# Math Notes

12 + 22 + 32 + … + 242 = 702

This is the only case in which the sum of the first k perfect squares is itself a square.

03/23/2020 UPDATE: Reader Pieter Post made a pyramid of 4900 golf balls in the Netherlands last summer:

It took him an hour and a half. (Thanks, Pieter.)

# Pretending the Truth

In a 1988 experiment with 2-year-olds, psychologist Alan Leslie asked each child to “fill” two toy cups with imaginary “juice” or “tea” from a bottle. Leslie then said, “Watch this!”, upended one of the cups, shook it, and replaced it next to the other cup. Then he asked the child to point to the “full cup” and the “empty cup.” Though both cups had been empty throughout, all 10 of the 10 subjects indicated that the “empty” cup was the one that had been inverted.

“This leads to pretending something that is true, namely, that the empty cup is empty,” Leslie wrote. “At first glance, this may seem ridiculous. But there is, of course, an important difference between the empty cup is empty and pretending (of) the empty cup ‘it is empty.'” Children distinguish between pretense and reality even when the content of those beliefs is the same.

“These examples help us realize that, far from being unusual and esoteric, cases of ‘non-counterfactual pretence’, that is, pretending something is true when it is true, are ubiquitous in young children’s pretence and indeed has an indispensable role in the child’s ability to elaborate pretend scenarios.”

(Alan M. Leslie, “Pretending and Believing: Issues in the Theory of ToMM,” Cognition on Cognition [1995], 193-220.)

# Misc

• The state sport of Maryland is jousting.
• North and South Dakota were established together, in 1889.
• NEAT TAILOR makes ALTERATION.
• Percentages are reversible: 25% of 16 is 16% of 25.
• “Success in research needs four Gs: Glück, Geduld, Geschick, und Geld [luck, patience, skill, and money].” — Paul Ehrlich

# Manly Faces

In studying the attractiveness of human facial features, University of St Andrews psychologist David Perrett found that femininity in female face shape was preferred across cultures and by both men and women. Surprisingly, he found that masculinity in male faces makes them less attractive — in fact, people prefer male faces with a slightly feminized shape. Evaluators said that images of male faces that had been artificially masculinized looked less kind, less emotional, colder, less honest, less cooperative, and less likely to be a good parent. Feminization of male faces had the opposite effect.

This has some basis in real life, where men with more masculine faces show more aggressive and more uncooperative behavior:

Men with masculine face proportions commit more fouls in ice-hockey games and end up with a greater number of time-out penalties than men with more feminine facial proportions. Off the pitch, in an experimental set-up in which players can gain resources, defend them, and steal from others, men with masculine facial proportions choose to retaliate aggressively when other players steal from them and then to steal right back. In contrast, men with feminine facial proportions are more likely to build defences against further infringement. Furthermore, men with high testosterone (that is, those likely to have a masculine facial appearance) have more troubled relationships, and show increased rates of infidelity, violence, and divorce. Masculine males, it seems, are more likely to behave like cads than be good dads.

Men with masculine features do tend to be perceived as stronger and more dominant, and they do tend to be physically strong. Some studies have found a preference for a slight degree of masculinity, but this effect is neither dramatic nor consistent. And “no one has found an overall preference for a high degree of facial masculinity.”

(From Perrett’s 2010 book In Your Face: The New Science of Human Attraction.)

# Centrists

The Cook pine, Araucaria columnaris, leans toward the equator. In 2017 botanist Matt Ritter of California Polytechnic State University noticed that pines growing in California and Hawaii leaned south; he called a colleague in Australia, who reported that the trees there leaned north.

“We got holy-smoked that there’s possibly a tree that’s leaning toward the equator wherever it grows,” Ritter told Nature.

In both hemispheres, the trees lean more sharply the farther they are from the equator. The average incline is 8 degrees, but one tree in South Australia leans 40 degrees.

The reason isn’t clear; it “may be related to an adaptive tropic response to the incidence angles of annual sunlight, gravity, magnetism, or any combination of these,” the authors write. But “It’s a shockingly distinct pattern,” Ritter said.

See Beef Tack.

(Jason W. Johns et al., “Worldwide Hemisphere-Dependent Lean in Cook Pines,” Ecology 98:9 [2017], 2482-2484.)

