Perfect Numbers

From Lee Sallows:

As the reader can check, the English number names less than “twenty” are composed using 16 different letters of the alphabet. We assign a distinct integral value to each of these as follows:

E   F   G   H   I   L   N   O   R   S   T   U   V   W   X   Z  
3   9   6   1  -4   0   5  -7  -6  -1   2   8  -3   7  11  10

The result is the following run of so called “perfect” numbers:

Z+E+R+O       =   10 + 3 – 6 – 7          =    0
O+N+E         =   –7 + 5 + 3              =    1
T+W+O         =    2 + 7 – 7              =    2
T+H+R+E+E     =    2 + 1 – 6 + 3 + 3      =    3
F+O+U+R       =    9 – 7 + 8 – 6          =    4
F+I+V+E       =    9 – 4 – 3 + 3          =    5
S+I+X         =   –1 – 4 + 11             =    6
S+E+V+E+N     =   –1 + 3 – 3 + 3 + 5      =    7
E+I+G+H+T     =    3 – 4 + 6 + 1 + 2      =    8
N+I+N+E       =    5 – 4 + 5 + 3          =    9
T+E+N         =    2 + 3 + 5              =   10
E+L+E+V+E+N   =    3 + 0 + 3 – 3 + 3 + 5  =   11
T+W+E+L+V+E   =    2 + 7 + 3 + 0 – 3 + 3  =   12

The above is due to a computer program in which nested Do-loops try out all possible values in systematically incremented steps. The above solution is one of two sets coming in second place to the minimal (lowest set of values) solution seen here:

 E   F   G   H   I   L   N   O   R   S   T   U   V   W   X   Z 
–2  –6   0  –7   7   9   2   1   4   3  10   5   6  –9  –4  –3

But why does the list above stop at twelve? Given that 3 + 10 = 13, and assuming that THREE, TEN and THIRTEEN are all perfect, we have T+H+I+R+T+E+E+N = T+H+R+E+E + T+E+N. But cancelling common letters on both sides of this equation yields E = I, which is to say E and I must share the same value, contrary to our requirement above that the letters be assigned distinct values. Thus, irrespective of letter values selected, if it includes THREE and TEN, no unbroken run of perfect numbers can exceed TWELVE. This might be decribed as a formal proof that THIRTEEN is unlucky.

But not all situations call for an unbroken series of perfect numbers. Sixteen distinct numbers occur in the following, eight positive, eight negative. This lends itself to display on a checkerboard:

sallows perfect numbers

Choose any number on the board. Call out the letters that spell its name, adding up their associated numbers when on white squares, subtracting when on black. Their sum is the number you selected.

(Thanks, Lee.)

| Language · Science & Math

Half Measures

https://archive.org/details/strand-1897-v-14/page/396/mode/2up?view=theater

When we read type we imagine that we read the whole of the type — but that is not so; we only notice the upper half of each letter. You can easily prove this for yourself by covering up the upper half of the line with a sheet of paper (being careful to hold the paper exactly in the middle of the letters), and you will not, without great difficulty, decipher a single word. Now place the paper over the lower half of a line, and you can read it without the slightest difficulty.

— George Lindsay Johnson, “Some Curious Optical Illusions,” Strand, October 1897

11/04/2024 UPDATE: In the experimental writing system Aravrit, devised by typeface designer Liron Lavi Turkenich, the upper half of each letter is Arabic and the lower half is Hebrew:

(Thanks to reader Djed F Re.)

| Language · Oddities

Canting Arms

https://commons.wikimedia.org/wiki/File:Hensbroek.svg

The village of Hensbroek in North Holland takes its name from the personal name Hein and the Dutch cognate of brook, i.e., “Henry’s brook.”

