Boys’ Club

https://commons.wikimedia.org/wiki/File:Athos_grenze_frangokastro_01.jpg

No women are allowed on Greece’s Mount Athos, the site of 20 Eastern Orthodox monasteries, because they would hinder the monks’ progress toward spiritual enlightenment. Mary alone represents her sex on the mountain.

The ban has been in place since an imperial decree in 1046, with a few colorful exceptions:

  • In the 1300s a Serbian emperor brought his wife to the peninsula to protect her from the plague. She was borne in a hand carriage the whole time, her feet never touching the ground.
  • French writer Maryse Choisy snuck in in the 1920s, disguised as a sailor. She published her adventure under the title Un mois chez les hommes (“A Month With Men”).
  • In 1953 Ohio Fulbright Program teacher Cora Miller landed briefly with two other women, creating a furor.
  • In 2008, five Moldovan migrants arrived by way of Turkey; four were women. The monks forgave them.

The rule extends even to hens, cows, nanny-goats, and sows, which means that dairy products and eggs have to be brought in from outside. Female cats, insects, and songbirds are admitted.

In 2003 the European Parliament passed a resolution saying the ban violated “the universally recognised principle of gender equality,” but it remains in place — even female sightseers must stay at least 500 meters offshore.

“Holes” and Factors

Here are the proper prime divisors of the first nine natural numbers (a proper prime divisor is a prime different from n that divides n evenly):

1: (none)
2: (none)
3: (none)
4: 2 × 2
5: (none)
6: 2 × 3
7: (none)
8: 2 × 2 × 2
9: 3 × 3

So, if we include repeated instances of a given factor:

  • 1, 2, 3, 5, and 7 have 0 proper prime divisors
  • 4, 6, and 9 have 2 proper prime divisors
  • 8 has 3 proper prime divisors

Mathematicians Ana Luzón and Manuel A. Morón of Universidad Politecnica de Madrid point out a coincidence: The numerals in each of these groups have the same basic shape — within each group it’s possible to transform one numeral into another by bending, shrinking, and expanding. So, for example, it’s possible to bend a numeral 1 made of clay into a 2 or a 7, but not into a 9 — we’re not allowed to poke a new hole in the clay or to affix one part of it to another.

Luzón and Morón write that if two of these nine numerals have the same number of proper prime divisors, then those two will “cut a sheet in the same number of pieces if you write them down with a scalpel.” And if the scalpel doesn’t cut the sheet into multiple pieces, then the number you’re writing is prime (except for 1).

Note: This works only if the numeral 4 is “closed” at the top, not open. So this post will make sense if you’re reading it on Futility Closet (which uses the “closed” font Georgia), but possibly not if you’re reading it in a different font elsewhere. Maybe this tells us how 4 “ought” to be written!

(Ana Luzón and Manuel A. Morón, “4 or 4? Mathematics or Accident?” Mathematics Magazine 75:4 [October 2002], 274.)

Evolution

Order a beer at Kayabukiya Tavern, in Japan’s Tochigi prefecture, and it will be brought to you by one of five monkeys. Owner Kaoru Otsuka started his business nearly 30 years ago, but business really took off when he brought his pet macaque to the premises and got it to hand a wet oshibori towel to a customer. Now the monkeys hand out oshibori and beer and perform on a makeshift stage.

Under animal welfare laws the monkeys work only two hours a day, and customers give them boiled soya beans as tips for their service. Now they’ve begun donning human masks and wigs — perhaps they’ll soon be opening a restaurant of their own:

Podcast Episode 151: Double-Crossing the Nazis

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In 1941, Catalonian chicken farmer Juan Pujol made an unlikely leap into the world of international espionage, becoming a spy first for the Germans, then for the British, and rising to become one of the greatest double agents of World War II. In this week’s episode of the Futility Closet podcast we’ll describe Pujol’s astonishing talent for deceiving the Nazis, which led one colleague to call him “the best actor in the world.”

We’ll also contemplate a floating Chicago and puzzle over a winding walkway.

See full show notes …

Math and Pancakes

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

If you apply one straight cut to a pancake, pretty clearly you’ll get 2 pieces. With two cuts, the most you can get is 4. What’s the greatest number you can produce with three cuts? If the cuts meet neatly in the center, you’ll get 6 pieces, but if you’re artfully sloppy you can make 7 (above). Charmingly, this leads us into the “lazy caterer’s sequence” — the maximum number of pieces you can produce with n straight cuts:

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, …

Generally it turns out that the maximum number for n cuts is given by the formula

\displaystyle p = \frac{n^{2} + n + 2}{2};

each number equals 1 plus a triangular number.

A related question is the pancake flipping problem. You’re presented with a spatula and an untidy stack of pancakes of varying sizes. You can insert the spatula at any point in the stack and flip all the pancakes above it. What’s the least number of flips required to sort the pancakes in order of size? Interestingly, no one has found a general answer. It’s possible to work out the solution for relatively small stacks (in which the number of pancakes is 1, 2, 3, …):

0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, …

But no one has found a formula that will tell how many flips will get the job done for a stack of any given size.

The problem has an interesting pedigree. Bill Gates worked on it at Harvard (PDF), and David X. Cohen, who went on to write for The Simpsons and Futurama, worked on a related problem at Berkeley in which the bottom of each pancake is burnt and the sort must be completed with the burnt sides facing down.

CCNY mathematician Jacob Goodman, who first hit on the pancake flipping problem while sorting folded towels for his wife, submitted it to the American Mathematical Monthly under the name Harry Dweighter (“harried waiter”). His household chores have produced at least one other publication: After some thoughtful work with a swivel-bladed vegetable peeler, he published “On the Largest Convex Polygon Contained in a Non-convex n-gon, Or How to Peel a Potato.”

(Thanks, Urzua.)

Riposte

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Chess master Wilhelm Steinitz was having a heated political argument.

His opponent said, “Do you think you understand politics because you can play chess?”

Steinitz said, “Do you think you understand politics because you can’t play chess?”

Edifice Wrecks

giarre, sicily

Between the 1950s and the 2000s, the Sicilian town of Giarre started a series of ambitious building projects as its politicians competed to create jobs and secure funds from the regional government. Unfortunately, there was no need for the buildings — Giarre’s population is only 27,000 — and today the seaside town hosts 25 half-built and abandoned constructions, including an amphitheater, a sports stadium, a polo ground, and a swimming pool.

“Giarre offers the extreme form of a condition found in most cities, making it a parable of urban planning,” writes social geographer Alastair Bonnett in Off the Map. “It is the epicentre not of merely an Italian but a global phenomenon of accreted unfinished visions.”

“Several companies started the projects without the intention of finishing them,” architect Salvo Patane told the BBC. “These were projects started so as not to lose funds that were available from the regional government. More than waste, this was bad politics.”

Community activist Claudia D’Aita wants to reconceive the abandoned constructions as a park — “a kind of open-air museum” — exhibiting a cautionary new architectural subgenre. They would call it the Archaeological Park of Sicilian Incompletion.

Tempest-Tost

In 2011 Royal Caribbean introduced a self-leveling pool table on its cruise ship Radiance of the Seas.

This video was shot during a storm in the Pacific — gyroscopes inside the table keep the playing surface level so a game can continue even as the ship rolls.

In a Word

ullage
n. the amount a container lacks of being full

Given a 5-gallon jug, a 3-gallon jug, and a limitless supply of water, how can you measure out exactly 4 gallons?

Click for Answer