Well Done

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Using a 7-quart and a 3-quart jug, how can you obtain exactly 5 quarts of water from a well?

That’s a water-fetching puzzle, a familiar task in puzzle books. Most such problems can be solved fairly easily using intuition or trial and error, but in Scripta Mathematica, March 1948, H.D. Grossman describes an ingenious way to generate a solution geometrically.

Let a and b be the sizes of the jugs, in quarts, and c be the number of quarts that we’re seeking. Here, a = 7, b = 3, and c = 5. (a and b must be positive integers, relatively prime, where a is greater than b and their sum is greater than c; otherwise the problem is unsolvable, trivial, or can be reduced to smaller integers.)

Using a field of lattice points (or an actual pegboard), let O be the point (0, 0) and P be the point (b, a) (here, 3, 7). Connect these with OP. Then draw a zigzag line Z to the right of OP, connecting lattice points and staying as close as possible to OP. Now “It may be proved that the horizontal distances from OP to the lattice-points on Z (except O and P) are in some order without repetition 1, 2, 3, …, a + b – 1, if we count each horizontal lattice-unit as the distance a.” In this example, if we take the distance between any two neighboring lattice points as 7, then each of the points on the zigzag line Z will be some unique integer distance horizontally from the diagonal line OP. Find the one whose distance is c (here, 5), the number of quarts that we want to retrieve.

Now we have a map showing how to conduct our pourings. Starting from O and following the zigzag line to C:

  • Each horizontal unit means “Pour the contents of the a-quart jug, if any, into the b-quart jug; then fill the a-quart jug from the well.”
  • Each vertical unit means “Fill the b-quart jug from the a-quart jug; then empty the b-quart jug.”

So, in our example, the map instructs us to:

  • Fill the 7-quart jug.
  • Fill the 3-quart jug twice from the 7-quart jug, each time emptying its contents into the well. This leaves 1 quart in the 7-quart jug.
  • Pour this 1 quart into the 3-quart jug and fill the 7-quart jug again from the well.
  • Fill the remainder of the 3-quart jug (2 quarts) from the 7-quart jug and empty the 3-quart jug. This leaves 5 quarts in the 7-quart jug, which was our goal.

You can find an alternate solution by drawing a second zigzag line to the left of OP. In reading this solution, we swap the roles of a and b given above, so the map tells us to fill the 3-quart jug three times successively and empty it each time into the 7-quart jug (leaving 2 quarts in the 3-quart jug the final time), then empty the 7-quart jug, transfer the remaining 2 quarts to it, and add a final 3 quarts. “There are always exactly two solutions which are in a sense complementary to each other.”

Grossman gives a rigorous algebraic solution in “A Generalization of the Water-Fetching Puzzle,” American Mathematical Monthly 47:6 (June-July 1940), 374-375.

| Puzzles · Science & Math

Sweet and Bubbly

In 2005, archivist George Redmonds discovered something surprising among English birth records of the 14th century: a girl named Diot Coke.

She was born in the West Riding of Yorkshire in 1379. Researchers at Britain’s National Archives believe that her first name is a diminutive of Dionisia and her last name a variation of Cook.

She might have done worse. Popular girls’ names of the time included Godelena, Helwise, Idony, and Avice.

| Language

Clatter Plot

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A puzzle by Lewis Carroll:

Two travelers, starting at the same time, went opposite ways round a circular railway. Trains start each way every 15 minutes, the easterly ones going round in 3 hours, the westerly in 2. How many trains did each meet on the way, not counting trains met at the terminus itself?

Click for Answer
| Puzzles

The Pizza Principle

In 1980, New York patent lawyer Eric Bram correctly predicted that the city’s transit fare would increase. He explained his reasoning to the New York Times: “Since the early ’60s, the price of a slice of pizza has matched, with uncanny precision, the cost of a New York subway ride. Right now, it is impossible for any discerning New Yorker to find a decent slice of pizza for less than 60 cents. The 50-cent fare was doomed.”

