None for All

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

If a lion hunts a herd of antelope, what rules govern the herd’s behavior? One intriguing possibility is known as selfish herd theory: Rather than acting to benefit the group as a whole, each member positions itself so that there’s at least one other animal between it and the predator. This produces a pattern known as a Voronoi tesselation — if each dot in the diagram above is an antelope, then the surrounding colored region is the area that’s closer to that antelope than to any of its neighbors. If a lion enters your cell, then you’re the antelope that’s going to get eaten.

This understanding helps to explain some herd behavior. Each animal wants to make its “domain of danger” as small as possible and to be as far as possible from the predator. Dominant animals tend to get prime positions near the center, subordinate animals get pushed to the fringes, and the whole formation evolves continuously as predator and prey move about.

Studies have shown that groups of fiddler crabs tend to take up Voronoi patterns fairly quickly when a predator first appears, and to huddle together when the danger increases as each tries to reduce its surrounding polygon. This actually leads some to move toward the predator as they try to reach the center and put others between the hunter and themselves. Those that violate the movement rules tend to get picked off, which reinforces the evolutionary strength of the strategy.

Skyward

https://en.wikipedia.org/wiki/File:An_Incident_on_the_Western_Front_Art.IWMART2660.jpg

An aviators’ drinking song from World War I, from James Gilbert’s 1978 anthology Skywriting:

A young aviator lay dying
At the end of a bright summer’s day.
His comrades had gathered around him
To carry his fragments away.

The aeroplane was piled on his wishbone,
His Lewis was wrapped round his head,
He wore a spark plug in each elbow,
‘Twas plain he would shortly be dead.

He spat out a valve and a gasket
As he stirred in the sump where he lay,
And then to his wondering comrades
These brave parting words did he say:

“Take the manifold out of my larynx
And the butterfly valve off my neck.
Remove from my kidneys the camrods;
There’s a lot of good parts in this wreck.

“Take the piston rings out of my stomach,
And the cylinders out of my brain.
Extract from my liver the crankshaft,
And assemble the engine again.

“Pull the longeron out of my backbone,
The turnbuckle out of my ear,
From the small of my back take the rudder —
There’s all of your aeroplane here.”

The Union-Closed Sets Conjecture

A family of sets is said to be union-closed if, given any two sets in the family, their union is as well. Here’s an example:

{}, {1,3}, {2}, {1,2,3}, {1,2,3,4}

Combine any two of those sets and you’ll get a member of the same family.

Now, provided our family is finite and consists of more than just the empty set, is there always an element that’s present in at least half of the sets? In the example above, the answer is yes: The element 2 appears in three of the five sets.

Will this always be the case? Strange to say, no one knows. Though the problem is almost absurdly simple to state, it has remained unsolved since Péter Frankl first posed it in 1979.

Henning Bruhn and Oliver Schaudt survey the state of the inquiry here. “The union-closed sets conjecture still has a bit of a journey ahead of it,” they conclude. “We hope it will be an exciting trip.”

(Thanks, Drake.)

Podcast Episode 93: The Old Flying Days

https://commons.wikimedia.org/wiki/File:%27A_Balloon_in_Mid-Air%27_by_Jules_Tavernier,_1875.jpg

In the early days of English aviation, journalist C.C. Turner seemed to be everywhere, witnessing bold new feats and going on some harrowing adventures of his own. In this week’s episode of the Futility Closet podcast we’ll sample Turner’s record of Edwardian aviation, including his own clumsy first attempt to fly an airplane and a record-setting balloon voyage to Sweden.

We’ll also ponder the nuances of attempted murder and puzzle over a motel guest’s noisemaking.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

Sources for our feature on early aviation in England:

Charles Cyril Turner, The Old Flying Days, 1927.

Charles Cyril Turner, The Marvels of Aviation, 1917.

This week’s lateral thinking puzzle was contributed by listeners J.C. and Brenna Lundberg, who found it in this collection.

Sources for listener mail:

Wikipedia, “Death of Sammy Yatim” (accessed Feb. 2, 2016).

Diana Mehta, “Toronto Cop Found Not Guilty of Murdering Sammy Yatim, But Is Found Guilty of Attempted Murder,” National Post, Jan. 25, 2016.