Magnificently, the municipal coat of arms interprets it instead as “hen’s breeches” — and depicts a chicken wearing trousers.

| History · Language

Hermae

https://commons.wikimedia.org/wiki/File:Herma_Demosthenes_Glyptothek_Munich_292.jpg
Image: Wikimedia Commons

Ancient Athenians would sometimes encounter a remarkable sculpture on roads and in public places: The bearded head of the god Hermes set on a squared pillar of stone, with male genitals carved at the appropriate height. Hermes had been a phallic god before becoming a guardian of merchants and travelers, and so was associated with fertility, luck, roads, and borders. The hermae served as a form of protection from evil and were sometimes anointed with oil or bedecked with garlands.

Plato writes that “figures of Hermes” were set up “along the roads in the midst of the city and every district town” “with the design of educating those of the countryside,” sometimes by bearing edifying inscriptions. The sculptures did not always depict Hermes — this one, from the Athenian Agora, honors Demosthenes. The practice spread eventually to Rome, where the figures were known as mercuriae.

(Peter Keegan, Graffiti in Antiquity, 2014.)

| History

“A Statesman”

A Statesman who attended a meeting of a Chamber of Commerce rose to speak, but was objected to on the ground that he had nothing to do with commerce.

‘Mr. Chairman,’ said an Aged Member, rising, ‘I conceive that the objection is not well taken; the gentleman’s connection with commerce is close and intimate. He is a commodity.’

— Ambrose Bierce, Fantastic Fables, 1899

| Society

Inspiration

https://commons.wikimedia.org/wiki/File:Rowan_Oak.JPG
Image: Wikimedia Commons

One summer afternoon in 1932, William Faulkner and his wife Estelle were sitting on the side porch of their home in Oxford, Mississippi.

She said, “Does it ever seem to you that the light in August is different from any other time of the year?”

He said, “That’s it!”, disappeared into the house, and returned a moment later.

“What he had done was to go to his worktable and draw four pen strokes through the title ‘Dark House,'” Estelle wrote later. “Above and slightly to the left he printed ‘Light in August.'”

| Literature

New Color

In a 1985 op-ed in the New York Times, writer Maggie Sullivan proposed some irregular verbs to match go, went, gone:

furlough, furlent, furlon: “All the soldiers were furlon except those the captain furlent last week.”
subdue, subdid, subdone: “Nothing else could have subdone him the way her violet eyes subdid him.”
frisbee, friswas, frisbeen: “Although he had never frisbeen before, after watching the tournament he friswas every day, trying to frisbee as the champions friswere.”
pay, pew, pain: “He had pain for not choosing a wife more carefully.”
conceal, console, consolen: “After the murder, Jake console the weapon.”
seesaw, sawsaw, seensaw: “While the children sawsaw, the old man thought of long ago when he had seensaw.”
fit, fat, fat: “The vest fat Joe, whereas the jacket would have fat a thinner man.”
ensnare, ensnore, ensnorn: “In the ’60s and ’70s, Sominex ads ensnore many who had never been ensnorn by ads before.”
displease, displose, displosen: “By the look on her face, I could tell she was displosen.”

Commemorate could emulate eat: “At the banquet to commemoreat Herbert Hoover, spirits were high, and by the end of the evening many other Republicans had been commemoreaten.”

(Maggie Sullivan, “You, Too, Can Strengthen English, and Write Good,” New York Times, May 4, 1985.)

| Language

Math Notes

If we have two numbers a and b such that ab + 1 is square, then it’s always possible to find a number c for which ac + 1 and bc + 1 are both square. For example, 8 × 3 + 1 = 25 = 52, and 8 × 21 + 1 = 169 = 132 and 3 × 21 + 1 = 64 = 82.

Proof:

If ab + 1 = m2, then set c = a + b + 2m. Now

ac + 1 = a2 + ab + 2am + 1 = a2 + 2am + m2 = (a + m)2

bc + 1 = ab + b2 + 2bm + 1 = b2 + 2bm + m2 = (b + m)2

Via Edward Barbeau, Power Play, 1997.

| Science & Math