Red and Black

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I propose a card game. I’ll shuffle an ordinary deck of cards and turn up the cards in pairs. If both cards in a given pair are black, I’ll give them to you. If both are red, I’ll take them. And if one is black and one red, then we’ll put them aside, belonging to no one.

You pay a dollar for the privilege of playing the game, and then we’ll go through the whole deck. When the game is over, if you have no more cards than I do, you pay nothing. But for every card that you have more than I, I’ll pay you 3 dollars. Should you play this game?

Click for Answer

Turning Points

http://commons.wikimedia.org/wiki/File:Lincoln-Warren-1865-03-06.jpeg

During the drive on Washington city in July 1863, Confederate sharpshooters were unknowingly presented with a particularly high-value target. Captain Robert E. Park wrote: ‘The sharpshooters and the Fifth Alabama, which supported them, were hotly engaged; some of this enemy, seen behind their breastworks, were dressed in civilians’ clothes, and a few had on linen coats. I suppose they were “Home Guards” composed of Treasury, Post Office and other Department clerks.’ Park’s ‘Home Guards’ were in fact President Abraham Lincoln and his retinue, who had left the White House to inspect the defences around Washington. A doctor standing a few feet away from Lincoln was hit, and only prompt action by a nearby Union officer in throwing the president to the ground prevented a sudden and dramatic change in the course of Civil War history.

— John Anderson Morrow, The Confederate Whitworth Sharpshooters, 2002

[After the Battle of Belmont,] Grant went into a cabin and lay down on a sofa to rest, but he was there only a moment. He got up almost immediately to see what was happening on deck. As he rose a musket ball cut cleanly through the boat’s wooden side and splintered the head of the sofa where he had been lying. And still we waste ink and paper in trying to prove that there is no such thing as luck!

— W.E. Woodward, Meet General Grant, 1928

Traffic Waves

In 2008, physicist Yuki Sugiyama of the University of Nagoya demonstrated why traffic jams sometimes form in the absence of a bottleneck. He spaced 22 drivers around a 230-meter track and asked them to proceed as steadily as possible at 30 kph, each maintaining a safe distance from the car ahead of it. Because the cars were packed quite densely, irregularities began to appear within a couple of laps. When drivers were forced to brake, they would sometimes overcompensate slightly, forcing the drivers behind them to overcompensate as well. A “stop-and-go wave” developed: A car arriving at the back of the jam was forced to slow down, and one reaching the front could accelerate again to normal speed, producing a living wave that crept backward around the track.

Interestingly, Sugiyama found that this phenomenon arises predictably in the real world. Measurements on various motorways in Germany and Japan have shown that free-flowing traffic becomes congested when the density of cars reaches 40 vehicles per mile. Beyond that point, the flow becomes unstable and stop-and-go waves appear. Because it’s founded in human reaction times, this happens regardless of the country or the speed limit. And as long as the total number of cars on the motorway doesn’t change, the wave rolls backward at a predictable 12 mph.

“Understanding things like traffic jams from a physical point of view is a totally new, emerging field of physics,” Sugiyama told Gavin Pretor-Pinney for The Wavewatcher’s Companion. “While the phenomenon of a jam is so familiar to us, it is still too difficult to truly understand why it happens.”

The Facts

“Boarding-House Geometry,” by Stephen Leacock:

Definitions and Axioms

All boarding-houses are the same boarding-house.
Boarders in the same boardinghouse and on the same flat are equal to one another.
A single room is that which has no parts and no magnitude.
The landlady of a boarding-house is a parallelogram — that is, an oblong angular figure, which cannot be described, but which is equal to anything.
A wrangle is the disinclination of two boarders to each other that meet together but are not in the same line.
All the other rooms being taken, a single room is said to be a double room.

Postulates and Propositions

A pie may be produced any number of times.
The landlady can be reduced to her lowest terms by a series of propositions.
A bee line may be made from any boarding-house to any other boarding-house.
The clothes of a boarding-house bed, though produced ever so far both ways, will not meet.
Any two meals at a boarding-house are together less than two square meals.
If from the opposite ends of a boarding-house a line be drawn passing through all the rooms in turn, then the stovepipe which warms the boarders will lie within that line.
On the same bill and on the same side of it there should not be two charges for the same thing.
If there be two boarders on the same flat, and the amount of side of the one be equal to the amount of side of the other, each to each, and the wrangle between one boarder and the landlady be equal to the wrangle between the landlady and the other, then shall the weekly bills of the two boarders be equal also, each to each.
For if not, let one bill be the greater. Then the other bill is less than it might have been — which is absurd.

From his Literary Lapses, 1918. See Special Projects.

