Catch 22

From reader Chris Smith:

Pick a three-digit number in which all the digits are different. Example: 314.

Now list every possible combination of two digits from the chosen number. In our example, these are 13, 14, 31, 34, 41, and 43.

Divide the sum of these two-digit numbers by the sum of the three digits in the original number, and you’ll always get 22. In our example, (13 + 14 + 31 + 34 + 41 + 43) / (3 + 1 + 4) = 176/8 = 22.

This works because 10a + b, 10a + c, 10b + a, 10b + c, 10c + a, and 10c + b sum to 22a + 22b + 22c = 22(a + b + c), so dividing by a + b + c will always give 22.

(Thanks, Chris.)

06/08/2024 Reader Tom Race points out that essentially the same trick can be performed using the entire number: If you add all six permutations of the original 3 digits, then divide that total by the sum of the 3 digits, the answer is always 222.

For example, using 561:

561 + 516 + 156 + 165 + 651 + 615 = 2664

5 + 6 + 1 = 12

2664 / 12 = 222

“This works because in the first sum each of the three digits (a, b and c) occurs twice in each of the three columns, so the sum is 222a + 222b + 222c = 222(a + b + c).” (Thanks, Tom.)

| Science & Math

Résumé

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

In the 1897 edition of Who’s Who, George Bernard Shaw listed his recreations as “cycling and showing off.”

To H.G. Wells he once wrote, “The longer I live the more I see that I am never wrong about anything, and that all the pains I have so humbly taken to verify my notions have only wasted my time.”

| Literature

Side Line

https://archive.org/details/APM10_2/page/n7/mode/2up?view=theater

The April 1, 1878, issue of the New York Daily Graphic announced that Thomas Edison had invented a “victuals machine” that would feed the human race:

I made all this food out of the dirt taken from the cellar and water that runs through these pipes. … I believe that in ten years my machines will be used to provide the tables of the civilized world. … I can make cabbages and oranges that have never felt the rain. Nature is full of surprises. Bananas and chocolate can be made out of the very same ingredients, and the methods of combining differ only a trifle.

The last paragraph revealed that the story was a hoax, but many readers didn’t get that far — several newspapers picked up the news, and some readers even tried to order the device. Reporter William Augustus Croffut, who’d concocted the tale, wrote diffidently to Edison on April 4 (above), “Did you see my hoax? And are you in a state of fiery wrath? Or how is it?”

| Hoaxes

Ease of Use

https://commons.wikimedia.org/wiki/File:Hangul_chosongul_fontembed.svg
Image: Wikimedia Commons

The Korean alphabet was designed expressly to increase literacy among the country’s uneducated lower classes, who found traditional Chinese characters hard to recognize and understand. King Sejong the Great promulgated the new letters in 1444 to permit the common people to express themselves conveniently in writing.

It’s said that “a wise man can acquaint himself with them before the morning is over, and even a stupid man can learn them in the space of 10 days.”

| Language

Capsized

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

The upside-down catfish, Synodontis nigriventris, is right side up. Or, rather, it’s adapted to spend most of its time upside down — its belly is darker than its back, and it swims fastest in this inverted position. The behavior may have evolved to help it reach food on the undersides of submerged branches or to breathe dissolved oxygen near the surface.

| Science & Math

A Twist

https://commons.wikimedia.org/wiki/File:Al-Jazari_Automata_1205.jpg

Here’s a surprise: The Book of Knowledge of Ingenious Mechanical Devices, a 1206 manuscript by the Turkish author Ismail al-Jazari, depicts a chain pump in the form of a Möbius strip. A rope bearing a chain of cups dips them successively into a water source at the bottom and then pours them into a course at the top. The single, continuous rope makes two passes through this route, describing the edges of a strip with a half twist so that the cups suspended between the loops are turned 180 degrees with each pass. This would permit the cups to last longer, since they’re worn more evenly, and even a broken cup might still convey some water with every second pass.

