A Fable

A paper kite, that soared on high,
While looking down was able to descry
A joyous butterfly,
And cried, “We faintly see your foolish efforts zealous;
Confess that you are rather jealous,
When you behold our elevated lot.”
“Jealous? Oh, surely not.
Your grandeur is a phantom, or an empty vision,
Your tethered flight awakes, believe me, but derision;
Can happiness be got from an existence ever fettered?
And how could life for me be bettered?
As I aspire I venture higher,
And never, solely for another’s idle leisure
And pleasure,
Lose freedom I desire.”

— Ivan Krylov

| Literature · Society

Succinct

A puzzle by Emily Cox and Henry Rathvon, from the October 1991 issue of Games magazine:

You’re visiting a familiar island on which each resident either invariably lies or invariably tells the truth. Three residents appear.

Resident 1: We’re all liars.
Resident 2: Just one of us is a truth teller.

The third resident doesn’t speak. Which of the three are telling the truth?

Click for Answer
| Puzzles

“The Lacquer Liquor Locker”

Now once upon a time the King of Astrakhan, at that,
Was sitting on his throne because his throne was where he sat;
And comfortably beside him, and magnificently stocked,
Was a lacquer liquor locker which a liquor lackey locked.

“My boy,” the King would often say with granulated voice,
“I think the 1640 is particularly choice.”
The boy would understand and so, endeavoring to please,
He’d try his luck at fitting several likely locker keys.

The King was always miuch annoyed because of this delay:
“See here, my lad, you’ve got to throw those other keys away.”
“This minute, Sire?” “This minute, sir!” And with a pox that pocked,
He cursed the keys which didn’t keep his liquor locker locked.

The lackey did as he was bid. Alackalasalack!
He threw them all so far away that no one threw them back.
A silly throw, as I can show, for he was simply shocked
To find he lacked the very one that left the liquor locked.

“O Sire, I’ve thrown them all away!” “Look here, my liquor lad!”
“It so befell I threw as well the one I wish I had.”
And since it was the kind of lock that isn’t quickly picked,
The lacquer liquid locker had the little lackey licked.

Unhappy page! In such a rage a king is hard to calm;
A butt or tun of ’51’s the proper kind of balm.
“I always liked my liquor locked, from brandy down to beer;
It might as well be lacquer now as liquor under here.”

Not magic of the magi nor the wisdom of the wise
Could either find the key again or ply where it applies.
The stricken King of Astrakhan soon sickened unto death,
Who tasted not of bitters but what most embittereth.

The little lackey lastly fell into a deep decline,
And evil over all the land to shrivel up the vine:
And now the only vintages of Astrakhan are crocked
In that lacquer liquor locker which a liquor lackey locked.

— David McCord

| Poems

Turnabout

One problem on the 1982 Scholastic Aptitude Test depicted a circle A adjoining a larger circle B. The problem read:

In the figure above, the radius of circle A is one-third the radius of circle B. Starting from position shown in figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A reach its starting point?

(A) 3/2
(B) 3
(C) 6
(D) 9/2
(E) 9

The College Board had expected the answer 3, but three students pointed out that the correct answer is 4. The Board explained, “The circumference of the large circle is three times the circumference of the small circle. If the small circle were to rotate along a straight line segment equal in length to the circumference of the large circle it would make three revolutions.”

“However, the motion of the small circle is not in a straight line, but rather around the large circle. This revolving action around the large circle contributes an extra revolution as circle A rolls around Circle B.” Here’s the same idea using coins of equal size:

https://commons.wikimedia.org/wiki/File:Coin_reverse_rotation_paradox.gif
Image: Wikimedia Commons

Math scores for some 300,000 students had to be recalculated. See The Pup Tent Problem.

| Science & Math

Co-Existence

https://commons.wikimedia.org/wiki/File:Fractal_Africa_1_by_Hamid_Naderi_Yeganeh.png
Image: Wikimedia Commons

Above: A fractal Africa, by Iranian mathematical artist Naderi Yeganeh. The areas of the octagons in adjacent tiers stand in the golden ratio, and their numbers increase according to the Fibonacci sequence.

Below: A tessellating Texas, by Wikimedia user Anachronist. “After having seen a plaza using Texas-shaped flagstone tiling many years ago, I decided to reproduce this tessellation.” Each Texas shape is a 12-sided polygon.

https://commons.wikimedia.org/wiki/File:Texas_tessellation.svg
Image: Wikimedia Commons
| Art · Science & Math

Self-Study

In Abraham Verghese’s 2009 novel Cutting for Stone, a character declares that 10213223 “is the only number that describes itself when you read it. ‘One zero, two ones, three twos, and two threes!'”

Refining that a bit, we can imagine a number N in which the first digit counts the number of zeroes in N, the second counts the number of ones, and so on. Such numbers are called autobiographical, and they’re remarkably few: 1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000.

But numbers needn’t look only inward — MIT mathematician Tanya Khovanova identified these “mutually praising” pairs, which inventory one another’s contents:

6300000100 and 7101001000
130 and 1101
2210 and 11200

“These numbers are too shy to advertise themselves,” she writes. “But Alice praises Bob, because Bob praises Alice. It’s a very advantageous flattery pattern!”

(Tanya Khovanova, “Autobiographical Numbers,” arXiv preprint arXiv:0803.0270 [2008]. Somewhat similar: John Conway’s “look and say” sequence.)

| Science & Math

Scrap Mettle

In 1882, Lewis Carroll invented a game he called MischMash:

The essence of this game consists in one player proposing a ‘nucleus’ (i.e., a set of two or more letters, such as GP, EMO, IMSE), and in the other trying to find a ‘lawful word’ (i.e., a word known in ordinary society and not a proper name), containing it. Thus, ‘maGPie,’ ‘lEMOn,’ and ‘hIMSElf’ are lawful words containing the nuclei GP, EMO, and IMSE.

He sent these nuclei to his godson. What English words contain these fragments?

OLS
VY
NGU
EWH
STEN

Click for Answer
| Puzzles

The Mind’s Eye

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

I can see the stairs down which Smerdyakov tumbles. Dostoevsky gives us a certain description. As a result, I know exactly what this staircase looks like. I know where it starts, how it gets darker and then turns to the left. All this is clear to me in the most concrete way and yet I also know that no one else ‘sees’ the staircase the way I do. But anyone who is receptive to this masterly narrative will ‘see’ the staircase in a most specific way and be convinced that he sees it as it really is.

— Hans-Georg Gadamer, The Relevance of the Beautiful and Other Essays, 1986

| Literature

Misc

https://commons.wikimedia.org/wiki/File:Taft_at_Mitchell.jpg

  • TAFT: FAT is a palindrome. (Dmitri Borgmann)
  • Valencia is the name of the third-largest city in both Venezuela and Spain.
  • Rearrange COORDINATES and you get DECORATIONS.
  • 32759 = (3 – (2 – 7))5 – 9
  • “Nonbeing must in some sense be, otherwise what is it that there is not?” — W.V.O. Quine
| Language · Oddities · Science & Math · Trivia