A curious puzzle by Dartmouth mathematician Peter Winkler: You’ve just joined the Coin Flippers of America, and fittingly the amount of your dues will be decided by chance. You’ll name a head-tail sequence of length 5, and then a coin will be flipped until that sequence appears in five consecutive flips. Your dues will be the total number of flips in U.S. dollars; for instance, if you choose HHHHH and it takes 36 flips to produce a run of five heads, then your annual dues will be $36. What sequence should you pick?

At first it seems that it shouldn’t matter — any fixed sequence should have the probability (1/2)⁵, or 1/32. But “Not so fast,” Winkler writes. “Overlapping causes problems.” It is true that in an infinite sequence of random flips, the average distance between one occurrence and the next of any fixed sequence is 1/32. But if you choose HHHHH (for example), one occurrence of this outcome gives a huge head start to the next — if the next flip is a tail, then you’re starting over cleanly, but if it’s a head then you’ve already produced the next occurrence.

“If X is the average time needed to get HHHHH starting fresh, the average of 1 + X and 1 is 32,” Winkler writes. “Solving for X yields a startlingly high 62 flips.” To get your expected dues down to $32, you need to pick a sequence where this “head start” effect doesn’t obtain. There are 10 such sequences; one is HHHTT.

(Peter Winkler, “Coin Flipping,” Communications of the ACM 56:11 [November 2013], 120.)

An observation by Oxford University mathematician Nick Trefethen:

A student leaves university in America with a transcript full of information. Even with grade inflation, there are thirty marks of A or A- or B+ or B to look at, each one attached to a different course like Advanced Calculus or 20th Century Philosophy or Introduction to Economics. Grade-point averages are constructed from these transcripts and reported to three digits of accuracy.

An Oxford graduate finishes with no transcript, just a degree result which may be a First, a II.1, a II.2, a Third, a Pass, or a Fail. Failures are more or less nonexistent, and the numbers last year [2000] for the other degrees were 691, 1925, 374, 39, and 3, respectively. The corresponding probabilities are 23%, 63%, 12%, 1%, and 0.1%.

If you add up these probabilities times their base 2 logarithms, all times minus one, you find out how much information there is in an Oxford degree. The result is: 1.37 bits of information.

(From Trefethen’s Index Cards, 2011.)

Early experimenters in electricity sometimes dealt in frogs’ thighs. Dissecting a frog creates an “injury potential” in its muscles, which can then be arranged in series to produce a kind of biological battery. Carlo Matteucci strung together 12 to 14 half-thighs to make a “frog battery” strong enough to decompose potassium iodide; he was able to induce some effect even with living frogs.

Matteucci did similar work with eel, pigeon, and rabbit batteries. In 1803 Giovanni Aldini used a galvanoscope made of frogs to detect current in a circuit that ran from an ox’s tongue to its ear through Aldini’s own body. The mechanisms underlying these results weren’t always clearly understood, but they formed important early strides in bioelectrochemistry.

On a dry summer day in California, physicist Julius Sumner Miller was driving slowly near the desert when a friend overtook him on the left. The friend’s wife, in the passenger seat, reached out to hand him a package of gum. Their hands were no less than 3 inches apart when “a terrific discharge took place which possessed the classical physiological effects. The shock was momentarily disabling, as a three-inch spark in air can well be.”

Miller published an inquiry about this in the American Journal of Physics and received a reply from R.F. Miller of B.F. Goodrich in Ohio. The motion of the cars had built up charges of different amounts; Goodrich had found that the accumulated charges can (or could) increase greatly as the wheel rotates, and “as soon as the tread charges are far enough removed, they will find a lower resistance path through the rim to ground rather than around the tread,” charging the vehicle.

Even at the time the phenomenon was well known; in his original letter Miller noted that gasoline trucks were required by law to carry a dragging chain or strap. But “the question as to how great a charge may accumulate is difficult to answer.”

(Julius Sumner Miller, “Concerning the Electric Charge on a Moving Vehicle,” American Journal of Physics, 21:4 [April 1953], 316.)

Carved into the brickwork of a cylindrical tower at Cambridge University’s New Museums Site is a great crocodile. It was commissioned by Pyotr Kapitza, who had moved to Cambridge from Russia expressly to work with Ernest Rutherford, the father of nuclear physics. Kapitza called his mentor “crocodile,” a title that Russians traditionally confer on great men (and also, Kapitza said, because Rutherford’s thunderous voice announced his approach, just as the crocodile in Peter Pan was announced by the ticking watch in its belly).

Eric Gill carved the animal into the side of the Mond Laboratory, which was erected in 1933 with Rutherford’s backing to support Kapitza’s work in low-temperature physics. Unfortunately, after a holiday in Russia the following year, Kapitza was barred from leaving the country, and he never returned to Cambridge.

