A Fateful Choice

A disease is spreading rapidly across the country. Half the people who have contracted it have died, and half have recovered on their own. A crash program to ward off the epidemic has produced two serums, A and B, but there’s been little time to test them. All three of the patients who were given serum A recovered, and so did 7 of the 8 patients who were given serum B. Unfortunately, you’ve just learned that you have the disease. If you get no treatment, your chances of surviving are 50-50. Both serums have a better record than that, but which one should you take?

“There doesn’t seem to be anything we can do other than appeal to our intuitive feelings on the matter,” writes University of Waterloo mathematician Ross Honsberger. “However, a very ingenious notion, the so-called ‘null hypothesis,’ permits a measure of analysis which, in this case, yields a definite preference.”

The key is to ask how likely it is that 3 out of 3 patients would have recovered if serum A were neither helping nor hindering them. An untreated patient has a 50-50 chance of recovery, so the answer is

$\displaystyle \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}.$

On the other hand, if serum B had no effect, then the chance of 7 recoveries out of 8 is

$\displaystyle 8 \left ( \frac{1}{2} \times \frac{1}{2} \cdots \frac{1}{2} \right ) = 8 \left ( \frac{1}{2} \right )^{8} = \frac{1}{32}.$

(Here the factor 8 reflects the fact that there are 8 different possible victims, and again the probability of dying is 1/2.)

So the available evidence suggests that it’s 4 times as likely that serum A has no effect as that serum B has no effect. Your best course is to take serum B.

(Ross Honsberger, “Some Surprises in Probability,” in his Mathematical Plums, 1979.)

October 3, 2023October 2, 2023 | Science & Math

The Ta Prohm Stegosaur

https://commons.wikimedia.org/wiki/File:Dinosaur_carving_at_Ta_Prohm_temple,_Siem_Reap,_Cambodia_(5534467622).jpg — Image: Wikimedia Commons

At Cambodia’s Ta Prohm temple, near Angkor Wat, visitors have noticed an unusual carving on a crumbling wall — it appears to resemble a stegosaur. Well, viewed from the right angle it does, and allowing for the large head, large ears, and horn. In fact, it better resembles a boar, a rhino, or a chameleon viewed against a leafy background.

Because the temple was built in the 12th century, some creationists have cited the carving as evidence that humans and dinosaurs once coexisted in the region. But it’s not even clear when the carving was made — the temple is a favorite spot for filmmakers, one of whom may have added it, or it may even be a deliberate hoax.

In any case, writes Smithsonian‘s Riley Black, “the temple carving can in no way be used as evidence that humans and non-avian dinosaurs coexisted. Fossils have inspired some myths, but close scrutiny of geological layers, reliable radiometric dating techniques, the lack of dinosaur fossils in strata younger than the Cretaceous, and other lines of evidence all confirm that non-avian dinosaurs became extinct tens of millions of years before there was any type of culture that could have recorded what they looked like.”

October 2, 2023October 1, 2023 | Oddities

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