In 1837 English journalist Albany Fonblanque wrote, “Sir Robert Peel was a smooth round peg, in a sharp-cornered square hole, and Lord Lyndenurst is a rectangular square-cut peg, in a smooth round hole.”

Which of these is the better fit? In other words, which is larger, the ratio of the area of a circle to a circumscribed square, or the area of a square to a circumscribed circle?

In two dimensions, these ratios work out to π/4 and 2/π, respectively, so a round peg fits better into a square hole than a square peg into a round hole.

But, strangely, Berkeley mathematician David Singmaster discovered in 1964 that this is true only in dimensions less than 9. For n ≥ 9 the n-dimensional unit cube fits more closely into the n-dimensional unit sphere than vice versa.

There’s a moral in there, but I don’t know what it is.

(David Singmaster, “On Round Pegs in Square Holes and Square Pegs in Round Holes,” Mathematics Magazine 37:5 [November 1964], 335-337.)

Spun Wool

Image: Flickr

The wordplay journal Word Ways has made a tradition of revising the familiar rhyme “Mary Had a Little Lamb” under various constraints. Some examples:

Alliteration: James Puder, WW February 1998

Astral Aries’ avatar, alabaster “Aly,”
Ann adopted; allies are Ann and Ann’s argali.
Ann, an able autodidact, academic angst avoids;
And arch Aly’s Argus-eyed act awes astonished anthropoids.

Pangram (uses all 26 letters): A. Ross Eckler, WW February 1989

Mary had a little lamb with fleece extremely white;
Instead of grazing, all alone, the lamb kept her in sight.
It followed her to school one day, which was against the rule;
The children thought it quite a joke to view a lamb in school.

Words formed of chemical element symbols: WW February 2008

In ClAsS ONe MoRn SHe TaKEs HEr PLaCe; TeAcHEr CrIEs “SHoO! RUN!”
HeAr THoSe LaSSiEs ScReAm “HoW CuTe!” ThIS AgNUS — PURe FUN!”

Four-letter words: Dave Morice, WW November 2006

Mary kept some tiny lamb with wool hued just like snow,
Each spot that this girl, Mary, went, that lamb went also (slow).
Once lamb went past home room with girl. That bent some rule last year.
This made kids loud, glad, made them play: they eyed lamb very near.

Three-letter and shorter words: Jeff Grant, WW May 2004

Amy had an ewe so wee, it was an icy hue,
And any way our Amy led, the ewe it did go too.
It ran in to her den one day (an act not in the law).
Oh, the fun for boy and gal! The ewe so wee all saw.

Child Abuse


Nominative determinism is the theory that people gravitate toward occupations that reflect their names. In 1994 New Scientist noted that a new book, Pole Positions: The Polar Regions and the Future of the Planet, had been written by one Daniel Snowman, and that another, London Under London: A Subterranean Guide, received just two weeks later, had been written by Richard Trench. Psychologist Jen Hunt of the University of Manchester pointed out an article on incontinence in the British Journal of Urology whose authors were A.J. Splatt and D. Weedon.

If the theory is valid, then the naming of children is more momentous than we think. Harry Truman’s vice president, Alben William Barkley, above, was originally named Willie Alben Barkley, and contended that no one named Willie Alben could be elected superintendent of the county poorhouse. He changed his name to Alben William.

“In fact,” he wrote in his autobiography, “I think one of the graver shortcomings of my long career as a lawmaker was my failure to introduce a bill making it mandatory for parents to postpone the naming of their children until the youngsters are old enough to pick out a name for themselves.”

The 7-11 Problem

A man goes into a 7-11 store, buys four items, and notices that the bill totals $7.11. Even more interestingly, the product of the four prices is 7.11. What are the prices?

The answer is $1.20, $1.25, $1.50, and $3.16. There’s no thunderbolt insight to find; the problem yields to patient consideration.

Who came up with it? The most common credit I’ve seen is Doug Brumbaugh of the University of Central Florida. I found it in Crux Mathematicorum; see problem M203 in the September 2006 issue for Richard K. Guy’s solution.

The Floating Water Bridge

When a high voltage difference is applied, a liquid bridge of deionized water can sustain itself between two beakers even when they’re separated by 25 millimeters.

British engineer William Armstrong first reported this in an 1893 lecture. The phenomenon is known to be founded in surface polarization, but it’s still not completely understood.

No Exit

On Feb. 23, 1950, a railroad signal worker discovered the badly mangled body of a man in a tunnel south of Salzburg, Austria. Among its torn clothes police found the diplomatic passport and service identification of U.S. Navy captain Eugene S. Karpe, who’d been returning to the United States after serving for three years as naval attaché in Rumania.

It appeared that Karpe had fallen from the door of the Arlberg-Orient Express as the train sped around a curve in the dark of night. The train car had very small windows, and the doors had been locked automatically before the train had entered the tunnel. A student testified that he’d had breakfast and lunch with Karpe on the day he was killed; Karpe had had an ordinary breakfast and only a bottle of mineral water for lunch, eliminating the theory that he’d been drunk.

Karpe was the second-highest-ranking American mysteriously killed in Austria since the end of World War II. The first had been found stabbed and beaten to death after having been seen in the company of four men wearing Russian uniforms. Karpe was a close friend of Robert Vogeler, who had just been convicted as a spy and saboteur in Bucharest and sentenced by a people’s court to five years in prison.

The Austrian police contended that Karpe’s death was not a suicide and didn’t appear to be an accident. Formally the case remains unsolved.

(From Scott Baron and James Wise Jr., Dangerous Games: Faces, Incidents, and Casualties of the Cold War, 2013.)

Just So

Postscript of a letter from Benjamin Franklin to the Abbé André Morellet, July 1779:

P.S. To confirm still more your piety and gratitude to Divine Providence, reflect upon the situation which it has given to the elbow. You see in animals, who are intended to drink the waters that flow upon the earth, that if they have long legs, they have also a long neck, so that they can get at their drink without kneeling down. But man, who was destined to drink wine, is framed in a manner that he may raise the glass to his mouth. If the elbow had been placed nearer the hand, the part in advance would have been too short to bring the glass up to the mouth; and if it had been nearer the shoulder, that part would have been so long that when it attempted to carry the wine to the mouth it would have overshot the mark, and gone beyond the head; thus, either way, we should have been in the case of Tantalus. But from the actual situation of the elbow, we are enabled to drink at our ease, the glass going directly to the mouth.

“Let us, then, with glass in hand, adore this benevolent wisdom; — let us adore and drink!”



This just caught my eye in an old issue of the Mathematical Gazette, a note from P.G. Wood. Suppose we’re designing a cylinder that’s closed at both ends and must encompass a given volume. What relative dimensions should we give it in order to minimize its surface area?

A young student thought, well, if we slice the cylinder with a plane that passes through its axis, the plane’s intersection with the cylinder will form a rectangle. And if we spin that rectangle, it’ll sweep out the surface area of the cylinder. So really we’re just asking: Among all rectangles of the same area, which has the smallest perimeter? A square. So the cylinder’s height should equal its diameter.

It turns out that’s right, but the student had overlooked something. The fact that the volume of the cylinder is fixed doesn’t imply that the area of the rectangle is fixed. We don’t know that.

Wood wrote, “We seem to have arrived at the right answer by rather dubious means.”

(P.G. Wood, “73.5 Interesting Coincidences?”, Mathematical Gazette 73:463 [1989], 33-33.)