(3 + 3 + 6) + (3² + 3² + 6²) + (3³ + 3³ + 6³) = 336

(By Mike Keith.)

07/20/2017 UPDATE: For more, see OEIS A240511, “Numbers that are equal to the sum of their digits raised to each power from 1 to the number of digits.” (Thanks, Dave.)

In 1972 biologists Colin Tayler and Graham Saayman were observing a group of Indo-Pacific bottlenose dolphins in a South African aquarium. One of them, a 6-month-old calf named Dolly, began to seek their attention by pressing feathers, stones, seaweed, and fish skins against the glass of the viewing chamber. If they ignored her she swam off and returned with a different object.

At the end of one observation session, one of the investigators blew a cloud of cigarette smoke against the glass as Dolly was looking in. “The observer was astonished when the animal immediately swam off to its mother, returned and released a mouthful of milk which engulfed her head, giving much the same effect as had the cigarette smoke,” the biologists reported. “Dolly subsequently used this behaviour as a regular device to attract attention.”

“Dolly didn’t ‘copy’ (she wasn’t really smoking) or imitate with intent to achieve the same purpose,” argues ecologist Carl Safina in Beyond Words: What Animals Think and Feel. “Somehow Dolly came up with the idea of using milk to represent smoke. Using one thing to represent something else isn’t just mimicking. It is art.”

(C.K. Tayler and G.S. Saayman, “Imitative Behaviour by Indian Ocean Bottlenose Dolphins [Tursiops aduncus] in Captivity,” Behaviour 44:3 [1973], 286-298.)

In Visual Thinking in Mathematics, M. Giaquinto writes, “Calculus grew out of attempts to deal with quantitative physical problems which could not be solved by means of geometry and arithmetic alone. Many of these problems concern situations which are easy to visualize. In fact visual representations are so useful that most books on calculus are peppered with diagrams.” But there’s an intriguing footnote: “Moshé Machover brought to my attention a notable exception: Landau (1934). It has no diagram, and no geometrical application.”

That’s Differential and Integral Calculus, by Edmund Landau, a professor of mathematics at Gottingen University. Machover is right — the 366-page volume contains not a single diagram. Landau writes, “I have not included any geometric applications in this text. The reason therefor is not that I am not a geometer; I am familiar, to be sure, with the geometry involved. But the exposition of the axioms and of the elements of geometry — I know them well and like to give courses on them — requires a separate volume which would have to precede the present one. In my lecture courses on the calculus, the geometric applications do, of course, make up a considerable portion of the material that is covered. But I do not wish to wait any longer to make generally available an account, rigorous and complete in every particular, of that which I have considered in my courses to be the most suitable method of treating the differential and integral calculus.”

The book was quite successful — the first English edition appeared in 1950, and subsequent editions have continued right up through 2001.

Which of these faces is male, and which female? In fact both photos show the same androgynous face; the only difference is the amount of contrast in the image. But most people see the face on the left as female and the one on the right as male.

Gettysburg College psychologist Richard Russell says, “Though people are not consciously aware of the sex difference in contrast, they unconsciously use contrast as a cue to tell what sex a face is. We also use the amount of contrast in a face to judge how masculine or feminine the face is, which is related to how attractive we think it is.”

Cosmetics may serve to make a female face more attractive by heightening this contrast. “Cosmetics are typically used in precisely the correct way to exaggerate this difference,” Russell says. “Making the eyes and lips darker without changing the surrounding skin increases the facial contrast. Femininity and attractiveness are highly correlated, so making a face more feminine also makes it more attractive.”

(Richard Russell, “A Sex Difference in Facial Pigmentation and Its Exaggeration by Cosmetics,” Perception 38:8 [August 2009], 1211-1219.)

The digits 1-9 can be arranged into a 3 × 3 magic square in essentially one way (not counting rotations or reflections) — the so-called lo shu square:

4    3    8

9    5    1

2    7    6

As in any magic square, each row, column, and diagonal produces the same total. But surprisingly (to me), the sum of the row products also equals the sum of the column products:

4 × 3 × 8 + 9 × 5 × 1 + 2 × 7 × 6 = 96 + 45 + 84 = 225

4 × 9 × 2 + 3 × 5 × 7 + 8 × 1 × 6 = 72 + 105 + 48 = 225

Even more surprisingly, the same is true of the Fibonacci sequence, if we arrange its first nine terms into a square array in the same pattern:

 3    2   21

34    5    1

 1   13    8

3 × 2 × 21 + 34 × 5 × 1 + 1 × 13 × 8 = 126 + 170 + 104 = 400

3 × 34 × 1 + 2 × 5 × 13 + 21 × 1 × 8 = 102 + 130 + 168 = 400

It turns out that this is true of any second-order linear recursion. (The sums won’t always be squares, though.)

From Edward J. Barbeau’s Power Play, 1997.

When chemists at the University of California at Berkeley discovered elements 97 and 98, they named them berkelium and californium. The New Yorker suggested that the school showed “a surprising lack of public-relations foresight”: “Now it has lost forever the chance of immortalizing itself in the atomic tables with some such sequence as universitium (97), ofium (98), californium (99), berkelium (100).”

The discoverers sent back a reply: “By using these names first, we have forestalled the appalling possibility that after naming 97 and 98 ‘universitium’ and ‘ofium’, some New Yorker might follow with the discovery of 99 and 100 and apply the names ‘newium’ and ‘yorkium’.”

The magazine answered, “We are already at work in our office laboratories on ‘newium’ and ‘yorkium’. So far we just have the names.”