Decision Time

When you’re driving and see an upcoming traffic light turn yellow, you face an urgent choice: stop quickly or try to run through the intersection before the light turns red. In 1962, Stanford aeronautics professor Howard Seifert worked out that you can choose either alternative if

$\displaystyle \left ( v_{0}T + \frac{1}{2}a^{+}T^{2} - s \right ) > d_{0} > \frac{1}{2}\left (v_{0}^{2} / a^{-} \right ),$

where your car’s initial speed is v₀ ft/sec, its maximum acceleration is a⁺ ft/sec², its maximum deceleration is a^– ft/sec², the duration of the yellow light is T seconds, and the intersection is s feet wide and d₀ feet away.

“[I]f the left or right inequality is reversed, you will not be able to run through or to stop, respectively,” he concluded. “It can be shown that there are situations where neither alternative will work and hazard and law violation are inevitable, as some palpitating drivers will testify.”

(Howard S. Seifert, “The Stop-Light Dilemma,” American Journal of Physics 30:3 [1962], 216-218.)

February 16, 2021February 14, 2021 | Science & Math

Podcast Episode 332: Princess Caraboo

In 1817 a young woman appeared in the English village of Almondsbury, speaking a strange language and seeking food and shelter. She revealed herself to be an Eastern princess, kidnapped by pirates from an exotic island. In this week’s episode of the Futility Closet podcast we’ll tell the story of Princess Caraboo, who was both more and less than she seemed.

We’ll also discover a June Christmas and puzzle over some monster soup.

See full show notes …

Disappearing Act

https://commons.wikimedia.org/wiki/File:Quantum_Man.jpg — Image: Wikimedia Commons

Julian Voss-Andreae studied physics at the University of Vienna before pursuing an art degree in the United States. His sculpture Quantum Man consists of 115 vertical steel sheets spaced by 1,000 short steel rods. The resulting figure looks solid when viewed from the front but almost disappears when viewed from the side, as light passes between the sheets.

“My interest is really nature,” he says. “One way to explore it is through science. Another is through intuitive sense and a search for metaphors.”

(Thanks, Ron.)

February 13, 2021 | Art · Science & Math

Handiwork

In their early studies of time and motion, engineers Frank Bunker Gilbreth and Lillian Moller Gilbreth divided all motions of the hand into 17 varieties:

https://commons.wikimedia.org/wiki/File:Therblig_(English).svg — Image: Wikimedia Commons

Magnificently, they called these therbligs, which is roughly Gilbreth spelled backward. “Transport empty” refers to receiving an item with an empty hand, “transport loaded” means moving an object with the hand, and so on. With careful study, a worker’s movements might be optimized to maximize speed and efficiency. (It was the Gilbreths, for example, who suggested that surgeons employ “caddies” to pass their instruments to them.)

This scheme makes an appearance in fiction: In their 1948 novel Cheaper by the Dozen, Frank and Lillian’s children Frank Jr. and Ernestine describe what it’s like to grow up in the home of an efficiency expert:

Suppose a man goes into a bathroom and shave. We’ll assume that his face is all lathered and that he is ready to pick up his razor. He knows where the razor is, but first he must locate it with his eye. That is ‘search’, the first Therblig. His eye finds it and comes to rest — that’s ‘find’, the second Therblig. Third comes ‘select’, the process of sliding the razor prior to the fourth Therblig, ‘grasp’. Fifth is ‘transport loaded’, bringing the razor up to his face, and sixth is ‘position’, getting the razor set on his face.

“There are eleven other Therbligs — the last one is ‘think’!”

February 12, 2021 | Science & Math

Inspiration

Berndnaut Smilde makes clouds. The Dutch artist has devised a way to combine water vapor with smoke to create miniature clouds that hover in enclosed spaces. He’s been deploying it in locations ranging from cathedrals to coal mines.

“I see them as temporary sculptures, made of almost nothing, balancing on the edge of materiality, an image of prospect in an empty space,” he told Rajesh Punj. “For me the work is about the idea of a cloud inside a space and what people project onto it. You can see them as a sign of misfortune or an element from a classical painting. There is something ungraspable about clouds: it might explain why people have been projecting so many meanings and myths upon clouds for centuries.”

Time named the technique one of the 50 best inventions of 2012.

February 12, 2021 | Art

Route Cause

A problem submitted by France and shortlisted for the 17th International Mathematical Olympiad, Burgas-Sofia, Bulgaria, 1975:

A lake has six ports. Is it possible to arrange a series of routes that satisfy the following conditions?

Each route must include exactly three ports.
No two routes may contain the same three ports.
Any tourist who wants to visit two different arbitrary ports has a choice of exactly two routes.

