Overheard

A puzzle by Princeton mathematician John Horton Conway:

Last night I sat behind two wizards on a bus, and overheard the following:

A: I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.

B: How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?

A: No.

B: Aha! AT LAST I know how old you are!

“This is an incredible puzzle,” writes MIT research affiliate Tanya Khovanova. “This is also an underappreciated puzzle. It is more interesting than it might seem. When someone announces the answer, it is not clear whether they have solved it completely.”

We can start by auditioning various bus numbers. For example, the number of the bus cannot have been 5, because in each possible case the wizard’s age and the number of his children would then uniquely determine their ages — if the wizard is 3 years old and has 3 children, then their ages must be 1, 1, and 3 and he cannot have said “No.” So the bus number cannot be 5.

As we work our way into higher bus numbers this uniqueness disappears, but it’s replaced by another problem — the second wizard must be able to deduce the first wizard’s age despite the ambiguity. For example, if the bus number is 21 and the first wizard tells us that he’s 96 years old and has three children, then it’s true that we can’t work out the children’s ages: They might be 1, 8, and 12 or 2, 3, and 16. But when the wizard informs us of this, we can’t declare triumphantly that at last we know how old he is, because we don’t — he might be 96, but he might also be 240, with children aged 4, 5, and 12 or 3, 8, and 10. So the dialogue above cannot have taken place.

But notice that if we increase the bus number by 1, to 22, then all the math above will still work if we give the wizard an extra 1-year-old child: He might now be 96 years old with four children ages 1, 1, 8, and 12 or 1, 2, 3, and 16; or he might be 240 with four children ages 1, 4, 5, and 12 or 1, 3, 8, and 10. The number of children increases by 1, the sum of their ages increases by 1, and the product remains the same. So if bus number b produces two possible ages for Wizard A, then so will bus number b + 1 — which means that we don’t have to check any bus numbers larger than 21.

This limits the problem to a manageable size, and it turns out that the bus number is 12 and Wizard A is 48 — that’s the only age for which the bus number and the number of children do not uniquely determine the children’s ages (they might be 2, 2, 2, and 6 or 1, 3, 4, and 4).

(Tanya Khovanova, “Conway’s Wizards,” The Mathematical Intelligencer, December 2013.)

Two similar puzzles: A Curious Conversation and A Curious Exchange.

Head Trauma

Unfortunate newspaper headlines, collected by Robert Goralski for Press Follies, 1983:

TOWN OKS ANIMAL RULE (Asheville Citizen)
TRAVIS MAN DIES AFTER ALTERATION (Sacramento Bee)
INDIAN OCEAN TALKS (The Plain Dealer)
JUVENILE COURT TO TRY SHOOTING DEFENDANT (Deseret News)
TRAIN ROLLS 0 MILES WITH NO ONE ABOARD (New York Times)
LAWMEN FROM MEXICO BARBECUE GUESTS (San Benito [Texas] News)
FLIES TO RECEIVE NOBEL PRIZE (New York Times)
CARTER TICKS OFF BLACK HELP (San Francisco Examiner)
MAULING BY BEAR LEAVES WOMAN GRATEFUL FOR LIFE (Herald-Dispatch, Huntington, W.Va.)
SILENT TEAMSTER GETS CRUEL PUNISHMENT: LAWYER (The Home News, Brunswick, N.J.)
MANCHESTER MAN BURSTS, HALTS TRAFFIC (Hartford Times)
SKELETON TIED TO MISSING DIPLOMAT (Philadelphia Evening Bulletin)
POET DOESN’T WANT AUDIENCE OF ILLERATES (Raleigh Times)
GLASS EYE IS NO HELP IN IDENTIFYING CORPSE (Deseret News)
FORMER MAN DIES IN CALIFORNIA (Freemont County [Calif.] Chronicle News)
MATH IMPROVEMENT INDICATES LEARNING IS TIED TO TEACHING (New York Times)
PAIR CHARGED WITH BATTERY (Denver Post)
TUNA RECALLED AFTER DEATH (Chicago Daily News)
TWO CONVICTS EVADE NOOSE; JURY HUNG (Oakland Tribune)
JERK INJURES NECK, WINS AWARD (Buffalo News)
TEACHERS’ HEAD GOES OFF TO JAIL (Sarasota Herald-Tribune)
SIX SENTENCED TO LIFE IN CLARKSVILLE (Nashville Banner)
POPE LAUNCHES TALKS TO END LONG DIVISION (Pomono Progress Bulletin)
A GRATEFUL NATION BURIES SAM RAYBURN (New York Herald Tribune)
SHOUTING MATCH ENDS TEACHER’S HEARING (Newsday)
DOCTOR TESTIFIES IN HORSE SUIT (Waterbury Republican)

