If two triangles ABC and abc are oriented so that lines Aa, Bb, and Cc meet at a point, then the pairs of corresponding sides (AB and ab; BC and bc; and AC and ac) will meet in three collinear points.
The converse is also true: If the pairs of corresponding sides intersect in three collinear points, then the lines joining corresponding vertices will meet in a point.
Scotland’s 1904 antarctic expedition made a unique contribution to science:
A number of emperor penguins, which were here very numerous, were captured. … To test the effect of music on them, Piper Kerr played to one on his pipes, — we had no Orpheus to warble sweetly on a lute, — but neither rousing marches, lively reels, nor melancholy laments seemed to have any effect on these lethargic phlegmatic birds; there was no excitement, no sign of appreciation or disapproval, only sleepy indifference.
— Rudmose Brown et al., The Voyage of the “Scotia,” 1906
As Ernest Shackleton was approaching Antarctica on December 18, 1914, “During the afternoon three adelie penguins approached the ship across the floe while Hussey was discoursing sweet music on the banjo. The solemn-looking little birds appeared to appreciate ‘It’s a Long Way to Tipperary,’ but they fled in horror when Hussey treated them to a little of the music that comes from Scotland.”
Canadian prime minister Mackenzie King took peculiar note of the relative position of clock hands. This began as early as 1918, when his diary shows that he began to notice moments when the hands overlapped (as at 12:00) or formed a straight line (as at 6:00). By the 1940s the diary sometimes refers to clock hands several times a day. On Aug. 25, 1943, when Franklin Roosevelt was visiting Ottawa, King wrote a whole “Memo re hands of clock”:
A year later, Nov. 2, 1944: “As I look at the clock from where I am standing as I dictate this sentence, the hands are both together at 5 to 11.”
Biographer Robert Macgregor Dawson writes, “What significance he attached to the occurrences is difficult to determine; there is no key to his interpretation.” But one clue comes later in 1944, when King records a conversation with Violet Markham: “As I … went to take the watch out of my pocket, to show her how the face had been broken, I looked at it and the two hands were exactly at 10 to 10. I mentioned it to her as an illustration of my belief that some presence was making itself known to me. That I was on the right line, and that the thought was a true one which I was expressing.” But the two had been discussing the death of King’s dog, so the meaning is still very obscure.
(From C.P. Stacey, A Very Double Life, 1976.)
In a joke issue of the Berichte der Deutschen Chemischen Gesellschaft in 1886, F.W. Findig offered an article on the constitution of benzene in which he finds that “zoology is capable of rendering the greatest service in clearing up the behavior of the carbon atom”:
Just as the carbon atom has 4 affinities, so the members of the family of four-handed animals possess four hands, with which they seize other objects and cling to them. If we now think of a group of six members of this family, e.g. Macacus cynocephalus, forming a ring by offering each other alternately one and two hands, we reach a complete analogy with Kekulé’s benzene-hexagon (Fig. 1).
Now, however, the aforesaid Macacus cynocephalus, besides its own four hands, possesses also a fifth gripping organ in the shape of a caudal appendix. By taking this into account, it becomes possible to link the 6 individuals of the ring together in another manner. In this way, one arrives at the following representation: (Fig. 2).
“It appears to me highly probable that a complete analogy exists between Macacus cynocephalus and the carbon atom,” Findig wrote. “In this case, each C-atom also possesses a caudal appendix, which, however, cannot be included among the normal affinities, although it takes part in the linking. Immediately this appendix, which I call the ‘caudal residual affinity’, comes into play, a second form of Kekulé’s hexagon is produced; this, being obviously different from the first, must behave differently.”
(From John Read, Humour and Humanism in Chemistry, 1947.)
A puzzle from Colin White’s Projectile Dynamics in Sport (2010): Suppose a billiard table has a length twice its width and that a rolling ball loses no energy to bounces or friction but simply caroms around the table forever. Call the angle between the launch direction and the long side of the table α. At what angle(s) should the ball be hit so that it will arrive back at the same point on the table and traveling in the same direction, so that its motion is cyclic, following the same path repeatedly?
