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
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.
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.”
That’s the final entry in a minutes book discovered in May 2015 at the YMCA Buffalo Niagara in Buffalo, N.Y. The management committee of the local railroad department had met there in December 1899.
Who knows what it means? University at Buffalo archivist-in-training Matthew Oliver found it while reorganizing the YMCA’s records. Details are here.
02/20/2017 A number of readers have written in with a likely answer: The reference is to 1 Samuel 7:12, “Then Samuel took a stone and set it up between Mizpah and Shen. He named it Ebenezer, saying, ‘Thus far the LORD has helped us.'” The Israelites erected the stone to commemorate their victory over the Philistines. This meeting took place during the Third Great Awakening, when the reference would have been well understood.
Reader Phil Moberg Jr. writes, “The ‘Railroad Ys,’ as they were known to those of us in the business, were a great improvement in the general living conditions to crews between runs, being a more than welcome change from the seedy flophouses and saloons that preceded them. The last of them in Southern New England closed in the early ’70s, with the building that housed the New Haven (CT) Railroad Y having been torn down late last year.”
(Thanks also to Delyth Yabar and Anthony Douglas.)
Reader Joe Antognini sent this in: Brazilian mathematician Inder Taneja has found a way to render every number from 1 to 11,111 by starting with either of these strings:
1 2 3 4 5 6 7 8 9
9 8 7 6 5 4 3 2 1
and applying any of the operations addition, subtraction, multiplication, division, and exponentiation. Brackets are permitted. For example:
6439 = 1 + 2 × (34 × 5 × 6 + 789)
and
6439 = 9 × (8 + 7 + 6) + 54 × (32 + 1)
Intriguingly, there’s one hole: There doesn’t seem to be a way to render 10958 from the increasing sequence.
Taneja’s paper is here. (Thanks, Joe.)
01/29/2017 UPDATE: Taneja tells me that, while it can’t be calculated using only basic operations, 10958 can be reached using factorials or square roots. Here are two factorials:
10958 = 1 + 2 + 3!! + (-4 + 5! + 6 – 7) × 89
10958 = 1 × 2 × (3!! – 4! × (5 + 6) + 7! – 8 – 9)
U.S. senator Alan Cranston once lost a copyright suit to Adolf Hitler. Cranston, who had begun his career in journalism, spotted an abridged translation of Mein Kampf in a New York bookstore in 1939. He had read the full text in German and was concerned that the English adaptation omitted Hitler’s anti-Semitism and ambitions to dominate Europe.
To publicize the truth, Cranston worked with a friend to publish an anti-Nazi version of the book. “I wrote this, dictated it [from Hitler’s German text] in about eight days, to a battery of secretaries in a loft in Manhattan,” Cranston told the Los Angeles Times in 1988. They produced a tabloid edition of 32 pages, reducing Hitler’s 270,000 words to 70,000 to yield a “Reader’s Digest-like version [showing] the worst of Hitler.”
At 10 cents apiece, Cranston’s version sold half a million copies in 10 days. But by that time the original was a best-seller in Germany, and the publishers sued Cranston for undercutting the market. In June the U.S. Circuit Court of Appeals in New York ordered the presses stopped. The truth had gotten out, Cranston said, but “we had to throw away half a million copies.”
Should parents be licensed? We ask teachers to study full-time for years and to pass qualifying exams before we let them educate children for six hours a day. And we carefully assess the suitability of adoptive and foster parents. But anyone has the right to become a biological parent without any training at all in child development.
Philosopher Peg Tittle writes, “How many children have been punished because they could not do what their parents mistakenly thought they should be able to do at a certain age — remember X, carry Y, say Z? How many have been disadvantaged because they grew up on junk food — for their bodies as well as their minds? How many have been neglected because their parents didn’t notice the seeds of some talent?”
Today’s children are tomorrow’s citizens, so the public has a legitimate concern in this. Psychiatrist Jack Westman writes, “The way children are parented plays a vital role in the quality of all our lives. We no longer can afford to avoid defining and confronting incompetent parenting.”
Psychologist Roger McIntire writes, “We already license pilots, salesmen, scuba divers, plumbers, electricians, teachers, veterinarians, cab drivers, soil testers, and television repairmen. … Are our TV sets and toilets more important to us than our children?”
(Peg Tittle, ed., Should Parents Be Licensed?, 2004.)
I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules — (1) Use mathematics as a shorthand language, rather than an engine of inquiry. (2) Keep to them until you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can’t succeed in 4, burn 3. This last I did often.
— Alfred Marshall, in a letter to A.L. Bowley, Jan. 27, 1906
In 1919 a bizarre catastrophe struck Boston’s North End: A giant storage tank failed, releasing 2 million gallons of molasses into a crowded business district at the height of a January workday. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Boston Molasses Disaster, which claimed 21 lives and inscribed a sticky page into the city’s history books.
We’ll also admire some Scandinavian statistics and puzzle over a provocative Facebook photo.
Our legal system assumes that a defendant is innocent until proven guilty beyond a reasonable doubt. But what constitutes a reasonable doubt? Law professors Ariel Porat and Alon Harel suggest that an “aggregate probabilities principle” might help to determine whether an accused party is innocent or guilty.
Suppose we’ve decided that the evidence must indicate a probability of 95 percent guilt before we’re willing to declare a defendant guilty. Mr. Smith is accused of two separate crimes, with a 90 percent probability of guilt in each case. Under the 95 percent rule he’d be acquitted of both crimes. But Porat and Harel point out that there’s a 10 percent chance that Smith is innocent of each crime, and aggregating the probabilities gives a 0.10 × 0.10 = 0.01 chance that Smith is innocent of both — that is, there’s a 99 percent chance that he’s guilty of at least one of the offenses.
On the other hand, consider Miller, who is also accused of two different crimes. Suppose that the evidence gives a 95 percent probability that he committed each crime. Normally he’d be convicted of both offenses, but aggregating the probabilities gives a 0.95 × 0.95 = 0.9025 chance that he’s guilty of both offenses, and hence he’d be acquitted of one.
In A Mathematical Medley (2010), mathematician George Szpiro points out that this practice can produce some paradoxical outcomes. Peter and Paul are each accused of a crime, each with a 90 percent chance of being guilty. Normally both would be acquitted. But suppose that each was accused of a similar crime in the past, Peter with a 90 percent chance of guilt and Paul with a 95 percent chance. Accordingly Peter was acquitted and Paul went to prison. But historically Peter has now been accused of two crimes, with a 90 percent chance of guilt in each case; according to the reasoning above he ought to be convicted of one of the two crimes and hence ought to go to jail today. Paul has also been accused of two crimes, with a 0.95 × 0.90 = 0.855 chance that he’s guilty of both. He’s already served one prison term, so the judge ought to acquit him today.
Szpiro writes, “Thus we have the following scenario: in spite of the evidence being identical, the previously convicted Peter is acquitted, while Paul, with a clean record, is incarcerated.”
(Ariel Porat and Alon Harel, “Aggregating Probabilities Across Offences in Criminal Law,” Public Law Working Paper #204, University of Chicago, 2008; George Szpiro, A Mathematical Medley, 2010.)