Squaring Words

In his 1864 autobiography Passages From the Life of a Philosopher, Charles Babbage describes an “amusing puzzle.” The task is to write a given word in the first rank and file of a square and then fill the remaining blanks with letters so that the same four words appear in order both horizontally and vertically. He gives this example with the word DEAN:

D E A N
E A S E
A S K S
N E S T

“The various ranks of the church are easily squared,” he writes, “but it is stated, I know not on what authority, that no one has yet succeeded in squaring the word bishop.”

By an unlikely coincidence I’ve just found that Eureka put this problem to its readers in 1961, and they found three solutions:

B I S H O P    B I S H O P    B I S H O P
I L L U M E    I N H E R E    I M P A L E
S L I D E S    S H A R P S    S P I N E T
H U D D L E    H E R M I T    H A N G A R
O M E L E T    O R P I N E    O L E A T E
P E S E T A    P E S T E R    P E T R E L

The first was found by A.L. Cooil and J.M. Dagnese; the second by A.R.B. Thomas; and the third by R.W. Payne, J.D.E. Konhauser, and M. Rumney.

12/10/2023 UPDATE: Reader Giorgos Kalogeropoulos has enlisted a database of 235,000 words to produce more than 100 bishop squares (click to enlarge):

Kalogeropoulos bishop squares

This is pleasing, because it’s a road that Babbage himself was trying to follow in the 19th century, laboriously cataloging the contents of physical dictionaries after an algorithm of his own devising — see page 238 in the book linked above. (Thanks, Giorgos.)

Dressing Old Words New

Trained on 11,000 early modern texts in English, French, and Latin, MonadGPT is a chatbot with the language and style (and limited knowledge) of the 17th century:

User: What are the planets of the solar system?

Monad-GPT: The six Planets are, Mercury, Venus, Mars, Jupiter, Saturn, and the Sun. The Moon is also a Planet; which being of so great Influence upon Earth, makes it to be most admired of all others by Astronomers.

User: Why has the Moon such an influence upon Earth?

Monad-GPT: But the Moon is so neer to us, as almost to be part of Earth; and therefore it much influences our Atmosphere. This Influence makes all that are in any way affected by Air, to be more under her influence than any other Planet. For if there were a little more or less Matter in the Air, than there is now, we should never have such Plants and Creatures as there are upon Earth: for every Plant and Creature hath its proper Temperature necessary for its Existence. If the Sun did not make up this shortage by his warmth from without, we should be destroyed.

User: Can you write a poem about the Moon?

Monad-GPT: O fairest Moon, that from the Heav’n dost shine, / In her own Lustre all the World does find: / As in thine Orb, where all is bright and clear, / The Sun himself is not more pure than thee.

User: How could we travel to the Moon? With a big canon?

Monad-GPT: But how to travel from Earth to the Moon, with our present Engines? A very hard Question: for we are like those that would go to Heaven in a Basket: We have no Wings, and God hath not given us any Angels to bear us up. Therefore I will leave this Enquiry to others; and end my Discourse of the Moon.

Via MetaFilter.

A Self-Descriptive Crossword Puzzle

From Lee Sallows:

Can you complete the ‘self-descriptive crossword puzzle’ at left below? As in the solution to a similar puzzle seen at right, each of its 13 entries, 6 horizontal, 7 vertical, consists of an English number name folowed by a space followed by a distinct letter. The number preceding each letter describes the total number of occurrences of the letter in the completed puzzle. Hence, in the example, E occurs thirteen times, G only once, and so on, as readers can check. Note that the self-description is complete; every distinct letter is counted.

Though far from easy, the self-descriptive property of the crossword enables its solution to be inferred from its empty grid using reasoning based on orthography only.

sallows self-descriptive crossword

Click for Answer

Words and Numbers

https://pixabay.com/illustrations/pi-circle-diameter-circumference-1338559/

Strategies to memorize π sometimes rely on devising a sentence with words of representative lengths. Isaac Asimov offered this one:

How I want a drink, alcoholic, of course, after the heavy lectures involving quantum mechanics!

Count the letters in each word and you get 3.14159265358979. That’s not the best strategy, though: What shall we do about zero? And it’s hard to reel off the digits impressively for friends when you have to stop and count the letters in each word.

Arthur Benjamin, a mathematician at Harvey Mudd College, suggests using a phonetic code instead:

1 = t or th or d
2 = n
3 = m
4 = r
5 = l
6 = sh, ch, or j
7 = k or hard g
8 = f or v
9 = p or b
0 = s or z

Once we’ve memorized this list, we can turn a whole string of digits into a single word simply by inserting vowels between the consonants. So, for example, 314 could be represented by meter, motor, meteor, matter, mother, etc. The consonants h, w, and y, which don’t appear in the list, can be used like additional vowels.

Using this method, it takes only three sentences to encode the first 60 digits of π:

3. 1415 926 5 3 58 97 9 3 2 384 6264
My turtle Pancho will, my love, pick up my new mover Ginger.

