Quickie

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From Martin Gardner: Each of the two equal sides of an isosceles triangle is one unit long. How long must the third side be to maximize the triangle’s area? There’s an intuitive solution that doesn’t require calculus.

Click for Answer

Good Boy

Elisabeth Mann Borgese taught her dog to type. In her book The Language Barrier she explains that her English setter, Arli, developed a vocabulary of 60 words and 17 letters, though “He isn’t an especially bright dog.” “[Arli] could write under dictation short words, three-letter words, four-letter words, two-letter words: ‘good dog; go; bad.’ And he would type it out. There were more letters but I never got him to use more than 17.”

She began in October 1962 by training all four of her dogs to distinguish 18 designs printed on saucers; Arli showed the most promise, so she focused on him. By January 1963 he could count to 4 and distinguish CAT from DOG. Eventually she gave him a modified typewriter with enlarged keys, which she taught him to nose mechanically by rewarding him with hamburger. “No meaning at all was associated with the words,” she writes, though he did seem to associate meaning with words that excited him. “When asked, ‘Arli, where do you want to go?’ he will unfailingly write CAR, except that his excitement is such that the ‘dance’ around the word becomes a real ‘stammering’ on the typewriter. ACCACCAAARR he will write. GGOGO CAARR.”

(And it’s always tempting to discover meaning where there is none. Once while suffering intestinal problems after a long flight Arli ignored his work when she tried to get him to type GOOD DOG GET BONE, and then he stretched, yawned, and typed A BAD A BAD DOOG. This was probably just a familiar phrase that he’d chosen at random; Borgese estimated its likelihood at 1 in 12.)

Arli did earn at least one human fan — at one point Borgese showed his output to a “well-known critic of modern poetry,” who responded, “I think he has a definite affinity with the ‘concretist’ groups in Brazil, Scotland, and Germany [and an unnamed young American poet] who is also writing poetry of this type at present.”

MENACE

In 1960, British researcher Donald Michie combined his loves of computation and biology to consider whether a machine might learn — whether by consulting its record of past experience it could perform tasks with progressively greater success.

To investigate this he designed a machine to play noughts and crosses (or tic-tac-toe). He called it the Machine Educable Noughts And Crosses Engine, which gives it the pleasingly intimidating acronym MENACE. MENACE consists of 304 matchboxes, each of which represents a board position. Each box contains a collection of beads representing available moves in that position, and after each game these collections are adjusted in light of the outcome (as described here). In this way the engine learns from its experience — over time it becomes less likely to play losing moves, and more likely to play winning (or drawing) ones, and it becomes a more successful player as a result.

University College London mathematician Matthew Scroggs describes the engine above, and he’s built an online version that you can try out for yourself — it really does get noticeably better as it plays.

Near and Far

Designed by Baroque architect Francesco Borromini in 1632, this gallery in Rome’s Palazzo Spada is a masterpiece of forced perspective — though it appears to be 37 meters long, in fact it’s only 8. The effect is produced by diminishing columns and a rising floor; the sculpture at the end, which Borromini contrived to appear life size, is only 60 centimeters high.

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Image: Wikimedia Commons

That Settles That

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The famous mathematician Stanislaw Ulam thought of the following paradox, which is now known as the Ulam Paradox: When President Richard Nixon was appointed to office, on the first day he met his cabinet he said to them: ‘None of you are yes-men, are you?’ And they all said, ‘NO!’

— Raymond Smullyan, A Mixed Bag, 2016

Horizons

holmes circles

I say that conceit is just as natural a thing to human minds as a centre is to a circle. But little-minded people’s thoughts move in such small circles that five minutes’ conversation gives you an arc long enough to determine their whole curve. An arc in the movement of a large intellect does not sensibly differ from a straight line. Even if it have the third vowel [‘I’, the first-person pronoun] as its centre, it does not soon betray it. The highest thought, that is, is the most seemingly impersonal; it does not obviously imply any individual centre.

— Oliver Wendell Holmes Sr., The Autocrat of the Breakfast-Table, 1858

Priorities

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“Once I saw a chimpanzee gaze at a particularly beautiful sunset for a full 15 minutes, watching the changing colors until it became so dark that he had to retire to the forest without stopping to pick a pawpaw for supper.” — Adriaan Kortlandt

Four Play

It’s a popular recreation to try to arrange four 4s into various expressions to generate the whole numbers, like so:

1 = 4 ÷ 4 + 4 – 4
2 = 4 – (4 + 4) ÷ 4
3 = (4 × 4 – 4) ÷ 4
4 = 4 + 4 × (4 – 4)
5 = (4 × 4 + 4) ÷ 4

In 1881 a writer to the London journal Knowledge noted that each of the first 20 integers except 19 can be generated using the operations +, -, ×, and ÷. In 1964 Martin Gardner found that if you use square roots, decimals, factorials, concatenations (444), and overline (.444 …) then every positive integer less than 113 becomes possible. (113 is surprisingly hard; it becomes possible if you use percents or the gamma function.)

In 2001 a team of mathematicians from Harvey Mudd College found that you can even get four 4s to approximate some notable constants if you use a whip and a chair:

\displaystyle e \approx \left ( 4!! \right ) \sqrt[4!!]{\frac{\sqrt{4!!}}{4!!!}}

\displaystyle \pi \approx \sqrt{\sqrt{4!\cdot 4 + \sqrt{\sqrt{4\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{.4}}}}}}}}}} \approx 3.1415932

That expression for e is accurate to 21 decimal places; it can be made arbitrarily accurate by repeatedly replacing 4 with 4!. The authors note that similar expressions can be derived using three 3s or five 5s.

Amazingly, they also approximated g, the acceleration due to gravity, with four 4s, as well as Avogadro’s number NA.

(A. Bliss, S. Haas, J. Rouse and G. Thatte, “Math Bite: Four Constants in Four 4s,” Mathematics Magazine 74:4 [October 2001], 272.)

Some Enchanted Evening

In Southeast Asia, fireflies synchronize their flashing. Observing them in Siam in the 1920s, naturalist Hugh Smith wrote, “Imagine a tenth of a mile of river front with an unbroken line of [mangrove] trees with fireflies on every leaf flashing in synchronism. … Then, if one’s imagination is sufficiently vivid, he may form some conception of this amazing spectacle.”

The phenomenon was so unexpected that some initially dismissed the reports as an illusion; Phillip Laurent “could hardly believe [his] own eyes, for such a thing to occur among insects is certainly contrary to all natural laws.”

Each male fly’s flashes are initially sporadic, but they adjust their timing according to those around them until they’re synchronized. This helps identify them to females of their own species. Biologist John Buck observed, “Centers of synchrony built up slowly, two individuals often flashing independently for up to half a minute (about fifty cycles) before the flashes coincided. At this point their rhythms locked together and continued in synchrony thereafter.”

In 2015 Robin Meier and Andre Gwerder used LEDs to artificially direct the speed and rhythm of thousands of flashing fireflies (above), using this technique to “explore the idea of free will and transform a machine into a living actor inside a colony of insects.”

(Ying Zhou, Walter Gall, and Karen Nabb, “Synchronizing Fireflies,” College Mathematics Journal 37:3 [May 2006], 187-193.)