The Hypergame Paradox

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Call a game finite if it terminates in finitely many moves. Now consider Hypergame, which has two rules:

  1. The first player names a finite game.
  2. The two players play that game.

Is Hypergame a finite game? It seems so: It consists of a single game-naming move, followed by a subgame with a necessarily finite number of moves. But what if the first player names Hypergame itself as the subgame, and the second player names Hypergame as the sub-sub-game, and so on?

In presenting this question to students and colleagues at Union College, mathematician William Zwicker found that many saw the catch and quickly pointed out that it leads to infinite play, thinking that this settles the matter. But the proof that Hypergame is finite seems sound. “I … have to convince them that mathematicians cannot simply abandon a proof once a counter-example has been found, for if the internal flaw in such a proof cannot be identified then the counterexample threatens the entire edifice of mathematical proof.” What is the answer?

(William S. Zwicker, “Playing Games with Games: The Hypergame Paradox,” American Mathematical Monthly 94:6, 507-514)

Unquote

“There are two things to aim at in life: first, to get what you want; and, after that, to enjoy it. Only the wisest of mankind achieve the second.” — Logan Pearsall Smith

Reverses

A Hungarian problem shortlisted for the 30th International Mathematical Olympiad, 1989:

Around a circular race track are n race cars, each at a different location. At a signal, each car chooses a direction and begins to drive at a constant speed that will take it around the course in 1 hour. When two cars meet, both reverse direction without loss of speed. Show that at some future moment all the cars will be at their original positions.

Click for Answer

Plant Food

man-eating tree

Exploring a tabooed piece of land on Mindanao, Mississippian planter W.C. Bryant discovered a 35-foot tree surrounded by bones and the smell of carrion. He was entering the circle to examine a skull when his guide suddenly pinioned his arms and pulled him backward “with the strength of a maniac.” Bryant followed the man’s gaze and saw that “the tree was reaching for him”:

The whole thing had changed shape and was horribly alive and alert. The dull, heavy leaves had sprung from their compact formation and were coming at him from all directions, advancing on the ends of long vine-like stems which stretched across like the necks of innumerable geese and, now that the old man had stopped his screaming, the air was full of hissing sounds.

The leaves did not move straight at their target, but with a graceful, side-to-side sway, like a cobra about to strike. From the far side, the distant leaves were peeping and swaying on their journey around the trunk and even the tree top was bending down to join in the attack. The bending of the trunk was spasmodic and accompanied by sharp cracks.

The effect of this advancing and swaying mass of green objects was hypnotic, like the charm movements of a snake. Bryant could not move, though the nearest leaf was within an inch of his face. He could see that it was armed with sharp spines on which a liquid was forming. He saw the heavy leaf curve like a green-mittened hand, and as it brushed his eyebrows in passing he got the smell of it — the same animal smell that hung in the surrounding air. Another instant and the thing would have had his eyes in its sticky, prickly grasp, but either his weakness or the brown man’s strength threw them both on their backs.

The charm was broken. They crawled out of the circle of death and lay panting in the grass while the malignant plant, cracking and hissing, yearned and stretched and thrashed to get at them.

That’s from “Escaped From the Embrace of the Man-Eating Tree,” in the American Weekly, Jan. 4, 1925. Interestingly, naturalist Williard Clute tracked down the author and published a followup in American Botanist that April:

“The author of this tale, having been questioned, replies under date of January 8, 1925, that ‘the tree is there and in the main the account is true. The circle at the foot of the tree was about 80 maybe 100 feet in diameter. The tree looked nothing like the drawings [in the paper]. It was round as a smoke stack — the trunk, I mean, and dark gray or ash-color. The whole tree was symmetrical and the tree and ground under it, was very inviting to a storm-beset or sun-depressed traveller. The clucking and hissing was, I judged, from a gluey consistency of, or on, the leaves. My impression was that if it reached me, it would fasten and hold me, thus it had done to apes, birds, and animals.'” Stay out of the Philippines, I guess.

Pleasure Gardening

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The 1957 edition of Exotica: Pictorial Cyclopedia of Indoor Plants included a species called Rumandia cocacoliensis of the family Alcoholiaceae.

The description read “Cuba-libre tall, stemless, succulent, with brown-frosty bloom often with lemon flavor; good in summer, keep cool.”

