Watching the Detectives

Police exist, and sometimes they scrutinize other members of the constabulary. We might say Police police police. If the observed officers are already being observed by a third set of officers, then we could say Police police police police police, that is, “Police observe police [whom] police police.”

The trouble is that if you say this sentence, “Police police police police police,” to an innocent friend, she might take you to mean “Police [whom] police police … police police.” Police police police police police has one verb, police, and two noun phrases, Police and police police police, and without some guidance there’s no way to tell which noun phrase is intended to begin and which to end the sentence.

It gets worse. Suppose we add two more polices: Police police police police police police police. Now do we mean “Police [whom] police observe observe police [whom] police observe”? Or “Police observe police [whom] police whom police observe observe”? Or something else again?

In general, McGill University mathematician Joachim Lambek finds that if police is repeated 2n + 1 times (n ≥ 1), then the numbers of ways in which the sentence can be parsed is \frac{1}{\left ( n + 1 \right )}\binom{2n}{n} , the (n + 1)st Catalan number.

Buffalo have their own troubles.

(J. Lambek, “Counting Ambiguous Meanings,” Mathematical Intelligencer 30:2 [March 2008], 4.)

| Language · Science & Math

The Mengenlehreuhr

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

Further to Saturday’s triangular clock post, reader Folkard Wohlgemuth points out that a “set theory clock” has been operating publicly in Berlin for more than 40 years. Since 1995 it has stood in Budapester Straße in front of Europa-Center.

The circular light at the top blinks on or off once per second. Each cell in the top row represents five hours; each in the second row represents one hour; each in the third row represents five minutes (for ease of reading, the cells denoting 15, 30, and 45 minutes past the hour are red); and each cell in the bottom row represents one minute. So the photo above was taken at (5 × 2) + (0 × 1) hours and (6 × 5) + (1 × 1) minutes past midnight, or 10:31 a.m.

Online simulators display the current time in the clock’s format in Flash and Javascript.

If that’s not interesting enough, apparently the clock is a key to the solution of Kryptos, the enigmatic sculpture that stands on the grounds of the CIA in Langley, Va. In 2010 and 2014 sculptor Jim Sanborn revealed to the New York Times that two adjacent words in the unsolved fourth section of the cipher there read BERLIN CLOCK.

When asked whether this was a reference to the Mengenlehreuhr, he said, “You’d better delve into that particular clock.”

| Puzzles · Science & Math

Extreme Measures

Zürich has a singularly violent way to welcome summer: They roast a snowman until its head explodes.

At the spring holiday Sechseläuten, traditionally celebrated on the third Monday in April, residents build an effigy of winter in the shape of a giant snowman known as the Böögg, pack it with explosives, and set it afire.

“It is believed the shorter the combustion, the hotter and longer summer will be,” writes Bob Eckstein in The History of the Snowman. “When the head of the snowman explodes to smithereens, winter is considered officially over.”

The shortest time on record is 5 minutes 7 seconds, in 1974. The longest, just last year, is 43 minutes 34 seconds.

| Oddities

Podcast Episode 137: The Mystery of Fiona Macleod

https://commons.wikimedia.org/wiki/File:Croix_celtiques_sur_Inisheer,_%C3%AEles_d%27Aran,_Irlande.jpg
Image: Wikimedia Commons

When the Scottish writer William Sharp died in 1905, his wife revealed a surprising secret: For 10 years he had kept up a second career as a reclusive novelist named Fiona Macleod, carrying on correspondences and writing works in two distinctly different styles. In this week’s episode of the Futility Closet podcast we’ll explore Sharp’s curious relationship with his feminine alter ego, whose sporadic appearances perplexed even him.

We’ll also hunt tigers in Singapore and puzzle over a surprisingly unsuccessful bank robber.

