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.)

| Science & Math

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.

See full show notes …

| History · Podcast · Technology

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.)

| Oddities

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).

| Science & Math

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
| Puzzles

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.)

| History

Storm Warning

https://www.google.com/patents/US1046533

How do you design a burglar alarm for people who can’t hear? Arnold Zukor came up with this solution in 1912 — when the burglar opens a door or window, a system of racks and gears opens a faucet and sprays the occupant through a nozzle mounted over the bed. “Alarm is thus given.”

If you leave your house during the day, you can disable the door alarm and direct the nozzle outside. Then if the window is opened, the flow of water will be visible from a distance, “indicating thereby the entrance of unauthorized persons into the building.”

| Technology

Unquote

Further excerpts from the notebooks of English belletrist Geoffrey Madan (1895-1947):

“Worldly faces never look so worldly as at a funeral.” — George Eliot

Wit is a new and apt relation of ideas: humour, of images.

Use of words “vision” and “supremely” an infallible sign of the uneducated.

“No great country was ever saved by good men, because good men will not go the length that may be necessary.” — Horace Walpole

“Precautions are always blamed. When successful, they are said to be unnecessary.” — Benjamin Jowett

“Some speak and write as if they wanted to say something: others as if they had something to say.” — Archbishop Richard Whately

“A rose has no back.” — Chinese reply if you apologize for turning your back

Reasonable to ask young people to be adventurous, to go to the North Pole, say: but religion asks them to start off, without being sure if there is a North Pole.

Three pieces of earnest advice from the Revd. H.J. Bidder, aged 86, after sitting silent, with a crumpled face, all through dinner, and once loudly asking the man opposite who I was:

1. Never drink claret in an East wind.
2. Take your pleasures singly, one by one.
3. Never sit on a hard chair after drinking port.

| Quotations

Grist

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

A few adventures of Dashiell Hammett, who worked for the Pinkerton Detective Agency before turning to fiction:

  • “I know a man who once stole a Ferris-wheel.”
  • “I was once engaged to discharge a woman’s housekeeper.”
  • “I was once falsely accused of perjury and had to perjure myself to escape arrest.”
  • “A man whom I was shadowing went out into the country for a walk one Sunday afternoon and lost his bearings completely. I had to direct him back to the city.”
  • “Three times I have been mistaken for a Prohibition agent, but never had any trouble clearing myself.”
  • “I know an operative who while looking for pickpockets at the Havre de Grace race track had his wallet stolen. He later became an official in an Eastern detective agency.”
  • “I know a detective who once attempted to disguise himself thoroughly. The first policeman he met took him into custody.”
  • “In 1917, in Washington, D.C., I met a young woman who did not remark that my work must be very interesting.”
  • “The chief of police of a Southern city once gave me a description of a man, complete even to a mole on his neck, but neglected to mention that he had only one arm.”

Interestingly, he notes that “the chief difference between the exceptionally knotty problem confronting the detective of fiction and that facing the real detective is that in the former there is usually a paucity of clues, and in the latter altogether too many.”

(“From the Memoirs of a Private Detective,” The Smart Set, March 1923.)

| Crime · Literature