Initial Velocity

In 2008, University of Michigan psychologist Jesse Chandler and his colleagues examined donations to disaster relief after seven major hurricanes and found that a disproportionately large number of donations came from people who shared an initial with the hurricane (e.g., people named Kate and Kevin after Hurricane Katrina).

It’s not clear why this is. It’s known that generally people attend to information with unusual care if it’s somehow relevant to themselves; in the case of a hurricane this may mean that they’re more likely to remember concrete information about victims and thus be more likely to donate.

Possibly they also feel more intense negative feelings (or a greater sense of responsibility) when the storm shares their initial. In that case, “Exposure to a same-initial hurricane makes people feel worse, and the most salient way to repair this feeling is the opportunity to donate money to Katrina.”

(Jesse Chandler, Tiffany M. Griffin, and Nicholas Sorensen, “In the ‘I’ of the Storm: Shared Initials Increase Disaster Donations,” Judgment and Decision Making 3:5 [June 2008], 404–410.)

A Pi Fractal

Jack Hodkinson, who created the Corpus-Christi-College-shaped prime number that I mentioned last week, has sent along a remarkable followup produced by string rewriting. Here are the rules:

The top row lists pixels of various colors; on each pass we’ll be replacing each of these with the 3 × 3 square beneath it. Start with a single color:

Following the associated rule produces a 3 × 3 array:

And so on:

After four iterations the rules produce a portrait of π:

And, charmingly, if we keep going, smaller πs begin to appear, due to the presence of dark pixels in the third image:

And so on forever. “After iteration 8, it’s turtles all the way down.”

For those who want more information, Jack explains the rewriting rules in detail here.

(Note: To conserve bandwidth I’ve had to reduce the last two images above — you can find the full-resolution PNGs in this imgur gallery.)

Post Haste

At the start of the 1892 story “Silver Blaze,” Sherlock Holmes and Watson set out on a train journey from Paddington to Swindon in a first-class train carriage.

“We are going very well,” says Holmes, looking out the window and glancing at his watch. “Our rate at present is fifty-three and a half miles an hour.”

“I have not observed the quarter-mile posts,” says Watson.

“Nor have I,” replies Holmes. “But the telegraph posts upon this line are sixty yards apart, and the calculation is a simple one.”

Is it? The speed itself is plausible — trains were allowed 87 minutes to travel the route, giving an average speed of 53.25 mph, and so the top running speed would have been higher than this. But A.D. Galbraith complained that the detective’s casual statement is “completely inconsistent with Holmes’ character.” Using the second hand of his watch, he’d had to mark the passage of two successive telegraph posts, probably a mile or more apart, and count the posts between them; an error of more than one second would produce an error of almost half a mile an hour. So Holmes’ scrupulous dedication to accuracy should have led him to say “between 53 and 54 miles an hour” or even “between 52 and 55.”

Guy Warrack, in Sherlock Holmes and Music, agreed: It would have been impossible to time the passage of the telegraph poles to the necessary precision using a pocket watch. But S.C. Roberts, in a review of the book, disagreed:

Mr. Warrack, if we may so express it, is making telegraph-poles out of fountain-pens. What happened, surely, was something like this: About half a minute before he addresssed Watson, Holmes had looked at the second hand of his watch and then counted fifteen telegraph poles (he had, of course, seen the quarter-mile posts, but had not observed them, since they were not to be the basis of his calculation). This would give him a distance of nine hundred yards, a fraction over half-a-mile. If a second glance at his watch had shown him that thirty seconds had passed, he would have known at once that the train was traveling at a good sixty miles an hour. Actually he noted that the train had taken approximately thirty-four seconds to cover the nine hundred yards; or, in other words, it was rather more than ten per cent (i.e., 6 1/2 from sixty). The calculation, as he said, was a simple one; what made it simple was his knowlege, which of course Watson did not share, that the telegraph poles were sixty yards apart.

In fact George W. Welch offered two different formulas that Holmes might have used:

First Method:–Allow two seconds for every yard, and add another second for every 22 yards of the known interval. Then the number of objects passed in this time is the speed in miles an hour. Proof:–Let x = the speed in miles per hour, y = the interval between adjacent objects. 1 m.p.h. = 1,760 yards in 3,600 seconds = 1 yard in 3,600/1,760 = 45/22 or 2.1/22 secs. = y yards in 2.1/22 y seconds x m.p.h. = xy yards in 2.1/22y seconds. Example:–Telegraph poles are set 60 yards apart. 60 × 2 = 120; 60 ÷ 22 = 3 (approx.); 120 + 3 = 123. Then, if after 123 seconds the observer is half-way between the 53rd and 54th poles, the speed is 53 1/2 miles an hour.