Jillian Bell, “Forcillo Attempted Murder Verdict Explained,” CBC News, Jan. 25, 2016.

Alyshah Hasham, “Forcillo Guilty of Attempted Murder in Shooting Death of Sammy Yatim,” Toronto Star, Jan. 25, 2016.

Wendy Gillis and Alyshah Hasham, “‘Mystery’ Charge Only One That Sticks in Sammy Yatim Slaying,” Toronto Star, Jan. 25, 2016.

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

As Advertised

http://blogs.reading.ac.uk/merl/2016/02/03/155-year-old-mouse-trap-claims-its-latest-victim/

The Museum of English Rural Life got a surprise on Wednesday — a 155-year-old mousetrap there managed to catch a mouse:

So, this retired rodent had managed to sneak past University of Reading security, exterior doors and Museum staff, and clambered its way up into our Store. Upon finding itself there it would have found the promised land; a mouse paradise laid before it full of straw, wood and textiles. Then, out of thousands of objects, it chose for its home the very thing designed to kill it some 150 years ago: a mouse trap.

The trap was patented in 1861; it bills itself as a “perpetual mouse trap” that “will last a lifetime.” More at the museum’s blog.

(Thanks, Djerrid.)

Foregone Conclusions

https://commons.wikimedia.org/wiki/File:The_White_Rabbit_(Tenniel)_-_The_Nursery_Alice_(1890)_-_BL.jpg

Here’s the opening of Alice’s Adventures in Wonderland:

Alice was beginning to get very tired of sitting by her sister on the bank, and of having nothing to do; once or twice she had peeped into the book her sister was reading.

Choose any of the first 12 words and tap your way forward in the sentence from that point, tapping one word for each letter. So, for example, if you choose the word Alice, which has five letters, you’d tap was, beginning, to, get, and land on very. Then do the same thing with that word, advancing four letters to land on by. If you keep this up you’ll always arrive at the word sister.

That’s from Martin Gardner; the same trick works with “Twinkle, Twinkle, Little Star,” the opening of the Bible, and countless other texts.

It’s less surprising than it seems — it’s based on a principle called the Kruskal count, proffered originally by Rutgers physicist Martin Kruskal as a card trick. In each case various tributaries merge into a common stream that arrives at a predictable destination. Here’s an analysis (PDF).

The Dark Side

https://pixabay.com/en/chess-game-board-intelligence-616836/

In a certain chess position, each row and each column contains an odd number of pieces. Prove that the total number of pieces on black squares is an even number.

Click for Answer

Right of Way

slippers and roosevelt

Theodore Roosevelt’s friend Jacob A. Riis remembers an encounter with Slippers, the White House cat, during a state dinner in 1906:

The dinner was over, and the President, with the wife of a distinguished Ambassador on his arm, led the procession from the state dining-room along the wide corridor to the East Room at the other end of the building, the ambassadors and plenipotentiaries and ministers following, according to their rank in the official world, all chatting happily with their ladies, seeing no cloud on the diplomatic horizon; when all of a sudden the glittering procession came to a halt. There, on the rug, in the exact middle of the corridor, lay Slippers, stretched at full length, and blinking lazily at the fine show which no doubt he thought got up especially to do him honor. The President saw him in time to avoid treading on him, and stopped. His first impulse was to pick Slippers up, but a little shiver of his lady and a half-suppressed exclamation, as he bent over the cat, warned him that she did not like cats, or was afraid, and for a moment he was perplexed. Slippers, perceiving the attention bestowed on him, rolled luxuriously on the rug, purring his delight. No thought of moving out of the path was in his mind.

There was but one other thing to do, and the man who found a way to make peace between Russia and Japan, did it quickly. With an amused bow, as if in apology to the Ambassadress, he escorted her around Slippers, and kept on his way toward the East Room. Whereupon the representatives of Great Britain, and of France, of Germany, and Italy, of all the great empires and of the little kingdoms clear down to the last on the long list, followed suit, paying their respects to Slippers quite as effectually as if the war-ships of their nations had thundered out a salute at an expenditure of powder that would have kept a poor man comfortable for a year, and certainly have scared even a White House cat almost to death.

(From St. Nicholas, January 1908.)