Which of the two shaded areas is larger, the central disc or the outer ring?
Surprisingly, they’re equal. Each of the concentric circles has a radius 1 unit larger than the last. So the area of the central disc is π × 32 square units, and the area of the outer ring is π × 52 – π × 42 = π × 32 square units. So the two areas are the same.
A brewery stored its beer in a cellar some distance from the bottling plant. The cellar was cooled by pipes that circulated a saline solution from a central cooling unit. The main pipe that connected this cooling unit and the cellar happened to pass near the cellar of a retailer.
The brewery’s owner eventually discovered that the retailer was using the saline solution to cool his own cellar. He sued the retailer for theft, but the judge ruled, “In accordance with Article 242 of the Criminal Code, theft is the unlawful appropriation of commodities belonging to another party. In the present case no theft has been committed, since the saline solution was not misappropriated; rather, it was returned in its entirety to the brewery’s main pipe.”
The brewery owner appealed the case, arguing, “The issue is not the theft of saline solution but the theft of energy. If the saline solution is used to cool the defendant’s cellar in addition to my own, I have to pay more for electricity to operate the central cooling unit.”
The court of appeals ruled: “The saline solution acquires heat from the retailer’s cellar; therefore, energy belonging to the brewery is not being stolen. On the contrary, the brewery is receiving gratuitous energy from the retailer.”
This story appeared in a German scientific monograph, “Questions of Thermodynamical Analysis,” by P. Grassman. In propounding it in May 1990, Quantum added, “We all agree the judge was wrong, but not everyone can correctly explain his error. Can you?”
Suppose there are an infinite number of Grim Reapers. Each has an appointed time to kill Fred if it finds him alive.
The last Grim Reaper (call it #1) is appointed to do this exactly one minute after noon. The next-to-last (#2) is appointed to do it one half minute after noon. And so on: If it finds him alive, Reaper n will kill Fred exactly 1/2(n-1) minutes after noon.
Thus there is no first Reaper. For any given Reaper, there are infinitely many others who precede it by moments.
Whatever happens, we know that Fred can’t survive this ordeal — to go on with his life he must still be alive at 12:01, and we know for certain that if he lives that long then Reaper #1 will kill him. But in order to survive to 12:01 he must still be alive at 30 seconds after 12 — and at that time Reaper #2 will kill him. And so on. It appears that no Reaper will ever get the chance to kill Fred, because each is preceded by another who will rob him of the opportunity.
So it’s impossible that Fred survives, but it’s also impossible that any Reaper kills him. Must we say that he dies for certain but of no cause?
(From José Benardete’s Infinity: An Essay in Metaphysics, 1964.)
Mathematician Norbert Wiener entered university at age 11 and earned a doctorate at 17, but he was 7 years old before he learned that Santa Claus does not exist. From his 1953 memoir Ex-Prodigy:
“At that time I was already reading books of more than slight difficulty, and it seemed to my parents that a child who was doing this should have no difficulty in discarding what to them was obviously a sentimental fiction. What they did not realize was the fragmentariness of the child’s world.”
In his 1909 autobiography Memories of My Life, Francis Galton remembers a boarding school to which he was sent at age 8:
“In that room was a wardrobe full of schoolbooks ready for issue. It is some measure of the then naïveté of my mind that I wondered for long how the books could have been kept so fresh and clean for nearly two thousand years, thinking that the copies of Caesar’s Commentaries were contemporary with Caesar himself.”
In Fragments of Genius, his 1989 survey of the feats of idiots savants, Michael Howe notes that a study of 8-year-olds who were exceptional chess players showed that they were perfectly normal in other spheres. “And the transcripts of interviews in which highly gifted young adults talk about their childhoods, supplemented by interviews with their parents, are full of testimonies to the extreme ordinariness of the individuals, outside their particular area of special talent.”
“The first author would like to acknowledge and thank Jesus Christ, through whom all things were made, for the encouragement, inspiration, and occasional hint that were necessary to complete this article. The second author, however, specifically disclaims this acknowledgement.”