Some are inspired: When the New York Times reported that a mansion-hunting Aristotle Onassis had visited Buster Keaton’s former estate, it chose the headline ARISTOTLE CONTEMPLATING THE HOME OF BUSTER.

Podcast Episode 52: Moving Day in New York

https://commons.wikimedia.org/wiki/File:Goethe_1791.jpg

For centuries, May 1 brought chaos to New York, as most tenants had to move on the same day, clogging the streets with harried people and all their belongings. In this episode of the Futility Closet podcast we’ll review the colorful history of “Moving Day” and wonder how it lasted through two centuries.

We’ll also recount some surprising escapes from sinking ships and puzzle over a burglar’s ingenuity.

Sources for our feature on Moving Day, New York City’s historic custom of changing residence on May 1:

Kenneth A. Scherzer, The Unbounded Community: Neighborhood Life and Social Structure in New York City, 1830-1875, 1992.

Elizabeth Blackmar, Manhattan for Rent, 1785-1850, 1991.

William Shepard Walsh, Curiosities of Popular Customs … Illustrated, 1897.

“Expressmen and Cartmen’s Charges — The Laws Relative Thereto,” New York Times, April 14, 1870.

“Rich Are Homeless This Moving Day,” New York Times, Oct. 1, 1919.

“Rain Adds to Gloom of City Moving Day,” New York Times, Oct. 2, 1919.

“May 1 Moving Rush a Thing of the Past,” New York Times, May 2, 1922.

In 1890 the New York Times published a list of the maximum prices that city ordinances permitted cartmen to charge:

http://www.baruch.cuny.edu/library/alumni/online_exhibits/digital/2007/moving_day/images/015_lg.gif

Sources for our feature on oddities in maritime disasters:

“Andrea Doria Tragedy Recalled by the Survivors,” Associated Press, July 24, 1981.

“A Remarkable Maritime Disaster,” Scientific American, Nov. 24, 1888.

“A Remarkable Collision,” New Zealand Herald, July 26, 1884.

This week’s lateral thinking puzzle was contributed by listener Ken Murphy.

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.

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.

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. And you can finally follow us on Facebook and Twitter. Thanks for listening!

Leisure Travel

https://www.google.com/patents/US400131

H.M. Small found it difficult to sleep in railway seats in 1889 — so he invented a hammock.

It’s actually possible to sleep at full length if the seat in front is pushed forward — but that might be going too far.

In a Word

https://en.wikipedia.org/wiki/File:Death_star1.png

interfector
n. a death-bringing planet

mundicidious
adj. likely or able to destroy the world

In 2012 an online petition urged the Obama administration to build a Death Star like the one in Star Wars. The campaign amassed 25,000 signatures, enough to require an official response, and it fell to Paul Shawcross, chief of the Science and Space Branch at the Office of Management and Budget, to reject the project. He gave three reasons:

  • The construction of the Death Star has been estimated to cost more than $850,000,000,000,000,000. We’re working hard to reduce the deficit, not expand it.
  • The Administration does not support blowing up planets.
  • Why would we spend countless taxpayer dollars on a Death Star with a fundamental flaw that can be exploited by a one-man starship?

Mileage

I leave my front door, run on a level road for some distance, then run to the top of a hill and return home by the same route. I run 8 mph on level ground, 6 mph uphill, and 12 mph downhill. If my total trip took 2 hours, how far did I run?

Click for Answer

Sweet Science

monod flan recipe

Sahara geology presented as a flan recipe, from French naturalist Théodore Monod’s Méharées: Explorations au vrai Sahara, 1937:

Take a flan-tray, which represents the basement (our Mauretanian and Tuareg granites).