He was right. In 1960, the fare was 15 cents, and so was a slice of pizza (a regular slice, mozzarella and tomato sauce, no toppings). In the early 1970s, both rose to 35 cents, and the two continued to rise together. By 2002, pizza had risen to $2 in midtown, while the fare lagged at $1.50; sure enough, the fare rose to $2 the following spring, after eight years without a change.

In 2003 the subway system switched from tokens to MetroCards, finding them more efficient in a digital age. “Who knows if the fundamentals of economics will hold?” Bram asked.

They did. As the price of pizza rose, the fare followed it, rising to $2.25 in 2009 and to $2.50 in 2011. “Don’t ask why,” wrote Clyde Haberman, who tracks all this in the Times. “It simply is so, and has been for decades.”

| Society

Easy Rider

http://www.google.com/patents/US1144150

As motorcycles grew more popular in the early 20th century, Russian inventor Frank Marcovsky designed a suit of armor to protect riders:

The suit in its entirety comprises a one-piece garment, having the body, legs and arms, and a detachable helmet or head piece, each of such portions being provided with inflatable cushioning elements adapted to be filled with compressed air, for the purpose of protecting the wearer from shocks or blows incidental to accidents or the use to which the armor is put.

The inflatable ribs can be filled at an air pump, forming a protective cushion that leaves the rider the full use of his limbs. “All exposed portions of the body of the wearer are amply protected against severe shocks of impact or blows incidental to accidental contact with the ground, other riders, fences, etc.” The patent was granted in 1915; I don’t know how it fared.

| Technology

Books and Benches

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Letter from Petrarch to Zanobi da Strada, April 1, 1352:

Let them teach who can do nothing better, whose qualities are laborious application, sluggishness of mind, muddiness of intellect, prosiness of imagination, chill of the blood, patience to bear the body’s labors, contempt of glory, avidity for petty gains, indifference to boredom. You see how far these qualities are from your character. Let them watch boys’ fidgety hands, their wandering eyes, their sotto voce whisperings who delight in that task, who enjoy dust and noise and the clamor of mingled prayers and tears and whimperings under the rod’s correction. Let them teach who love to return to boyhood, who are shy of dealing with men and shamed by living with equals, who are happy to be set over their inferiors, who always want to have someone to terrify, to afflict, to torture, to rule, someone who will hate and fear them. That is a tyrannical pleasure, such as, according to the story, pervaded the fierce spirit of that old man of Syracuse, to be the evil solace of his deserved exile. But you, a man of parts, merit a better occupation. Those who instruct our youth should be like those ancient authors who informed us in our own early age; as those who first aroused our young minds with noble examples, so should we be to our successors. Since you can follow the Roman masters, Cicero and Virgil, would you choose Orbillius, Horace’s ‘flogging-master’? What is more, neither grammar nor any of the seven liberal arts is worth a noble spirit’s attention throughout life. They are means, not ends …

Zanobi, a poor Florentine schoolmaster, was so affected by his friend’s words that he gave up teaching and became a government official. Three years later he was crowned poet laureate of Pisa, annoying Petrarch, who in 1341 had been crowned the first laureate since antiquity in Rome.

| Society

A Magic Mystery

collison bimagic square

In 1991, David Collison sent this figure to Canadian magic-square expert John Henricks, with no explanation, and then died.

It’s believed to be the first odd-ordered bimagic square ever discovered. Each row, column, and diagonal produces a sum of 369. The square remains magic if each number is squared, with a magic sum of 20,049.

No one knows how Collison created it.

UPDATE: Wait, Collison’s wasn’t the first — G. Pfeffermann published a 9th-order bimagic square as a puzzle in Les Tablettes du Chercheur in 1891. (Thanks, Baz.)

| Science & Math

Ruin Marble

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This brecciated limestone, quarried near Florence, has a curious property: When it’s polished it produces the image of an ancient city.