See full show notes …

| Literature · Podcast

Triangular Time

http://rmm.ludus-opuscula.org/PDF_Files/Pretz_BinaryClock_5_7(5_2016)_low.pdf

Aachen University physicist Jörg Pretz has devised a binary clock in the shape of a triangular array of 15 lamps. Here’s how to read it:

  • When lit, the top lamp denotes 6 hours.
  • Each lamp on on the second row denotes 2 hours.
  • Each lamp on on the third row denotes 30 minutes.
  • Each lamp on on the fourth row denotes 6 minutes.
  • Each lamp on on the fifth row denotes 1 minute.

So the clock above shows 6 hours + (2 × 2 hours) + (2 × 30 minutes) + (3 × 1 minute) = 11:03. The lamps’ color, red, shows that it’s after noon, or 11:03 p.m. The same array displayed in green would mean 11:03 a.m. A few more examples:

http://rmm.ludus-opuscula.org/PDF_Files/Pretz_BinaryClock_5_7(5_2016)_low.pdf

The time value assigned to each lamp is the total time value of the row below if that row contained one additional lamp.

On each row the lamps light up from left to right, so a row with n lamps can display n + 1 states (all lamps off to all lamps on). So for a triangular array with n lamps on the bottom row, the total number of states is

(n + 1) × ((n – 1) + 1) × ((n – 2) + 1) × · · · × (1 + 1) = (n + 1)!

That is, it’s a factorial of a natural number. And by a happy coincidence, the total number of minutes in 12 hours is such a factorial (720 = 6!).

“Thus the whole concept works because our system of time divisions is based on a sexagesimal system, dating back to the Babylonians, rather than a decimal system, as proposed during the French Revolution.”

There’s more info in Pretz’s article, and you can play with the clock using this applet.

(Jörg Pretz, “The Triangular Binary Clock,” Recreational Mathematics Magazine, March 2016.)

| Science & Math

“The Song of the Yellow Cork”

A golden cork is, mirror-wise,
shown by a polished shelf;
yet, even if endowed with eyes,
it could not see itself.

This is because it stands aligned
with its reflected view;
but if it sideways is inclined,
such is no longer true.

O man, suppose you did reflect
straight up, let’s say, in space:
Would this not have the same effect
as in the stated case?

— Christian Morgenstern, 1905

| Poems

The Perplexed Cellarman

dudeney cellarman puzzle

One last puzzle from Henry Dudeney’s Canterbury Puzzles:

Abbott Francis sends for his cellarman and complains that a particular bottling of wine is not to his taste. He asks how many bottles he had produced. The cellarman tells him that there had been 12 large and 12 small bottles, and that 5 of each have been drunk. The abbot replies that three men are waiting at the gate, and orders the cellarman to give each of them some combination of full and empty bottles so that each man receives the same quantity of wine and combination of bottles.

How can the cellarman do this? He has seven large and seven small bottles full of wine, and five large and five small bottles that are empty. A large bottle holds twice as much wine as a small one, but a large bottle when empty is not worth two small ones — hence the abbot’s order that each man must take away the same number of bottles of each size.

Click for Answer
| Puzzles

New Sounds

The Italian Futurist painter Luigi Russolo had no training as a composer, but in 1913 he argued that music had become “a fantastic world superimposed on the real one,” a collection of “gentle harmonies” that pursued “purity, limpidity and sweetness of sound” but had nothing to do with the real world.

He proposed that “this limited circle of pure sounds must be broken, and the infinite variety of ‘noise-sound’ conquered.” “We find far more enjoyment in the combination of the noises of trams, backfiring motors, carriages and bawling crowds than in rehearing, for example, the ‘Eroica’ or the ‘Pastoral’.”

Accordingly he invented a new set of experimental instruments, the intonarumori, or “noise makers.” There were 27 varieties, all acoustic. Typically a performer turned a handle that rattled or bowed a set of strings, and the surrounding box and horn amplified the sound.

When Russolo and Filippo Tommaso Marinetti debuted their “noise orchestra” in April 1914, it caused a riot, but Russolo was undisturbed. “I am not a musician,” he wrote. “I have therefore no acoustical predilections, nor any works to defend.”

| Art