  1. Place some pastry in the flan-tray in irregular masses (A) — these are the Precambrian mountain chains, the Saharides.
  2. Level this off with a knife (B) so that the folds, as in erosional peneplanation of the Sahara, are seen only in the ravines which cross the plain; the mountains are now vigorously planed down.
  3. First event: a tap (from which, fortunately, jam flows) floods the garnished mould (C). Similarly the sea at the beginning of the Palaeozoic invaded the Saharan basement, which it then partly occupied, until the middle Carboniferous — what an enormous amount of jam! All this time the Sahara is under water, and sandstones, limestones, conglomerates and shales were deposited — all the sediments of the Tuareg and Mauritanian plateaux.
  4. A new event (the djinns must have been at work here) — the bottom of the flan-tray experiences an uplift; the dish, pastry and jam emerge (D). This is the time of the coal measures; the sea retreats, and the Sahara is left high and dry, basking in the sun.
  5. But whoever says dry land, implies erosion; the sediments rise up, are corroded, and the spoon cuts so deeply that it exposes the jam, pastry, and sometimes even the metal of the flan-tray (E).
  6. And while this continues for millions of years, erosion is unable to evaporate its own debris and the eroded sediments are not washed away to the sea — they just accumulate, and what is lost in some districts is gained by others, whilst gradual infilling continues (F).
  7. Then one fine day, while iguanodons are blundering around in Picardy, and swarms of ammonites are scudding around in the Parisian sea, a second tap is turned on again and adds another layer, this time of cream (for convenience of explanation) (G). The sea re-invades a good part of the Sahara and deposits the usual sediments — Cretaceous and Eocene.
  8. A new retreat of the sea and a new continental phase occur, with customary erosion and deposition (H).
  9. Gradually, the country comes to be like it is today; sprinkle with granular sugar (fresh-water Quaternary deposits), and icing sugar dunes (I).
  10. And there we are! Serve hot or chilled.

“Very well — that will teach me to invent foolish nonsense for my neophyte when it is so easy to explain the influence of Saharidian tectonics on the orientation of Hercynian virgations, the suggestion of angular discordnce separating the basal congomerate of the continental beds from the post-Visean argillites, or more simply the origin of the bowlingite included in the pigeonite andesite with diabase facies of Telig. But I doubt that he would understand it any better …”

Unquote

“Knowledge is knowing that a tomato is a fruit. Wisdom is knowing that a tomato doesn’t belong in a fruit salad.” — Miles Kington

Boom Town

https://books.google.com/books?id=A7pZAAAAYAAJ&pg=PA3

In the summer of 1903, the United States Cartridge Company of Tewksbury, Mass., noticed a stain on the floor of its gunpowder magazine. Apparently the dynamite magazine next door had been leaking nitroglycerine. The company asked the dynamite’s owner, American Powder Mills, to attend to the matter, and on July 29 Cartridge’s powder was loaded onto three wagons and moved a few hundred feet away, and an unlucky foreman named Goodwin entered the building, poured a solution on the stain, and began to sweep it with a broom. The spot began to smoke.

The ensuing blast killed 20 people and flattened a score of houses. “Buildings were shaken and windows broken in hundreds of places within a radius of fifteen miles,” reported New England Magazine. “People as far away as Dedham on the south and the mid-New Hampshire towns on the north, felt the shock and guessed at reckless blasts or earthquakes.”

It appears that the fire had caused the dynamite magazine to explode, which set off the three wagons of gunpowder, which set off a third magazine, leased by the Dupont Powder Company. “The ruin caused by the accident was appalling in its perfection,” notes the report. “Three acres of ground were entirely laid waste, the trees and bushes in a considerable radius being torn and blasted as by a breath from a huge furnace.”

The magazines had been built 30 years earlier, when the area had been remote, and the town had grown up around them. “The only safe assumption is that sooner or later every magazine is bound to explode, and must therefore be kept a safe distance from dwelling houses and other buildings.”

(Thanks, Meredith.)

Black and White

rockwell chess problem

By A.F. Rockwell. White to mate in two moves.

Click for Answer
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