Draw any quadrilateral and connect the midpoints of its sides.
You’ll always get a parallelogram.
Triangles, too, have perfection at heart.
Author: Greg Ross
Unquote
“I know of no rule which holds so true as that we are always paid for our suspicion by finding what we suspect.” — Thoreau
Actual Size
The familiar Mercator projection is useful for navigation, but it exaggerates the size of regions at high latitudes. Greenland, for example, appears to be the same size as South America, when in fact it’s only one eighth as large.
An equal-area projection such as the Mollweide, below, distorts the shapes of regions but preserves their relative size. This reveals some surprising facts: Russia is larger than Antarctica, Mexico is larger than Alaska, and Africa is just mind-bogglingly huge — larger than the former Soviet Union, larger than China, India, Australia, and the United States put together.
The Deliveryman’s Problem
A puzzle from L. Despiau’s Select Amusements in Philosophy and Mathematics, 1801:
Distribute among 3 persons 21 casks of wine, 7 of them full, 7 of them empty, and 7 of them half full, so that each of them shall have the same quantity of wine, and the same number of casks.
Yes or No?
Will you either answer no to this question or pay me a million dollars?
(Raymond Smullyan)
Equal Opportunity
She frowned and called him Mr.
Because in sport he Kr.
And so in spite
That very night
This Mr. Kr. Sr.
— Anonymous
Better Safe
For several years during the Cold War, New York police guarded the Soviet consulate at 9 East 91st Street in Manhattan. Officers manned a pale blue guard post 24 hours a day. “It’s like being a prisoner of war stuck in a telephone booth,” one said.
The Soviets left in 1980, and the police department accordingly canceled the guard, but two months later the 23rd precinct received a call from an Officer Cowans who said that Inspector Whitmore of police intelligence had ordered the guard to be reactivated. So the police resumed their vigil over the now-disused building.
Five months later, in May 1982, the police happened to mention the consulate duty in a report. “What booth?” asked a bewildered intelligence official. It turned out that Officer Cowans and Inspector Whitmore did not exist; the police had been guarding an empty building around the clock for five months, right through Christmas, for no reason.
They closed up shop and removed the booth. “Whoever did this was someone who wanted to break chops or who stood to gain from it,” Lt. Robert McEntire told the New York Times. “We’re not sure which, and we probably never will be.”
Misc
- The telephone number 266-8687 spells both AMOUNTS and CONTOUR.
- 38856 = (38 – 85) × 6
- CARTHORSE is an anagram of ORCHESTRA.
- The French for paper clip is trombone.
- “The oldest books are only just out to those who have not read them.” — Samuel Butler
Gilbreath’s Conjecture
Doodling on a napkin in 1958, mathematician Norman L. Gilbreath noticed something odd. First he wrote down the first few prime numbers in a row. Then, on each succeeding row, he recorded the (unsigned) difference between each pair of numbers in the row above:
The first digit in each row (except the first) is 1. Will this always be true, no matter how many prime numbers we start with? It’s been borne out in computer searches extending to hundreds of billions of rows. But no one knows for sure.
A Matter of Interpretation
A French gentleman made a will in which, among other bequests, he left handsome sums of money to his two nephews, Charles and Henri. The sums were equal in amount. When the testator died and the will came to be proved, the nephews expected to receive two hundred thousand francs each as their specific bequests. But the executors disputed this, and said that each legacy was for one hundred thousand francs.
The legatees pointed to the word deux.
‘No,’ said the executors, ‘there is a comma or apostrophe between the d and the e, making it d’eux.’
‘Not so,’ rejoined Charles and Henri; ‘that is only a little blot of ink, having nothing to do with the actual writing.’
Let us put the two interpretations in juxtaposition:
À chacun deux cent milles francs.
À chacun d’eux cent milles francs.The first form means, ‘To each two hundred thousand francs,’ whereas the other has the very different meaning, ‘To each of them a hundred thousand francs.’ This little mark (‘) made all the difference.
The paper had been folded before the ink was dry. A few spots of ink had been transposed from one side of the fold to the other, and the question was whether the apparent or supposed apostrophe was one such spot.
The legatees had very strong reasons–two hundred thousand strong–for wishing that the little spot of ink should be proved merely a blot; but their opponents had equally strong reasons for wishing that the blot should be accepted as an apostrophe, an intended and component element in the writing.
The decision was in favor of the legatees, but was only reached after long and expensive litigation.
— William Shepard Walsh, Handy-Book of Literary Curiosities, 1892