A few quotations by Rutherford:

• “Don’t let me catch anyone talking about the Universe in my department.”
• “An alleged scientific discovery has no merit unless it can be explained to a barmaid.”
• “We’re like children who always want to take apart watches to see how they work.”
• “We’ve got no money, so we’ve got to think.”
• “When we have found how the nucleus of atoms is built up we shall have found the greatest secret of all — except life.”

Paul Langevin and Rutherford served together as research assistants at Cavendish Laboratory. Asked afterward whether they were friendly, Langevin said, “One can hardly speak of being friendly with a force of nature.”

Amazingly, the notion of a black hole was first posited in 1783, by the English natural philosopher John Michell. In a paper read before the Royal Society that November, he wrote:

Let us now suppose the particles of light to be attracted in the same manner as all other bodies with which we are acquainted; that is, by forces bearing the same proportion to their vis inertiae (or mass), of which there can be no reasonable doubt, gravitation being, as far as we know, or having any reason to believe, an universal law of nature. … [I]f the semi-diameter of a sphere of the same density as the Sun were to exceed that of the Sun in the proportion of 500 to 1, a body falling from an infinite height towards it, would have acquired at its surface greater velocity than that of light, and consequently supposing light to be attracted by the same force in proportion to its vis inertiae, with other bodies, all light emitted from such a body would be made to return towards it by its own proper gravity.

“From these quotations it is clear that Michell in 1783 understood many of the basic principles of black hole physics which are in daily use almost 200 years later,” writes Cambridge physicist Gary Gibbons. Indeed, Michell’s talent doomed him to obscurity: His breakthroughs were lost on his contemporaries and forgotten by the time the world could appreciate them. His notion of a “dark star” was rediscovered only in the 1970s. The American Physical Society says, “[H]e remains virtually unknown today, in part because he did little to develop and promote his own path-breaking ideas.”

(Gary Gibbons, “The Man Who Invented Black Holes,” New Scientist, June 28, 1979.) (Thanks, Alejandro.)

Biologist and mathematician D’Arcy Thompson advanced a strange new idea in his 1917 book On Growth and Form: He found that if you draw the outline of an animal or plant on an ordinary Cartesian grid, and then you put the grid through some mathematical transformation (stretching it, for example, so that its squares become rhombuses), very often the resulting shape is that of a related real creature.

What can that mean? Thompson doesn’t really say. He thought that the biologists of his day overemphasized evolution in explaining the form and structure of living things; he preferred to look for physical and especially mathematical laws. But he didn’t present his ideas as principles that might be tested, so his book has (so far) remained only a notable curiosity.

“This theory cries out for causal explanation, which is something the great man eschewed,” writes zoologist Wallace Arthur. “Perhaps the time is close when comparative developmental genetics will be able to provide such an explanation.”

(Wallace Arthur, “D’Arcy Thompson and the Theory of Transformations,” Nature Reviews Genetics, May 2006, 401-406.)

In 1880 Charles Hinton (inventor of the baseball gun) turned his attention to the fourth dimension, that unseen world whose behavior seems so baffling to ordinary thinkers.

In his 1888 book A New Era of Thought, he announced a unique way to think about it, a set of 81 colored cubes that correspond to the 81 parts of a 3 × 3 × 3 × 3 hypercube. By creating a set of wooden cubes, painting them according to Hinton’s instructions, and working through the prescribed exercises, the reader could learn to visualize the fourth dimension as intuitively as the third:

The square, in moving in the unknown direction, traces out a succession of squares, the assemblage of which makes the cube in layers. So also the cube, moving in the unknown direction, will at any point of its motion, still be a cube; and the assemblage of cubes thus placed constitutes the tessaract in layers. We suppose the cube to change its colour directly it begins to move. Its colour between 1 and 2 we can easily determine by finding what colours its different parts assume, as they move in the unknown direction.

Hinton’s method drew few adherents, but he was sure that it worked — he had proved it for himself. “The particular problem,” he wrote, “at which I have worked for more than ten years, has been completely solved. It is possible for the mind to acquire a conception of higher space as adequate as that of our three-dimensional space, and to use it in the same manner.”

He moved on to other things, but he’s left us one permanent calling card — Hinton coined the word tesseract.

Here’s a striking sign of the pervasive influence of the Industrial Revolution: It darkened England’s moths. Before 1811, the peppered moth, Biston betularia, had a white body. But as soot darkened trees, lighter-bodied insects became more visible to birds and other predators. By 1848 the frequency of dark-bodied moths in industrial regions had increased dramatically, one of the first documented instances of Darwin’s principle of natural selection. American geneticist Sewall Wright called it “the clearest case in which a conspicuous evolutionary process has actually been observed.”

Somewhat related: A curious wartime observation by Gertrude Stein, in Alsace, from The Autobiography of Alice B. Toklas:

Another thing that interested us enormously was how different the camouflage of the french looked from the camouflage of the germans, and then once we came across some very neat camouflage and it was american. The idea was the same but as after all it was different nationalities who did it the difference was inevitable. The colour schemes were different, the designs were different, the way of placing them was different, it made plain the whole theory of art and its inevitability.