Cyclic Billiards

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?

	Select Click for Answer
When the ball hits a cushion, it bounces back, with the angle of incidence equal to the angle of reflection. White writes, “Why not reflect the table instead?” Conceive that the ball travels in a straight line, and that each time it reaches a cushion it travels across a boundary and onto a new table. “So, we are now in a position to severely limit the allowable values of the angle, α, even if we cannot completely define those angles. The condition for cyclic motion is that the ball travels a certain number of tables up (not including the table the ball starts on), say, p tables, and so many tables to the right, say q tables. Two degenerate cases would be α = 0, in which case p = 0, and α = π/2 when q = 0. For the cases in between, 0 < α < π/2 and p > 0 and q > 0 but they must both be whole integers. In summary, we can now state that the ball will cycle on the billiard table if, and only if: i. α = 0° ii. α = π/2 = 90° iii. tan α = p/q (i.e. is a rational number) “As I stated, this is not a complete solution, but an interesting insight into a heuristic method. In justification, even if the perfect solution could be provided, it would be of no practical use to players of the game!”

Click for Answer

August 11, 2016August 21, 2016 | Puzzles

Sorted

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.

August 10, 2016August 10, 2016 | History · Society

Parallelogram Puzzle

parallelogram puzzle

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?

	Select Click for Answer
Draw EC. Now parallelogram ABCD and triangle EDC share a common base (DC), and they have the same altitude (a perpendicular from E to DC). So triangle EDC has half the area of parallelogram ABCD. But likewise, parallelogram EFGD and triangle EDC share a common base (ED), and they have the same altitude (a perpendicular from C to ED). So triangle EDC has half the area of parallelogram EFGD. Since both parallelograms have twice the area of the same triangle, their own areas must be equal. So the area of EFGD is 20 square units. (From Alfred S. Posamentier, Math Wonders to Inspire Teachers and Students, 2003.)

Click for Answer

August 10, 2016August 21, 2016 | Puzzles

Reconceptions

reconceptions

“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.)

August 9, 2016 | Art · Science & Math

Anamonics

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.

August 9, 2016August 8, 2016 | Entertainment · Language

Unquote

“There may now exist great men for things that do not exist.” — Samuel Burckhardt

August 8, 2016 | Quotations

Podcast Episode 116: Notes and Queries

In this week’s episode of the Futility Closet podcast we’ll explore some curiosities and unanswered questions from Greg’s research, including the love affair that inspired the Rolls Royce hood ornament, a long-distance dancer, Otto von Bismarck’s dogs, and a craftily plotted Spanish prison break.

We’ll also run after James Earl Ray and puzzle over an unsociable jockey.

See full show notes …

August 8, 2016October 5, 2017 | Podcast

More Morals

la rochefoucauld

Maxims of François VI, Duc de La Rochefoucauld (1613–1680):

“We commonly slander more thro’ Vanity than Malice.”
“We have more Laziness in our Minds than in our Bodies.”
“There are few People but what are ashamed of their Amours when the Fit is over.”
“We should not judge of a Man’s Merit by his great Qualities, but by the Use he makes of them.”
“He who is pleased with Nobody, is much more unhappy than he with whom Nobody is pleased.”
“There are some disguised Falsehoods so like Truths, that ‘twould be to judge ill not to be deceived by them.”
“Men sometimes think they hate Flattery, but they hate only the Manner of Flattering.”
“Acquired Honor is Surety for more.”
“Innocence don’t find near so much Protection as Guilt.”
“‘Tis our own Vanity that makes the Vanity of others intolerable.”
“‘Tis a common Fault to be never satisfied with ones Fortune, nor dissatisfied with ones Understanding.”
“Envy is more irreconcilable than Hatred.”
“‘Tis better to employ our Understanding, in bearing the Misfortunes that do befall us, than in foreseeing those that may.”
“A good Head finds less Trouble in submitting to a wrong Head than in conducting it.”
“Folly attends us close thro’ our whole Lives; and if anyone seems wise, ’tis merely because his Follies are proportionate to his Age, and Fortune.”

And “As ’tis the Characteristic of a great Genius to say much in a few Words, small Geniuses have on the contrary the Gift of speaking much and saying nothing.”

August 7, 2016August 6, 2016 | Quotations

Moessner’s Theorem

moessner's theorem

Write out the positive integers in a row and underline every fifth number. Now ignore the underlined numbers and record the partial sums of the other numbers in a second row, placing each sum directly beneath the last entry that it contains.

Now, in this second row, underline and ignore every fourth number, and record the partial sums in a third row. Keep this up and the entries in the fifth row will turn out to be the perfect fifth powers 1⁵, 2⁵, 3⁵, 4⁵, 5⁵ …

If we’d started by ignoring every fourth number in the original row, we’d have ended up with perfect fourth powers. In fact,

For every positive integer k > 1, if every kth number is ignored in row 1, every (k – 1)th number in row 2, and, in general, every (k + 1 – i)th number in row i, then the kth row of partial sums will turn out to be just the perfect kth powers 1^k, 2^k, 3^k …

This was discovered in 1951 by Alfred Moessner, a giant of recreational mathematics who published many such curiosa in Scripta Mathematica between 1932 and 1957.

(Ross Honsberger, More Mathematical Morsels, 1991.)

August 6, 2016August 5, 2016 | Science & Math

Decoy

In March 1945 the Japanese painted the giant image of an American B-29 on the Tien Ho airfield in China. They gave it a burning engine and a 300-foot wingspan, so that when viewed from a great altitude it would look like a stricken bomber flying at several thousand feet. Their hope was that this would induce high-flying Allied planes to drop down to investigate, bringing them within range of their anti-aircraft guns. I don’t know whether it worked.

The Atlantic has a collection of similar deceptive exploits from World War II.

August 5, 2016August 7, 2016 | History · Oddities

An idler's miscellany of compendious amusements

Author: Greg Ross