In the U.S. presidential election of 1884, Republican James G. Blaine was accused of having sold his influence in Congress and of manipulating stocks. Democrat Grover Cleveland had fathered a child out of wedlock and had paid a substitute $150 to take his place in the Civil War. One journalist wrote:
Mr. Blaine has been delinquent in office but blameless in private life, while Mr. Cleveland has been a model of official integrity but culpable in his personal relations. We should therefore elect Mr. Cleveland to the public office which he is so qualified to fill and remand Mr. Blaine to the private station which he is so admirably fitted to adorn.
The people agreed, narrowly electing Cleveland and breaking a six-election losing streak for the Democrats.
Point E lies on segment AB, and point C lies on segment FG. The area of parallelogram ABCD is 20 square units. What’s the area of parallelogram EFGD?
“A Kiss and Its Consequences,” English carte de visite, 1910.
In 1965, Caltech computer scientist Donald Knuth privately circulated a theorem that, “under special circumstances, 1 + 1 = 3”:
Proof. Consider the appearance of John Martin Knuth, who exhibits the following characteristics: Weight 8 lb. 10 oz. (3912.23419125 grams) (3) Height 21.5 inches (0.5461 meters) (4) Voice loud (60 decibels) (5) Hair dark brown (Munsell 5.0Y2.0/11.8) (6) Q.E.D.
He conjectured that the stronger result 1 + 1 = 4 might also be true, and that further research on the problem was contemplated. “I wish to thank my wife Jill, who worked continuously on this project for nine months. We also thank Dr. James Caillouette, who helped to deliver the final result.”
(From Donald E. Knuth, Selected Papers on Fun & Games, 2011.)
A Scrabble player needs a way to recognize the potential in any collection of tiles. If your rack contains the seven letters AIMNSTU, for example, what eighth letter should you be watching for to create an acceptable eight-letter word?
If you arrange your seven letters into the word TSUNAMI, and if you’ve memorized the corresponding phrase COASTAL HARM, then you have your answer: Any of the letters in that phrase will produce an acceptable eight-letter word:
TSUNAMI + C = TSUNAMIC
TSUNAMI + O = MANITOUS
TSUNAMI + A = AMIANTUS
TSUNAMI + S = TSUNAMIS
TSUNAMI + T = ANTISMUT
TSUNAMI + L = SIMULANT
TSUNAMI + H = HUMANIST
TSUNAMI + R = NATURISM
TSUNAMI + M = MANUMITS
TSUNAMI: COASTAL HARM is an example of an anamonic (“anagram mnemonic”), a tool that tournament players use to memorize valuable letter combinations. Devising useful anamonics is itself an art form in the Scrabble community — one has to create a memorable phrase using a constrained set of letters. Some are memorable indeed:
GERMAN: LOST TO ALLIES
NATURE: VISIT GOD’S SCHOOL
SENIOR: OLD MVP JOGS WITH A CRUTCH
WAITER: A MAN RAN PANS
“One of the first anamonics I ever read, back in 1998, was PRIEST: EVERYONE COMPLAINED OF THE SODOMY,” wrote Jeff Myers in Word Ways in May 2007. “I couldn’t believe it. The letters in that phrase — no more and no less — could combine with PRIEST to make 7-letter words.”
When the word list TWL06 appeared, PERITUS became a legal word. That’s PRIEST + U, so the mnemonic phrase now needed to include a U. “One simple fix is: EVERYONE COMPLAINED OF YOUTH SODOMY,” wrote Myers. “Now maybe even more startling.”
John Chew maintains canonical lists of anamonics using the official Tournament Word List and the alternate SOWPODS list.
“There may now exist great men for things that do not exist.” — Samuel Burckhardt
