List the first 2N positive integers (here let N = 4):
1, 2, 3, 4, 5, 6, 7, 8
Divide them arbitrarily into two groups of N numbers:
1, 4, 6, 7
2, 3, 5, 8
Arrange one group in ascending order, the other in descending order:
1, 4, 6, 7
8, 5, 3, 2
Now the sum of the absolute differences of these pairs will always equal N2:
| 1 – 8 | + | 4 – 5 | + | 6 – 3 | + | 7 – 2 | = 16 = N2
(Presented by Vyacheslav Proizvolov as a problem in the 1985 All-Union Soviet Student Olympiads.)
Why do we hold graduation ceremonies? You don’t have to attend the ceremony to collect your degree; your education would be just as complete without it. Why do we maintain this seemingly needless ritual?
Philosopher Elijah Millgram argues that the ceremony provides a motivation to persist through an otherwise uninspiring mountain of work. Education is a “jam-yesterday-jam-tomorrow” good — we value it when it’s in our future or in our past, but the experience itself is often stressful, difficult, or boring. “On any given occasion, a student is likely to be plowing through a hard-to-read book, or writing a difficult paper, or trying not to doze off in lecture,” he writes. “The education is all these things, and is correctly understood to be a great (and an intrinsic) good; but it is hard to stay focused on its value just because one does not see it, moment to moment.”
The graduation ceremony, with its colors, music, and ritual, provides a highly visible “dummy goal” that helps to motivate students to complete their requirements. Without it, the vague prospect of “jam tomorrow” — the promised satisfaction of holding a degree — might be too little to spur some students to finish their studies. “If students had to make their own moment-to-moment decisions as to whether to read the book, or write the paper, or stay in the lecture, on the basis of its intangible contribution to their education,” Millgram writes, “they would be all too likely to put off the unpleasant tasks to some other time.”
(Elijah Millgram, “Virtue for Procrastinators,” in Chrisoula Andreou and Mark D. White, eds., The Thief of Time, 2010.)
“Not to be born is, past all prizing, best,” wrote Sophocles. Does this mean that life is not worth living? For surely that judgment must be made from “inside” a lived life, whose subjective judgments are always open to question.
In four U.S. states a severely disabled child can sue a doctor for “wrongful life” for bringing him into the world. In 1980 the California Court of Appeal wrote:
The reality of the ‘wrongful-life’ concept is that such a plaintiff both exists and suffers, due to the negligence of others. It is neither necessary nor just to retreat into meditation on the mysteries of life. We need not be concerned with the fact that had defendants not been negligent, the plaintiff might not have come into existence at all. The certainty of genetic impairment is no longer a mystery. In addition, a reverent appreciation of life compels recognition that plaintiff, however impaired she may be, has come into existence as a living person with certain rights.
The rest of us, it seems, must find a way to be philosophical. After all, Lionel Tollemache wrote, “If there is more pain than pleasure in life, were not Hyder Ali and Napoleon, who put so many human sufferers out of existence, deserving of praise as beneficent heroes?”
In 1948 Melvin Wellman discovered this pretty anagram:
ELEVEN + TWO = TWELVE + ONE
And Dave Morice found this:
THIRTEEN + TWENTY – ONE = (NINETY / TWO) – TEN – THREE
Lee Sallows discovered two similar specimens in Spanish:
UNO + CATORCE = CUATRO + ONCE
DOS + TRECE = TRES + DOCE
These can be combined to make more:
UNO + DOS + TRECE + CATORCE = TRES + CUATRO + ONCE + DOCE
UNO + TRES + DOCE + CATORCE = DOS + CUATRO + ONCE + TRECE
Index entries in Thomas De Quincey’s Collected Writings, 1896:
Aldermen not necessarily gluttons
Anecdotes, on eating peas with a knife
Bed, early retirement to, of the Ancients
Christenings, Royal, often hurried
Coffee, atrocious in England
Cookery, English, the rudest of barbarous devices
Devonshire men good-looking
Fleas in Greece
Greece, Ancient, its people a nation of swindlers
Horses, weeping
Johnson, Dr, at dinner, an indecent spectacle
Leibnitz, died partly from the fear of not being murdered
Lisbon earthquake and its effect on the religion of Germany
Muffins, eating, a cause of suicide
Music, English obtuseness to good
Pig-grunting, mimicry of
Rhinoceros, first sale of a
Servants, England the paradise of household
Solon, what did he do for Homer?
Spitting, art of
Talk, too much in the world
Toothache, that terrific curse
Waterton’s adventure with a crocodile
Women, can die grandly
“If once a man indulges himself in murder, very soon he comes to think little of robbing; and from robbing he comes next to drinking and Sabbath-breaking, and from that to incivility and procrastination,” he wrote in Murder Considered as One of the Fine Arts. “Once begun upon this downward path, you never know where you are to stop. Many a man has dated his ruin from some murder or other that perhaps he thought little of at the time.”
A and B are playing a simple game. Between them are nine tiles numbered 1 to 9. They take tiles alternately from the pile, and the first to collect three tiles that sum to 15 wins the game. Does the first player have a winning strategy?
“It is well to read everything of something and something of everything.” — Henry Brougham
“Try to learn something about everything and everything about something.” — Inscribed on Thomas Huxley’s memorial
“The whole secret of life is to be interested in one thing profoundly and in a thousand things well.” — Horace Walpole
On Feb. 9, 1855, the residents of Devon in southern England awoke to find a bewildering set of footprints in the newfallen snow. “These are to be found in fields, gardens, roads, house-tops, & other likely and unlikely places, deeply embedded in snow,” ran one contemporary account. “The shape was a hoof.”
In this episode of the Futility Closet podcast we’ll examine the surviving descriptions of the odd marks and consider the various explanations that have been offered. We’ll also revisit the compassionate Nazi fighter pilot Franz Stigler and puzzle over how to sneak into Switzerland across a guarded footbridge.
Rousseau asks: Suppose that you and I hunt a stag. This requires long hours lying in wait along the trail, and it entails the risk that the stag will not appear. If, after some time has passed, a hare appears, either of us could seize it, abandoning the hope of capturing the stag but getting an immediate meal for himself. To hunt the stag successfully we have to agree to trust one another, knowing that each of us has an immediate reason to betray the other.
This gets even harder when many people are involved. David Hume writes, “Two neighbors may agree to drain a meadow, which they possess in common; because ’tis easy for them to know each others mind, and each may perceive that the immediate consequence of failing in his part is the abandoning of the whole project. But ’tis difficult, and indeed impossible, that a thousand persons shou’d agree in any such action.”
How did our society ever get off the ground when cooperation requires a faith in one another that simple self-reliance does not? It seems that our very rationality makes such a leap harder: Animals such as social insects work harmoniously together, but “the agreement of these creatures is natural,” writes Thomas Hobbes. “That of men is by covenant only, which is artificial.”
A conduit carries 50 identical wires under a river, but their ends have not been labeled — you don’t know which ends on the west bank correspond to which on the east bank. To identify them, you can tie together the wires in pairs on the west bank, then row across the river and test the wires on the east bank to discover which pairs close a circuit and are thus connected.
Testing wires is easy, but rowing is hard. How can you plan the work to minimize your trips across the river?