During World War II, Alan Turing enrolled in the infantry section of the Home Guard so that he could learn to shoot a rifle. After completing this section of his training he stopped attending parades, as he had no further use for the service. Summoned to account for this, he explained that he was now an excellent shot and this was why he had joined.

“But it is not up to you whether to attend parades or not,” said Colonel Fillingham. “When you are called on parade, it is your duty as a soldier to attend.”

“But I am not a soldier.”

“What do you mean, you are not a soldier! You are under military law!”

“You know, I rather thought this sort of situation could arise,” Turing said. “I don’t know I am under military law. If you look at my form you will see that I protected myself against this situation.”

It was true. On his application form Turing had encountered the question “Do you understand that by enrolling in the Home Guard you place yourself liable to military law?” He could see no advantage in answering yes, so he answered no, and the clerk had filed the form without looking at it.

“So all they could do was to declare that he was not a member of the Home Guard,” remembered Peter Hilton. “Of course that suited him perfectly. It was quite characteristic of him. And it was not being clever. It was just taking this form, taking it at its face value and deciding what was the optimal strategy if you had to complete a form of this kind. So much like the man all the way through.”

(From Andrew Hodges, Alan Turing: The Enigma, 1992.)

Climbing Chains

Princeton mathematician John Horton Conway investigated this curious permutation:

3n ↔ 2n

3n ± 1 ↔ 4n ± 1

It’s a simple set of rules for creating a sequence of numbers. In the words of University of Calgary mathematician Richard Guy, “Forwards: if it divides by 3, take off a third; if it doesn’t, add a third (to the nearest whole number). Backwards: if it’s even, add 50%; if it’s odd, take off a quarter.”

If we start with 1, we get a string of 1s: 1, 1, 1, 1, 1, …

If we start with 2 or 3 we get an alternating sequence: 2, 3, 2, 3, 2, 3, …

If we start with 4 we get a longer cycle that repeats: 4, 5, 7, 9, 6, 4, 5, 7, 9, 6, …

And if we start with 44 we get an even longer repeating cycle: 44, 59, 79, 105, 70, 93, 62, 83, 111, 74, 99, 66, 44, …

But, curiously, these four are the only loops that anyone has found — start with any other number and it appears you can build the sequence indefinitely in either direction without re-encountering the original number. Try starting with 8:

…, 72, 48, 32, 43, 57, 38, 51, 34, 45, 30, 20, 27, 18, 12, 8, 11, 15, 10, 13, 17, 23, 31, 41, 55, 73, 97, …

Paradoxically, the sequence climbs in both directions: Going forward we multiply by 2/3 a third of the time and by roughly 4/3 two-thirds of the time, so on average in three steps we’re multiplying by 32/27. Going backward we multiply by 3/2 half the time and by roughly 3/4 half the time, so on average in two steps we’re multiplying by 9/8. And every even number is preceded by a multiple of three — half the numbers are multiples of three!

What happens to these chains? Will the sequence above ever encounter another 8 and close up to form a loop? What about the sequences based on 14, 40, 64, 80, 82 … ? “Again,” writes Guy, “there are many more questions than answers.”

(Richard K. Guy, “What’s Left?”, Math Horizons 5:4 [April 1998], 5-7; and Richard K. Guy, Unsolved Problems in Number Theory, 2004.)

Pow!

Cartoon laws of physics:

1. Any body suspended in space will remain in space until made aware of its situation. Daffy Duck steps off a cliff, expecting further pastureland. He loiters in midair, soliloquizing flippantly, until he chances to look down. At this point, the familiar principle of 32 feet per second per second takes over.
2. Any body in motion will tend to remain in motion until solid matter intervenes suddenly. Whether shot from a cannon or in hot pursuit on foot, cartoon characters are so absolute in their momentum that only a telephone pole or an outsize boulder retards their forward motion absolutely. Sir Isaac Newton called this sudden termination of motion the stooge’s surcease.
3. Any body passing through solid matter will leave a perforation conforming to its perimeter. Also called the silhouette of passage, this phenomenon is the specialty of victims of directed-pressure explosions and of reckless cowards who are so eager to escape that they exit directly through the wall of a house, leaving a cookie-cutout-perfect hole. The threat of skunks or matrimony often catalyzes this reaction.
4. The time required for an object to fall twenty stories is greater than or equal to the time it takes for whoever knocked it off the ledge to spiral down twenty flights to attempt to capture it unbroken. Such an object is inevitably priceless, the attempt to capture it inevitably unsuccessful.

There are 10 laws altogether, including “9. Everything falls faster than an anvil.” As early as 1956 Walt Disney was describing the “plausible impossible.” In Who Framed Roger Rabbit, Eddie Valiant says, “Do you mean to tell me you could’ve taken your hand out of that cuff at any time?” Roger answers, “Not at any time! Only when it was funny!”

Altamura Man

About 150,000 years ago, a Neanderthal man was exploring the Lamalunga Cave in southern Italy when he fell into a sinkhole. Too badly injured to climb out again, he died of dehydration or starvation. Over the ensuing centuries, water running down the cave walls gradually incorporated the man’s bones into concretions of calcium carbonate. Undisturbed by predators or weather, they lay in an immaculate state of preservation until cave researchers finally discovered them in 1993.