# Estimating Payments

In 1996 Peyman Milanfar, a reader of Mathematics Magazine, presented a quick way to estimate monthly payments on a loan, passed down from his grandfather, who had been a merchant in 19-century Iran:

$\displaystyle \textup{Monthly payment} = \frac{1}{\textup{Number of months}} (\textup{Principal} + \textup{Interest})$

The interest is calculated as

$\displaystyle \textup{Interest} = \frac{1}{2} \textup{Principal} \times \textup{Number of years} \times \textup{Annual interest rate}.$

The exact formula given in finance textbooks is

$\displaystyle C = \frac{r(1 + r)^{N}P}{(1 + r)^{N} - 1},$

where C is the monthly payment, r is the monthly interest rate (1/12 the annual interest rate), N is the total number of months, and P is the principal. Rendered in that notation, the folk formula becomes

$\displaystyle C_{f} = \frac{1}{N} \left ( P + \frac{1}{2}PNr \right ).$

“In many cases, Cf is a surprisingly good approximation to C,” particularly when the principal is fixed, the monthly interest rate is sufficiently low, and the total number of months is sufficiently high, Milanfar writes. For example, for a four-year loan of $10,000 at an annual rate of 7% compounded monthly, the precise formula gives a monthly payment of$239.46, while the folk formula gives \$237.50.

“While its origins remain a mystery, the method is still in use among merchants all around Iran, and perhaps elsewhere.”

(Peyman Milanfar, “A Persian Folk Method of Figuring Interest,” Mathematics Magazine 69:5 [1996], 376.)

# The Tonal System

In 1859, far ahead of its application in computing, engineer John W. Nystrom proposed that we adopt base 16 for arithmetic, timekeeping, weights and measures, coinage, and even music.

“It is evident that 12 is a better number than 10 or 100 as a base, but it admits of only one more binary division than 10, and would, therefore, not come up to the general requirement,” he wrote. “The number 16 admits binary division to an infinite extent, and would, therefore, be the most suitable number as a base for arithmetic, weight, measure, and coins.”

He named the 16 digits an, de, ti, go, su, by, ra, me, ni, ko, hu, vy, la, po, fy, and ton, and invented new numerals for the upper values. Numbers above this range would be named using these roots, so 17 in decimal would be tonan (“16 plus 1”) in Nystrom’s system. And he devised some wonderfully euphonious names for the higher powers:

 Base 16 Number Tonal Name Base 10 Equivalent 10 ton 16 100 san 256 1000 mill 4,096 1,0000 bong 65,536 10,0000 tonbong 1,048,576 100,0000 sanbong 16,777,216 1000,0000 millbong 268,435,456 1,0000,0000 tam 4,294,967,296 1,0000,0000,0000 song 1612 1,0000,0000,0000,0000 tran 1616 1,0000,0000,0000,0000,0000 bongtran 1620

So the hexadecimal number 1510,0000 would be mill-susanton-bong.

The system was never widely adopted, but Nystrom was confident in its rationality. “I know I have nature on my side,” he wrote. “If I do not succeed to impress upon you its utility and great importance to mankind, it will reflect that much less credit upon our generation, upon scientific men and philosophers.”

His book is here.

# Limerick

A Maths Master, teaching at Rye,
Bought his pupils a succulent π.
But we’re sorry to state
That 3/8
With 6=7 knows why.

Punch, Sept. 29, 1937, via William R. Ransom, One Hundred Mathematical Curiosities, 1953

(I read this as “three overate, with sick sequels, heaven knows why.”)

# Venn Primes

The classic three-circle Venn diagram on the left has threefold rotational symmetry, and the more complex five-ellipse diagram on the right (discovered by Branko Grünbaum in 1975) has fivefold symmetry. Pleasingly, it turns out that a Venn diagram with n curves having an n-fold rotational symmetry exists if and only if n is prime.

(The diagram below has four curves and fourfold symmetry, but properly speaking it’s not a Venn diagram because it doesn’t represent all possible intersections of the sets.)

(Stan Wagon and Peter Webb, “Venn Symmetry and Prime Numbers: A Seductive Proof Revisited,” American Mathematical Monthly 115:7 [2008], 645-648; Frank Ruskey, Carla D. Savage, and Stan Wagon, “The Search for Simple Symmetric Venn Diagrams,” Notices of the AMS 53:11 [2006], 1304-1311.)

# Building Tangents

Here’s a way to find tangents to an ellipse from a point outside it, say E. Use E to draw any two chords CD and FG. Now lines CF and DG will meet at H, and CG and DF will meet at J. The line HJ intersects the ellipse at A and B, and EA and EB are the tangents we sought.

In 2001 David Bloom of Brooklyn College wrote, “I owe the above to a course I took in 1958, taught by O. Zariski. The result seemed so beautiful that I’ve never forgotten it.”

(“Miscellanea,” College Mathematics Journal 32:4 [September 2001], 317-318.)