Bird Talk

Birder William Young notes that hobbyists who look for wild birds tend to identify species as much by their songs and calls as by their plumage. One way to memorize the calls is to translate them into familiar words and phrases. “Just as many people cannot remember lyrics to popular songs without singing the melody,” he writes, “many birders cannot remember bird songs and calls without thinking of mnemonic phrases.” Examples:

White-throated sparrow: Old Sam Peabody Peabody Peabody
Black-throated green warbler: trees, trees, murmuring trees
Black-throated blue warbler: I’m so la-zy
Olive-sided flycatcher: Quick, free beer!
White-eyed vireo: Pick up the beer check quick
Song sparrow: Maids maids maids pick up the tea kettle kettle kettle
American goldfinch: potato chip
Barred owl: Madame, who cooks for you?
Brown pigeon: Didja walk? Didja walk?
American robin: cheerily, cheer-up, cheerily
White-crowned sparrow: Poor JoJo missed his bus
Ovenbird: teacher, TEACHER, TEACHER
Red-eyed vireo: Here I am. Where are you?
Common yellowthroat: Which is it? Which is it? Which is it?
MacLeay’s honeyeater: a free TV
Common potoo: POO-or me, O, O, O, O
Inca dove: no hope
Brown quail: not faair, not faair
Little wattlebird: fetch the gun, fetch the gun

The California quail says Chicago, the long-tailed manakin says Toledo, and the rufous-browed peppershrike says I’M-A-RU-FOUS-PEP-PER-SHRIKE. “Once when I was staying at [birding author Graham Pizzey’s] home, a Willie-wagtail sang outside my bedroom window around 3 A.M. and seemed to say I’m trying to an-NOY you.” Young’s full article appears in the Winter 2003 issue of Verbatim.

In a Word

guttatim
adv. drop by drop

supernaculum
adv. to the last drop

stillatitious
adj. falling in drops

quantulum
n. a small amount or portion

Sea Music

The lovely Irish folk tune Port na bPúcaí (“The Music of the Fairies”) had mystical beginnings — it’s said that the people of the Blasket Islands heard ethereal music and wrote an air to match it, hoping to placate unhappy spirits. Seamus Heaney’s poem “The Given Note” tells of a fiddler who took the song “out of wind off mid-Atlantic”:

Strange noises were heard
By others who followed, bits of a tune
Coming in on loud weather
Though nothing like melody.

Recent research suggests that, rather than fairies, the islanders may have been hearing the songs of whales transmitted through the canvas hulls of their fishing boats. Humpback whales pass through Irish waters each winter as they migrate south from the North Atlantic, and their songs seem to resemble the folk tune.

Ronan Browne, who plays the air above on Irish pipes, writes, “In the mid 1990s I went rooting through some cassettes of whale song and there in the middle of the Orca (Killer Whale) section I heard the opening notes of Port na bPúcaí!”

No one can say for certain whether the one inspired the other, of course, but if it didn’t it’s certainly a pleasing coincidence.

(Thanks, James.)

A Late Contribution

A ghost co-authored a mathematics paper in 1990. When Pierre Cartier edited a Festschrift in honor of Alexander Grothendieck’s 60th birthday, Robert Thomas contributed an article that was co-signed by his recently deceased friend Thomas Trobaugh. He explained:

The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom’s simulacrum remarked, ‘The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.’ Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K0 obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom’s simulacrum had been so insistent, I knew he wouldn’t let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper. To Tom, I could have explained why he must be listed as a coauthor.

Thomason himself died suddenly five years later of diabetic shock, at age 43. Perhaps the two are working again together somewhere.

(Robert Thomason and Thomas Trobaugh, “Higher Algebraic K-Theory of Schemes and of Derived Categories,” in P. Cartier et al., eds., The Grothendieck Festschrift Volume III, 1990.)

“The Mostly German Philosophers Love Song”

By Colorado classics teacher Jeremy Boor:

MP3, lyrics, and chords are on his website.

Cross Purposes

ferland crossword grid

The daily New York Times crossword puzzle fills a grid measuring 15×15. The smallest number of clues ever published in a Times puzzle is 52 (on Dec. 23, 2008), and the largest is 86 (on Jan. 21, 2005).

This set Bloomsburg University mathematician Kevin Ferland wondering: What are the theoretical limits? What are the shortest and longest clue lists that can inform a standard 15×15 crossword grid, using the standard structure rules (connectivity, symmetry, and 3-letter words minimum)?

The shortest is straightforward: A blank grid with no black squares will be filled with 30 15-letter words, 15 across and 15 down.

The longest is harder to determine, but after working out a nine-page proof Ferland found that the answer is 96: The largest number of clues that a Times-style crossword will admit is 96, using a grid such as the one above.

In honor of this result, he composed a puzzle using this grid — it appears in the June-July 2014 issue of the American Mathematical Monthly.

(Kevin K. Ferland, “Record Crossword Puzzles,” American Mathematical Monthly 121:6 [June-July 2014], 534-536.)

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