3 38 327 950 2 8841 971
My movie monkey plays in a favorite bucket.

69 3 99 375 1 05820 97494
Ship my puppy Michael to Sullivan’s backrubber!

Now we just need a way to remember the sentences …

(Arthur T. Benjamin, “A Better Way to Memorize Pi: The Phonetic Code,” Math Horizons 7:3 [February 2000], 17.)

Proof Without Words

schaer proof

In the January 2001 issue of the College Mathematics Journal, University of Calgary mathematician Jonathan Schaer offers this simple proof that arctan 1 + arctan 2 + arctan 3 = π. The sum of the angles of this large triangle is 180°. And the diagram shows that its lower left angle is arctan 3, its lower right angle is arctan 2, and its top angle is part of an isosceles right triangle. So arctan 1 + arctan 2 + arctan 3 = 180°, or π radians. “No words, even no symbols!”

(“Miscellanea,” College Mathematics Journal 32:1, 68-71.)

Last Words

On Sept. 8, 1880, an explosion tore through the Seaham Colliery, a coal mine in County Durham in the North of England, filling both shafts with debris and trapping scores of men in the burning seams. Most of the 164 dead were found where they had been working, suggesting that death had come quickly, but some of those farther from the shafts appear to have survived for hours or even longer in pockets of air — the oil in some of their lamps was exhausted. Some of these men had had time to leave messages to those who might find them:

September 8 1880
E.Hall and J.Lonsdale died at half-past 3 in the morning.W.Murray and W.Morris and James Clarke visited the rest on half-past nine in the morn and all living in the incline,
Yours truly,
W.Murray, Master-Shifter

Five o’ clock, we have been praying to God

The Lord has been with us, we are all ready for heaven – Ric Cole, half past 2 o’ clock Thursday

Bless the Lord we have had a jolly prayer meeting, every man ready for glory. Praise the Lord. Sign.R.Cole

Dear Margaret,
There was 40 of us altogether at 7am. Some was singing hymns but my thoughts was on my little Michael that him and I would meet in heaven at the same time. Oh Dear wife, God save you and the children, and pray for me … Dear wife, Farewell. My last thoughts are about you and the children. Be sure and learn the children to pray for me. Oh what an awful position we are in ! Michael Smith, 54 Henry Street

The last had been scratched on a tin water bottle with a rusty nail by Michael Smith, who had left his dying infant son to go to work in the mine that morning. The son died on the same day.

(From Durham Records Online.)

A Word for Everything

In the 17th century natural philosopher John Wilkins set out to create a universal language for scholars, to replace Latin, whose sometimes arbitrary features made it difficult to learn. His solution, laid out in the 1668 Essay Towards a Real Character, is a system of symbols that could classify every thing and idea in the world, somewhat like the Linnaean system in biology. There are 40 main genera, each of which supplies the first two letters of a word. Each genus is divided into “differences,” which supply the next letter. And a “species” gives the fourth letter. So, for example, Zi indicates the genus “beasts,” or mammals; Zit specifies the difference “rapacious beasts of the dog kind”; and Zita gives the species “dogs.” The symbols weren’t necessarily meant to be spoken, though Wilkins later assigned phonetic values to the various characters — rather they were meant to provide an unambiguous way of classifying the world so that scholars (and, perhaps, diplomats, travelers, and merchants) could communicate clearly.

Wilkins presented the system to the Royal Society, who were impressed but concluded that it was impractical — its descriptions of similar (and confusable) items might differ by a single letter, and it would be difficult to remember all the distinctions, which seemed to invite trouble. Borges later lampooned it with his (fictional) Celestial Emporium of Benevolent Knowledge, which divides animals into 14 categories:

  1. Those that belong to the emperor
  2. Embalmed ones
  3. Those that are trained
  4. Suckling pigs
  5. Mermaids (or Sirens)
  6. Fabulous ones
  7. Stray dogs
  8. Those that are included in this classification
  9. Those that tremble as if they were mad
  10. Innumerable ones
  11. Those drawn with a very fine camel hair brush
  12. Et cetera
  13. Those that have just broken the flower vase
  14. Those that, at a distance, resemble flies

(Thanks, Alex.)

Last Word

https://commons.wikimedia.org/wiki/File:Avro_Avro_691_Lancastrian_3_G-AGWH_cn_1280_%27Stardust%27_BSAA_(British_South_American_Airways)_(15215624954).jpg

In August 1947 a British South American Airways airliner en route from Buenos Aires to Santiago crashed into Mount Tupungato in the Argentine Andes. The wreckage wasn’t located for 50 years, but it’s believed today that, hindered by the jet stream, the pilots had started their descent before they’d cleared the mountaintops and crashed into Tupungato.

The last Morse code message received by the Santiago airport was “ETA SANTIAGO 17.45 HRS STENDEC.” The operator didn’t recognize the last word and asked that it be sent again. The flight transmitted STENDEC twice more and then was lost. The meaning of that last transmission is unknown — despite much speculation, it’s never been definitively explained.