— Michael I. Hartley and Dimitri Leemans, “Quotients of a Universal Locally Projective Polytope of Type {5, 3, 5},” Mathematische Zeitschrift 247:4 (2004), 663-674
I think if there’s one thing that you could truly say it the most beautiful sight you can possibly see as a human, it is watching sunrise over the Earth, because imagine, you’re looking at blackness out the window, black Earth, black space, and then as the Sun comes up, the atmosphere acts as a prism, and it splits the light into the component colors. It splits the white light of the Sun into the component colors, so you get this rainbow effect, and it starts with this deep indigo eyelash, just defining the horizon, and then as the Sun rises higher, you get these reds and oranges and blues in this rainbow. … You never got tired of looking at those.
— Astronaut Mike Mullane, quoted in Ariel Waldman, What’s It Like in Space?, 2016
Astronomer Clyde Tombaugh assembled his first telescope from spare parts on his family’s Kansas farm — the crankshaft of a 1910 Buick, a cream-separator base, and mechanical components from a straw spreader. He used this to make sketches of Jupiter and Mars that so impressed the astronomers at Lowell Observatory that they gave him a job there.
Years later, after he had made his name by discovering Pluto, the Smithsonian Institution asked if it could exhibit this early instrument. He told them he was still using it — he was making observations from his backyard near Las Cruces, N.M., until shortly before his death in 1997.
“Its mirror was hand-ground and tested in a storm cellar,” wrote Peter Manly in Unusual Telescopes in 1991. “It’s not the most elegant-looking optical instrument I’ve ever used, but it is one of the better planetary telescopes around. … Because of its role in the history of astronomy, I would classify this as one of the more important telescopes in the world.”
The following story is true. There was a little boy, and his father said, ‘Do try to be like other people. Don’t frown.’ And he tried and tried, but could not. So his father beat him with a strap; and then he was eaten up by lions.
Reader, if young, take warning by his sad life and death. For though it may be an honour to be different from other people, if Carlyle’s dictum about the 30 millions be still true, yet other people do not like it. So, if you are different, you had better hide it, and pretend to be solemn and wooden-headed. Until you make your fortune. For most wooden-headed people worship money; and, really, I do not see what else they can do. In particular, if you are going to write a book, remember the wooden-headed. So be rigorous; that will cover a multitude of sins. And do not frown.
— Oliver Heaviside, “Electromagnetic Theory,” in The Electrician, Feb. 23, 1900
(When asked the population of England, Thomas Carlyle had said, “Thirty million, mostly fools.”)
From Chapter 12 of Ken Follett’s novel The Pillars of the Earth:
‘My stepfather, the builder, taught me how to perform certain operations in geometry: how to divide a line exactly in half, how to draw a right angle, and how to draw one square inside another so that the smaller is half the area of the larger.’
‘What is the purpose of such skills?’ Josef interrupted.
‘Those operations are essential in planning buildings,’ Jack replied pleasantly, pretending not to notice Josef’s tone. ‘Take a look at this courtyard. The area of the covered arcades around the edges is exactly the same as the open area in the middle. Most small courtyards are built like that, including the cloisters of monasteries. It’s because these proportions are most pleasing. If the middle is bigger, it looks like a marketplace, and if it’s smaller, it just looks as if there’s a hole in the roof. But to get it exactly right, the builder has to be able to draw the open part in the middle so that it’s precisely half the area of the whole thing.’
How is this done? Inscribe a diamond within a square and then rotate it 45 degrees:
Edward Mann Langley, founder of the Mathematical Gazette, posed this problem in its pages in 1922:
ABC is an isosceles triangle. B = C = 80 degrees. CF at 30 degrees to AC cuts AB in F. BE at 20 degrees to AB cuts AC in E. Prove angle BEF = 30 degrees.
(Langley’s description makes no mention of D; perhaps this is at the intersection of BE and CF.)
A number of solutions appeared. One, offered by J.W. Mercer in 1923, proposes drawing BG at 20 degrees to BC, cutting CA in G. Now angle GBF is 60 degrees, and angles BGC and BCG are both 80 degrees, so BC = BG. Also, angles BCF and BFC are both 50 degrees, so BF = BG and triangle BFG is equilateral. But angles GBE and BEG are both 40 degrees, so BG = GE = GF. And angle FGE is 40 degrees, so GEF is 70 degrees and BEF is 30 degrees.