Second Method:–When time or space will not permit the first method to be used, allow one second for every yard of the known interval, and multiply by 2.1/22 the number of objects passed in this time. The product is the speed in miles an hour. Example:–Telegraph poles are set 60 yards apart. After 60 seconds the observer is about 10 yards beyond the 26th pole. 26.1/6 × 2 = 52.1/3; 26.1/6 divided by 22 = 1.1/6 (approx.); 52.1/3 = 1.1/6 = 53 1/2. Therefore the speed is 53 1/2 miles an hour. The advantage of the first method is that the time to be used can be worked out in advance, leaving the observer nothing to do but count the objects against the second hand of his watch.

Julian Wolff suggested examining the problem “in the light of pure reason.” The speed in feet per second is found by determining the number of seconds required to travel a known number of feet. Holmes says that the posts are 60 yards apart, so 10 intervals between poles is 1800 feet, and the speed in covering this distance is 1800/T feet per second. Multiply that by 3600 gives feet per hour, and dividing the answer by 5280 gives the speed in miles per hour. So:

$\displaystyle \textup{miles per hour} = \frac{\frac{1800}{T}\times 3600}{5280}=\frac{1227.27}{T}$

So to get the train’s speed in miles per hour we just have to divide 1227.27 by the number of seconds required to travel 1800 feet. And “1227 is close enough for all ordinary purposes, such as puzzling Watson, for instance.”

(From William S. Baring-Gould, ed., The Annotated Sherlock Holmes, 1967.)

Planet Packing

What’s the shortest string of letters that contains the words ONE, TWO, and THREE, each spelled out in order but not necessarily using adjacent letters? It can be done in eight letters — THRWONEE is one example — and it turns out that no shorter solution is possible.

In 2001, A. Ross Eckler set out to do the same thing with the names of the planets, from MERCURY through PLUTO. He got down as far as 26 letters, MNVESARCPJLUPITHOURYANUSER, and to my knowledge no one has found a shorter solution.

Dana Richards offered a discussion of the problem from a computing perspective later that year. He found that Eckler’s task is related to a problem in Garey and Johnson’s 1979 Computers and Intractability.

“Why would planet packing be found in a serious computer science book?” he writes. “It turns out to be an important problem with applications to data compression, DNA sequencing, and job scheduling. … The first practical thing is to abandon all hope of solving the problem with a fast algorithm that always gets the optimal answer.”

(A. Ross Eckler, “Planet Packing,” Word Ways 34:2 [May 2001], 157.)

09/23/2017 UPDATE: Reader Mikko Ratala has found a 25-letter solution: JVSMEURANEPLICTUERNTYESOH. “The string is not unique solution as you can, for example, change the order of the first four letters as you wish.”

Taxicab Geometry

What’s the shortest distance between the points in the lower left and upper right? In our familiar Euclidean geometry, it’s the green line. But in taxicab geometry, an intriguing variant devised by Hermann Minkowski in the 19th century, distance is reckoned as the sum of the absolute differences of Cartesian coordinates — basically the distance that a taxicab would drive if this were a city grid. In that case, the shortest distance between the two points is 12, and it’s shown equally well by the red, blue, and yellow lines. Any of these routes will cover the same “distance” in taking you from one point to the other.

This way of considering things is intriguing in the abstract, but it has some practical value as well. “Taxicab geometry is a more useful model of urban geography than is Euclidean geometry,” writes Eugene F. Krause in Taxicab Geometry. “Only a pigeon would benefit from the knowledge that the Euclidean distance from the Post Office to the Museum [below] is $\sqrt{8}$ blocks while the Euclidean distance from the Post Office to the City Hall is $\sqrt{9}=3$ blocks. This information is worse than useless for a person who is constrained to travel along streets or sidewalks. For people, taxicab distance is the ‘real’ distance. It is not true, for people, that the Museum is ‘closer’ to the Post Office than the City Hall is. In fact, just the opposite is true.”