(Julyan H.E. Cartwright and Diego L. González, “Mobius Strips Before Mobius: Topological Hints in Ancient Representations,” Mathematical Intelligencer 38:2 [June 2016], 69-76.)

| Science & Math · Technology

In a Word

https://commons.wikimedia.org/wiki/File:Hans_Richter_Korrespondenzkarte_mit_Notenzitat_und_Unterschrift_1898.jpg

tracasserie
n. a state of disturbance or annoyance

infamation
n. reproach

alienigenate
adj. born in a foreign country

baragouin
n. language so altered in sound or sense as not to be generally understood

‘It is a fact,’ wrote Stephen Spender, after trying to write a book about interwar Berlin, ‘that all the best German jokes are unconscious.’ He instanced the expostulation of the German conductor Hans Richter after a difficult rehearsal with the London Philharmonic Orchestra: ‘Up with your damned nonsense will I put twice, or perhaps once, but sometimes always, by God, never!’

— Paul Johnson, Humorists, 2011

| Language

Full Service

In his 1967 book Beyond Language, Dmitri Borgmann points out that every permutation of the three words ONE, MAY, and SAW produces a valid English sentence:

  1. ONE MAY SAW. (An individual has the privilege of performing the action of sawing some object, such as a wooden log.)
  2. ONE SAW MAY. (One person saw the girl whose first name is ‘May’.)
  3. MAY ONE SAW? (Is one permitted to saw wood?)
  4. MAY SAW ONE. (A girl named ‘May’ saw some object, previously mentioned, that is regarded as belonging to a group of objects of like character.)
  5. SAW ONE, MAY! (Cut a log of wood in half, May, by sawing through it!)
  6. SAW MAY ONE! (Saw a log of wood for May, Buster!)

In Word Ways, David Morice notes that BILL, PAT, and SUE can produce 12 valid three-word sentences, distinguished by capitalization and comma placement. Each item in the first group corresponds in meaning to one in the second:

Bill, pat Sue.     Pat Sue, Bill.
Bill, sue Pat.     Sue Pat, Bill.
Sue, bill Pat.     Bill Pat, Sue.
Sue, pat Bill.     Pat Bill, Sue.
Pat, bill Sue.     Bill Sue, Pat.
Pat, sue Bill.     Sue Bill, Pat.

(David Morice, “Kickshaws,” Word Ways 26:2 [May 1993], 105-117.)

| Language

Footwork

Albert Einstein used to say that he went to his office at the Institute for Advanced Study “just to have the privilege of walking home with Kurt Gödel.” The two would meet at Einstein’s home each day between 10 and 11 and undertake the half-hour walk to the institute. At 1 or 2 in the afternoon they’d walk back, discussing politics, philosophy, and physics. Biographer Palle Yourgrau estimates that these walks consumed 30 percent of Einstein’s workday.

Einstein’s secretary, Helen Dukas, wrote in 1946, “I know of one occasion when a car hit a tree after its driver suddenly recognized the face of the beautiful old man walking along the street.”

Gödel caused no such problems. “I have so far not found my ‘fame’ burdensome in any way,” he wrote to his mother. “That begins only when one becomes so famous that one is known to every child in the street, as is the case of Einstein.”

(From A World Without Time: The Forgotten Legacy of Godel and Einstein, 2009.)

| Science & Math

“Moral Thermometers”

Utopian socialist Robert Owen opposed corporal punishment, so when he took over the textile mill at New Lanark, Scotland, in 1800, he kept order with a “silent monitor”: Over each worker’s machine was hung a block whose successive sides were painted white, yellow, blue, and black:

The 2,500 toys had their positions arranged every day, according to the conduct of each worker during the preceding day: white indicating superexcellence; yellow, moderate goodness; blue, a neutral condition of morals; and black, exceeding naughtiness.

These ratings were assigned by the departmental overseer, whose own rating was assigned by an under-manager. The final say lay with Owen, to whom workers could appeal, and the daily ratings were recorded in a “book of character” maintained by each department.

This sounds draconian, but combined with Owen’s generous nature it seemed to work. “As time went on,” wrote one biographer, “the yellows and whites gained on the darker hues; and in the later stages of Owen’s management the signs were almost entirely white, with a sprinkling of yellows.”

| Society