Two poles stand vertically on level ground. One is 10 feet tall, the other 15 feet tall. If a line is drawn from the top of each pole to the bottom of the other, the two lines intersect at a point 6 feet above the ground. What’s the distance between the poles?
A letter from “J. A. McM.,” West Lynn, Mass., to Mark Twain, April 17, 1907:
Apropos of your very entertaining little book on ‘English as she is Taught’ — the following true story fits in well — A teacher asked her class of boys to tell the difference between herself and a clock. A bright little urchin in the rear row raised his hand and said — ‘You have a face and the clock has a face, and you have got hands and the clock has got hands, and — and (reflecting) the clock has got a pendooleum and you aint.’
On the envelope Twain wrote, “Preserve this. Frame it. It is the second time in 40 years that a stranger has done me a courtesy & charged me nothing for it.”
A paradox attributed to Proclus Lycaeus (412-485):
Consider two nonparallel lines, AQ and BP. BP is perpendicular to AB; AQ isn’t. Find the midpoint of AB and mark AC = BD = AB/2. Now if AQ and BP are going to intersect, it can’t happen on AC or BD; if it did, say at a point R, then that would give us a triangle ARB where the sum AR + RB < AB, which is impossible.
But now we can connect CD and follow the same process: CE and DF can't intersect for the same reason. EG and FH are likewise ruled out, and so on up the line forever.
This seems to mean that two nonparallel lines will never intersect. That can’t be right, but where is the error?
(From Alfred Posamentier, Magnificent Mistakes in Mathematics, 2013.)
In 1943, Colorado broom factory worker Mary Babnik Brown saw an advertisement in a Pueblo newspaper soliciting blond hair, at least 22 inches long, that had not been treated with chemicals or hot irons. Brown had never cut her hair, which she combed twice a day and washed twice a week with pure soap. When her samples were deemed acceptable, she cut off all 34 inches and sent it in, considering this her contribution to the war effort, though “I cried for two months.”
At the time she was told that her hair would be used in meteorological instruments. It wasn’t until 1987, the year of her 80th birthday, that she learned that it had been used in the Norden bombsight, a top-secret instrument that guided bombs to their targets. Engineers had determined that fine blond human hair worked ideally in crosshairs, but the technology was a closely guarded secret, so the donors weren’t told how their contributions would be used. “I couldn’t believe it when they told me,” Brown said. “All I knew was that they needed virgin hair.”
She did get some compensation: Pueblo declared Nov. 22, 1991, “Mary Babnik Brown Day,” she was inducted into the Colorado Aviation Historical Society’s hall of fame, and Ronald Reagan sent her this letter:
Since the Futility Closet book came out, a number of readers have asked whether I could provide signed copies. I think I’ve now worked out a way to do this for those who are interested. I can send signed copies to U.S. residents for $25 each, and to those elsewhere for a comparable price once we’ve worked out the shipping. If you’re interested, please write to me at email@example.com. (And thanks for asking — I hadn’t expected this!)
John Cage’s 4’33″ is commonly described as “four and a half minutes of silence,” but in fact it’s the opposite — Cage hoped to lead the audience to hear the ambient sounds of the concert hall as music, to accept as art sounds that they wouldn’t normally consider in that way.
“What they thought was silence, because they didn’t know how to listen, was full of accidental sounds,” he said of the piece’s 1952 premiere. “You could hear the wind stirring outside during the first movement. During the second, raindrops began pattering on the roof, and during the third the people themselves made all kinds of interesting sounds as they talked or walked out.”
In a broad sense 4’33″ was Cage’s most significant work, but the notion of a dedicated piece of art with no substance does introduce some perplexing puzzles. The work debuted as a piano piece with a specified length, but Cage later said that “the work may be performed by any instrumentalist or combination of instrumentalists and last any length of time,” and indeed he produced varying scores in different notations. Can all of these be said to be the same piece?
The “In Futurum” movement for solo piano from Czech composer Erwin Schulhoff’s 1919 Fünf Pittoresken consists entirely of rests, but directs the performer to play “the entire song with as much expression and feeling as you like, always, right to the end!” (French pianist Philippe Bianconi wondered, “Should I just sit there?”) And Alphonse Allais’s 1897 Funeral March for the Obsequies of a Deaf Man, below, consists of 24 blank measures. Could an unwitting audience member distinguish either of these from Cage’s work?
A puzzle by philosopher Patricia Werhane of Loyola University of Chicago: Suppose that a pianist were engaged to perform 4’33″ but had to withdraw at the last moment, and in desperation the stage manager sat in his place. Would this be a performance of Cage’s work? Would it be a musical performance?
Now more than 60 years old, Cage’s idea may still be too novel for a wide public. When BBC Radio 3 broadcast the first U.K. orchestral performance of 4’33″ in 2004, the network had to turn off an emergency backup system that would have interpreted the silence as dead air — and begun playing music.