	Select Click for Answer
Yes. The thing to notice is that we can’t have three routes in the pattern (A,B,C), (A,B,D), (A,C,D), because in that case it’s impossible to fulfill the third condition: If the remaining ports are E and F, then we’ve left ourselves only one permissible route (not two) that visits A and E. That’s (A, E, F). Any other three-port route that includes A and E would “overload” another segment. This means that if two of the routes are (A,B,C) and (A,B,D), then the second one that covers A,C must be (without loss of generality) (A,C,E). With that in mind the other routes can be worked out uniquely: (A,D,F), (A,E,F), (B,D,E), (B,E,F), (B,C,F), (C,D,E), (C,D,F).

Click for Answer

February 11, 2021February 10, 2021 | Puzzles

Guilty

“The reuse of this sentence is forbidden.” — Stephen M. Katcher

February 11, 2021February 10, 2021 | Poems

Lucky Seven

A puzzle from the National Security Agency’s Puzzle Periodical, posed by NSA mathematician David G. in March 2017:

Each die of a pair of non-identical dice has six faces, but some numbers are missing, others are duplicated, and some faces may have more than six spots.

The dice can roll every number from 2 to 12.

What is the largest possible probability of rolling a 7?

	Select Click for Answer
David’s solution: If one die has X different numbers and the other die has Y different numbers, there are at most XY possible totals. Since we need 11 totals from 2 to 12, XY must be at least 11. In particular, at least one die must have at least four different numbers. If the first die has six different numbers, then any number on the second die has at most a 1/6 probability of getting the right number on the first die to get a total of 7. Therefore, the probability of rolling a 7 is at most 1/6. If the first die has five different numbers, with one appearing twice, the second die must have at least three different numbers. There is only one possible number on the second die which totals a 7 with the duplicated number on the first die. The second die has at most four copies of the number which totals a 7 with the duplicated number and has two ways to roll a 7; all other numbers on the second die have only one way to roll a 7. This is a maximum total of 10 rolls for a probability of 10/36. If each die has four different numbers, they can be split 2,2,1,1 or 3,1,1,1. Each number on each die can pair with at most one number on the other die to make a 7. The largest possible sum of products, which gives the number of ways to roll a 7, is 3 × 3 + 1 × 1 + 1 × 1 + 1 × 1 = 12. The maximum probability is thus 12/36. If the first die has four different numbers and the second has three, there are only 12 different pairs of values, and thus there must be at most two different value pairs which total 7 in order to have 11 possible totals. Therefore, the largest possible sum of products is 3 × 4 + 1 × 1 = 13, not 3 × 4 + 1 × 1 + 1 × 1 = 14. And 13 is possible, with one die having 1,2,6,6,6,7 and the other having 1,1,1,1,3,5 (or 1,2,2,2,6,7 and 1,3,5,5,5,5); this gives a probability of 13/36.

Click for Answer

February 10, 2021February 10, 2021 | Puzzles

In a Word

amphiscians
n. inhabitants of the tropics

angustation
n. the condition of being narrowed, constricted, limited, or confined

caducity
n. frailty, transitoriness

avolation
n. the act of flying away

This is the song of the last male Kauaʻi ʻōʻō, recorded in the Alaka’i Wilderness Preserve on Kauaʻi in 1987. Rats, pigs, hurricanes, and disease-carrying mosquitoes had reduced the species to a single pair by 1981, and the female was not found after Hurricane Iwa in 1982. The male was last seen in 1985. This appears to be his song, overheard two years later, the last trace of a vanishing species.

February 10, 2021 | Death · Language · Science & Math

One World

In 1922, Baltic German teacher and linguist Edgar de Wahl offered a new language to facilitate communication between people from different nations. He called it Occidental, and designed it so that many of its words will be recognizable by those who already know a Romance language:

Li material civilisation, li scientie, e mem li arte unifica se plu e plu. Li cultivat europano senti se quasi in hem in omni landes queles have europan civilisation, it es, plu e plu, in li tot munde. Hodie presc omni states guerrea per li sam armes. Sin cessa li medies de intercomunication ameliora se, e in consequentie de to li terra sembla diminuer se. Un Parisano es nu plu proxim a un angleso o a un germano quam il esset ante cent annus a un paisano frances.

“Material civilization, science, and even art unify themselves more and more. The educated European feels himself almost at home in all lands that have European civilization, that is, more and more, in the entire world. Today almost all states war with the same armaments. Without pause the modes of intercommunication improve, and in consequence from that the world seems to decrease. A Parisian is now closer to an Englishman or a German than he was a hundred years before to a French peasant.”

It gained a small community of speakers in the 1920s and 1930s and had largely died out by the 1980s — but in recent years it’s seen a resurgence on the Internet as Interlingue.

February 9, 2021February 8, 2021 | Language