viewed from the side, it would look like a triangle — like the given solution, but with a line at the bottom.

That second solution looks like this:

elevation puzzle cylinder solution

Here’s an interactive plot of the first solution, and here’s the second. Click and drag these figures to examine them. In both cases the side elevation is an inverted V, so the published solution is upside down (and I’m not crazy!).

My thanks to Drake Thomas and to Catalin for their work on all this, and thanks again to everyone else who wrote in about it. I’ve actually gone through the subsequent issues of the journal to see whether an erratum notice was ever published, and I don’t find one. I wonder how much mail this generated in 1958!

September 26, 2023September 27, 2023 | Puzzles