“One is amused,” wrote Parisian naturalist Cyprien Prosper Brard, “to observe in it kinds of ruins; there it presents a Gothic castle half destroyed; here ruined walls; in another place, old bastions; and what still adds to the delusion is, that in these natural paintings there exists a kind of ærial perspective, very sensibly perceptible. The lower part, or what forms the first plane, has a warm, and bold tone; the second follows it, and weakens as it increases in distance; the third becomes still fainter, while the upper part presents in the distance, a whitish zone, and finally, as it reaches the top, blends itself, as it were, with the clouds.”

In his 1832 Introduction to Mineralogy, John Comstock wrote, “At a certain distance, slabs of this marble so nearly represent drawings done in bistre, on a ground of yellowish brown, that it would be difficult to convince one to the contrary.”

| Oddities

Better Late

On Dec. 7, 1968, Richard Dodd of Winamac, Ind., returned a book to the University of Cincinnati medical library, noting that it was overdue.

It certainly was. The book, Medical Reports of Effects of Water — Cold and Warm — as a Remedy in Fever and Febrile Diseases, Whether Applied to the Surface of the Body or Used Internally, had been checked out by Dodd’s great-grandfather in 1823. It was 145 years late.

Dodd, whose grandfather and great-grandfather had both attended medical school at Cincinnati, had received the book as part of an inheritance. The library decided not to fine him, which is a good thing — librarian Cathy Hufford calculated that the fee would have come to $22,646.

| Oddities

Prose Poetry

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In scenes of pathos, Charles Dickens often wrote unconsciously in blank verse. These paragraphs from The Old Curiosity Shop, arranged as poetry, show a strong iambic rhythm:

And now the bell — the bell
She had so often heard by night and day
And listened to with solid pleasure,
E’en as a living voice —
Rung its remorseless toll for her,
So young, so beautiful, so good.

Decrepit age, and vigorous life,
And blooming youth, and helpless infancy,
Poured forth — on crutches, in the pride of strength
And health, in the full blush
Of promise — the mere dawn of life —
To gather round her tomb. Old men were there,
Whose eyes were dim
And senses failing —
Granddames, who might have died ten years ago,
And still been old — the deaf, the blind, the lame, the palsied,
The living dead in many shapes and forms,
To see the closing of this early grave!
What was the death it would shut in,
To that which still would crawl and creep above it!

Along the crowded path they bore her now;
Pure as the new-fallen snow
That covered it; whose day on earth
Had been so fleeting.
Under that porch where she had sat when Heaven
In mercy brought her to that peaceful spot,
She passed again, and the old church
Received her in its quiet shade.

Richard Horne presented these stanzas in New Spirit of the Age in 1844. “Throughout the whole of the above, only two unimportant words have been omitted — in and its; ‘granddames’ has been substituted for ‘grandmothers,’ and ‘e’en’ for ‘almost.’ All that remains is exactly as in the original, not a single word transposed, and the punctuation the same to a comma.”

“It is not an affectation in me, nor have I the least desire to write them in that metre,” Dickens wrote to an inquirer that April, “but I run into it, involuntarily and unconsciously, when I am very much in earnest. I even do so, in speaking.”

“I am not prepared to say that this may not be a defect in prose composition, but I attach less importance to it than I do to earnestness. And considering that it is a very melodious and agreeable march of words, usually; and may be perfectly plain and free; I cannot agree with you that it is likely to be considered by discreet readers as turgid or bombastic, unless the sentiments expressed in it, be of that character. Then indeed it matters very little how they are attired, as they cannot fail to be disagreeable in any garb.”

But he seems to have grown self-conscious about it. In 1846 he wrote to John Forster regarding The Battle of Life, “If in going over the proofs you find the tendency to blank verse (I cannot help it, when I am very much in earnest) too strong, knock out a word’s brains here and there.”

| Literature