Aptitude

To earn some money during college, Raymond Smullyan applied for a job as a salesman. He had to take an examination, and one of the questions asked whether he had any objection to telling a small lie now and then. Smullyan did object, but he was afraid that he wouldn’t get the job if he said so. So he lied and said no.

“Later on, I realized I was in a kind of paradox!” Smullyan wrote later. “Did I object to the lie I told the sales company? I realized that I did not! Then since I didn’t object to that particular lie, it therefore followed that I don’t object to all lies, hence my answer ‘No’ was not a lie, but the truth! So was I lying or not?”

(From his book A Mixed Bag, 2016.)

Hot and Cold

The vortex tube is a bit of a magic trick: When a stream of compressed gas is injected into the chamber, it accelerates to a high rate of rotation and moves toward the nozzle on the right. Because of the nozzle’s shape, though, only the quickly rotating outer shell of this gas can escape; the rest moves back through the center of the vortex and escapes through the opening on the left.

The result, perplexingly, is that even though the tube has no moving parts, it emits hot air (up to 200°C) on the right and cold air (down to -50° C) on the left.

Could this principle be used to air-condition a home or vehicle? “That’s what everyone thinks when they first hear about it,” engineer Leslie Inglis told Popular Science in 1976. “I always tell them that they wouldn’t buy a toaster for the kitchen if they had to buy the generator to produce the electricity. You’ve got to think of this as a compressed-air appliance.”

Podcast Episode 169: John Harrison and the Problem of Longitude

Ships need a reliable way to know their exact location at sea — and for centuries, the lack of a dependable method caused shipwrecks and economic havoc for every seafaring nation. In this week’s episode of the Futility Closet podcast we’ll meet John Harrison, the self-taught English clockmaker who dedicated his life to crafting a reliable solution to this crucial problem.

We’ll also admire a dentist and puzzle over a magic bus stop.

See full show notes …

The Trinity Hall Prime

On Thursday, Numberphile published this video, which features a startling wall hanging in the Senior Combination Room at Trinity Hall, Cambridge: Junior research fellow James McKee devised a 1350-digit prime number whose image forms a likeness of the college’s coat of arms. (The number of digits is significant, as it’s the year that Bishop William Bateman founded the college.)

It turns out that finding such “prime” images is easier than one might think. In the video description, McKee explains: “Most of the digits of p were fixed so that: (i) the top two thirds made the desired pattern; (ii) the bottom third ensured that p-1 had a nice large (composite) factor F with the factorisation of F known. Numbers of this shape can easily be checked for primality. A small number of digits (you can see which!) were looped over until p was found that was prime.'”

Indeed, on the following day, Cambridge math student Jack Hodkinson published his own prime number, this one presenting an image of Corpus Christi College and including his initials and date of birth:

Hodkinson explains that he knew he wanted a 2688-digit prime, and the prime number theorem tells us that approximately one in every 6200 2688-digit numbers is prime. And he wasn’t considering even numbers, which reduces the search time by half: He expected to find a candidate in 100 minutes, and in fact found eight overnight.

(Thanks, Danesh.)

Unto the Breach

In 2004, engineers Richard Clements and Roger Hughes put their study of crowd dynamics to an unusual application: the medieval Battle of Agincourt, which pitted Henry V’s English army against a numerically superior French army representing Charles VI. In their model, an instability arises on the front between the contending forces, which may account for the relatively large proportion of captured soldiers:

[P]ockets of French men-at-arms are predicted to push into the English lines and with hindsight be surrounded and either taken prisoner or killed. … Such an instability might explain the victory by the weaker English army by surrounding groups of the stronger army.

This description is consistent with the three large mounds of fallen soldiers that are reported in contemporary accounts of the battle. If the model is accurate then perhaps French men-at-arms succeeded in pushing back the English in certain locations, only to be surrounded and slaughtered, rallying around their leaders. By contrast, modern accounts perhaps incorrectly describe a “wall” of dead running the length of the field.

“Interestingly, the study suggests that the battle was lost by the greater army, because of its excessive zeal for combat leading to sections of it pushing through the ranks of the weaker army only to be surrounded and isolated.” The whole paper is here.

(Richard R. Clements and Roger L. Hughes. “Mathematical Modelling of a Mediaeval Battle: The Battle of Agincourt, 1415,” Mathematics and Computers in Simulation 64:2 [2004], 259-269.)