This is a great boon for paleoanthropologists — “Altamura Man” is one of the most complete Paleolithic skeletons ever discovered in Europe — but there’s a downside: The bones have become so deeply involved in their matrix of limestone that no one has found a way to remove them without destroying them. So, for now, all research must be carried out in the cave.

Good Turns

In order to get a license, London taxicab drivers must pass a punishing exam testing their memory of 25,000 streets and every significant business and landmark on them. “The Knowledge” has been called the hardest test of any kind in the world; applicants must put in thousands of hours of study to pass a series of progressively difficult oral exams that take, on average, four years to complete. The guidebook for prospective cabbies says:

To achieve the required standard to be licensed as an ‘All London’ taxi driver you will need a thorough knowledge, primarily, of the area within a six-mile radius of Charing Cross. You will need to know: all the streets; housing estates; parks and open spaces; government offices and departments; financial and commercial centres; diplomatic premises; town halls; registry offices; hospitals; places of worship; sports stadiums and leisure centres; airline offices; stations; hotels; clubs; theatres; cinemas; museums; art galleries; schools; colleges and universities; police stations and headquarters buildings; civil, criminal and coroner’s courts; prisons; and places of interest to tourists. In fact, anywhere a taxi passenger might ask to be taken.

Interestingly, licensed London cabbies show a significantly larger posterior hippocampus than non-taxi drivers. Psychologist Hugo J. Spiers writes, “Current evidence suggests that it is the acquisition of this spatial knowledge and its use on the job that causes the taxi driver’s posterior hippocampus to grow larger.” Apparently it’s not actually driving the streets, or learning the information alone, that causes the change — London bus drivers don’t show the same effect; nor do doctors, who must also acquire vast knowledge; nor do cabbies who fail the exam. Rather it seems to be the regular use of the knowledge that causes the change: Retired cabbies tend to have a smaller hippocampus than current drivers.

While driving virtual routes in fMRI studies, cabbies showed the most hippocampal activity at the moment a customer requested a destination. One cabbie said, “I’ve got an over-patched picture of Peter Street. It sounds daft, but I don’t view it from ground level, it was slightly up and I could see the whole area as though I was about 50 foot up. And I saw Peter Street, I saw the market and I knew I had to get down to Peter Street.” Non-cabbie volunteers also showed the most activity when they were planning a route. “Thus,” writes Spiers, “the engagement of the hippocampus appears to depend on the extent to which someone thinks about what the possible streets they might want to take during navigation.”

(Hugo J. Spiers, “Will Self and His Inner Seahorse,” in Sebastian Groes, ed., Memory in the Twenty-First Century, 2016.)

Game Set

Multiply any two of these numbers together and add 1 and you’ll always get a perfect square:

1 + 1 × 3 = 4
1 + 1 × 8 = 9
1 + 1 × 120 = 121
1 + 3 × 8 = 25
1 + 3 × 120 = 361
1 + 8 × 120 = 961

Summing Up

In 1932, at the end of a 60-year career studying hydrodynamics, Sir Horace Lamb addressed the British Association for the Advancement of Science.

“I am an old man now,” he said, “and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather more optimistic.”

Orderly

```193939
939391
393919
939193
391939
and  919393
```

… are all prime.

Misc

• The Fall has had 66 members.
• Roundly defeated is squarely defeated.
• One pound of U.S. dimes, quarters, and half dollars, in any combination, is worth \$20.
• Reverse the digits in any multiple of 11 and you’ll get another multiple of 11.
• Bertrand Russell’s recipe for longevity: “Choose your parents wisely.”

One for All?

Suppose that there’s a power outage in your neighborhood. If someone calls the electric company, they’ll send someone to fix the problem. This puts you in a dilemma: If someone else makes the call, then you’ll benefit without having to do anything. But if no one calls, then you’ll all remain in the dark, which is the worst outcome:

This is the “volunteer’s dilemma,” a counterpart to the famous prisoner’s dilemma in game theory. Each participant has a greater incentive for “free riding” than acting, but if no one acts, then everyone loses.

A more disturbing example is the murder of Kitty Genovese, who was stabbed to death outside her New York City apartment in 1964. According to urban lore, many neighbors who were aware of the attack chose not to contact the police, trusting that someone else would make the call but hoping to avoid “getting involved.” Genovese died of her wounds.

In a 1988 paper, game theorist Anatol Rapaport noted, “In the U.S. Infantry Manual published during World War II, the soldier was told what to do if a live grenade fell into the trench where he and others were sitting: to wrap himself around the grenade so as to at least save the others. (If no one ‘volunteered,’ all would be killed, and there were only a few seconds to decide who would be the hero.)”

The Guinness Book of World Records lists the Yaghan word mamihlapinatapai as the “most succinct word.” It’s defined as “a look shared by two people, each wishing that the other would initiate something that they both desire but which neither wants to begin.”

(From William Poundstone, Prisoner’s Dilemma, 1992.)