Martin Gardner published this puzzle in his “Mathematical Games” column in Scientific American in February 1979. You’re blindfolded and sitting before a lazy susan. On each corner is a glass. Some are right side up and some upside down. On each turn you can inspect any two glasses and, if you choose, reverse the orientation of either or both of them. After each turn the lazy susan will be rotated through a random angle. When all four glasses have the same orientation, a bell will sound. How can you reach this goal in a finite number of turns?
Author: Greg Ross
Cameo
Grip, the talking raven in Dickens’ Barnaby Rudge, was based on a real bird, a pet who lived in the family’s Marylebone home, where she buried items in the garden, terrorized the dog, bit the children, and tore at the family carriage. She died in 1841 after ingesting some lead-based paint, and Dickens wrote her into the novel, which appeared later that year.
Interestingly, when Edgar Allan Poe reviewed the story for Graham’s Magazine, he remarked that “The raven … might have been made more than we see it … Its croaking might have been prophetically heard in the course of the drama.” In 1842, when the two authors met in Philadelphia, Poe was said to be “delighted” that Grip had been based on a real bird.
Poe’s famous poem appeared three years later, and scholars generally agree that Grip had inspired the “ebony bird” — in Dickens’ novel the raven had repeated the phrases “Never say die” and “Nobody” and is described as “tapping at the door” and “knocking softly at the shutter.”
Today Grip’s remains are on display in Philadelphia’s Parkway Central Library, where they may inspire yet more writings.
Noted
Band Practice
Drill a hole straight through the center of a sphere, leaving a band in the shape of a napkin ring. Suppose the height of this band is h. Now take a sphere of a different size and drill a hole through that, contriving the hole’s width so that its depth is again h. Remarkably, the two napkin rings will have the same volume.
As the sphere expands, the band must grow both wider and thinner, and it turns out that these two effects exactly cancel one another. It’s called the napkin ring problem.
Passing Through
Just a striking image: German astronomer Johann Friedrich Julius Schmidt prepared a plaster moon for the Field Columbian Museum in 1898. From the Field Museum Library.
Flat Devotion
William Linkhaw sang so badly that a grand jury indicted him for disrupting his church’s services. At trial in August 1872, a witness imitated Linkhaw’s singing style and provoked “a burst of prolonged and irresistible laughter, convulsing alike the spectators, the Bar, the jury and the Court.”
It was in evidence that the disturbance occasioned by defendant’s singing was decided and serious; the effect of it was to make one part of the congregation laugh and the other mad; that the irreligious and frivolous enjoyed it as fun, while the serious and devout were indignant.
Linkhaw protested that he felt a duty to worship God. The jury fined him a penny, but the North Carolina Supreme Court set aside the verdict, observing that Linkhaw had had no malicious intent. Justice Thomas Settle wrote, “It would seem that the defendant is a proper subject for the discipline of his church, but not for the discipline of the Courts.”
In 1906 a wit wrote, “although the proof did show / That Linkhaw’s voice was awful / The judges found no valid ground / For holding it unlawful.”
A Pi Diet
Students beginning with the compass learn to draw this rosette, sometimes called the Flower of Life. If the arcs and the circle have the same radius, a, then the area of one petal is
and the unshaded area of the circle is
Remarkably, though the area we sought is bounded entirely by the arcs of circles, the final expression is independent of π.
Related: When two cylinders of radius r meet at right angles, the volume of their intersection is 16r3/3 — again, no sign of π.
(J.V. Narlikar, “A Pi-Less Area,” Mathematical Gazette 65:431 [March 1981], 32-33.)
10/01/2024 UPDATE: Some deft rearranging shows that the unshaded area of the circle is just the area of an inscribed regular hexagon:
So the absence of π isn’t that surprising. Thanks to readers Catalin Voinescu (who sent this diagram) and Gareth McCaughan for pointing this out.
“Through the Looking-Glass”
C.S. Kipping published this unusual problem in Chess Amateur in 1923. In each position, White is to mate in two moves.
Fish Story
Then there is the other secret. There isn’t any symbolysm. The sea is the sea. The old man is an old man. The boy is a boy and the fish is a fish. The shark are all sharks no better and no worse. All the symbolism that people say is shit. What goes beyond is what you see beyond when you know. A writer should know too much.
— Ernest Hemingway, letter to Bernard Berenson, 1952
Topsy Turvy
A reversible print by John Kay, 1789.