It was indexed without a page number, and disappeared from subsequent editions.

Tomb Evader

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When you die, perhaps you will cease to exist. Or perhaps you’ll be reincarnated, or your soul will go to heaven, or to hell. But in none of these cases will “you” be placed in a casket and lowered into the ground; you will never find yourself in the grave. What all these conceptions have in common is that the dead cannot be buried — they are either elsewhere, or nowhere.

Socrates says, “I cannot make Crito believe that I am the same Socrates who have been talking and conducting the argument; he fancies that I am the other Socrates whom he will soon see, a dead body — and he asks, How shall he bury me? … I shall not remain, but go away and depart; and then he will suffer less at my death, and not be grieved when he sees my body being burned or buried. I would not have him sorrow at my hard lot, or say at the burial, Thus we lay out Socrates, or, Thus we follow him to the grave or bury him. … Be of good cheer, then, my dear Crito, and say that you are burying my body only, and do with that whatever is usual, and what you think best.”

(Palle Yourgrau, “Can the Dead Really Be Buried?”, Midwest Studies in Philosophy 24:1, 46-68.)

Ill Will

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A person with Münchausen syndrome feigns illness to get attention. In order to get this diagnosis, she must be faking symptoms of a condition that she doesn’t really have. But any such faked symptoms are an authentic sign of Münchausen syndrome. Does this mean it’s impossible to have the disorder?

(By Washington University philosopher Roy Sorensen.)

Turnabout

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A Swedish minister having assembled the chiefs of the Susquehanna Indians, made a sermon to them, acquainting them with the principal historical facts on which our religion is founded — such as the fall of our first parents by eating an apple, the coming of Christ to repair the mischief, his miracles and suffering, etc. When he had finished an Indian orator stood up to thank him.

‘What you have told us,’ says he, ‘is all very good. It is indeed bad to eat apples. It is better to make them all into cider. We are much obliged by your kindness in coming so far to tell us those things which you have heard from your mothers. In return, I will tell you some of those we have heard from ours.

‘In the beginning, our fathers had only the flesh of animals to subsist on, and if their hunting was unsuccessful they were starving. Two of our young hunters, having killed a deer, made a fire in the woods to boil some parts of it. When they were about to satisfy their hunger, they beheld a beautiful young woman descend from the clouds and seat herself on that hill which you see yonder among the Blue Mountains.

‘They said to each other, “It is a spirit that perhaps has smelt our broiling venison and wishes to eat of it; let us offer some to her.” They presented her with the tongue; she was pleased with the taste of it and said: “Your kindness shall be rewarded; come to this place after thirteen moons, and you will find something that will be of great benefit in nourishing you and your children to the latest generations.” They did so, and to their surprise found plants they had never seen before, but which from that ancient time have been constantly cultivated among us to our great advantage. Where her right hand had touched the ground they found maize; where her left had touched it they found kidney-beans; and where her backside had sat on it they found tobacco.’

The good missionary, disgusted with this idle tale, said: ‘What I delivered to you were sacred truths; but what you tell me is mere fable, fiction, and falsehood.’

The Indian, offended, replied: ‘My brother, it seems your friends have not done you justice in your education; they have not well instructed you in the rules of common civility. You saw that we, who understand and practise those rules, believed all your stories; why do you refuse to believe ours?’

— Benjamin Franklin, “Remarks Concerning the Savages of North America,” 1784

World of Wonders

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It is the one great weakness of journalism as a picture of our modern existence, that it must be a picture made up entirely of exceptions. We announce on flaring posters that a man has fallen off a scaffolding. We do not announce on flaring posters that a man has not fallen off a scaffolding. Yet this latter fact is fundamentally more exciting, as indicating that that moving tower of terror and mystery, a man, is still abroad upon the earth. That the man has not fallen off a scaffolding is really more sensational; and it is also some thousand times more common. But journalism cannot reasonably be expected thus to insist upon the permanent miracles. Busy editors cannot be expected to put on their posters, ‘Mr. Wilkinson Still Safe,’ or ‘Mr. Jones, of Worthing, Not Dead Yet.’ They cannot announce the happiness of mankind at all. They cannot describe all the forks that are not stolen, or all the marriages that are not judiciously dissolved. Hence the complex picture they give of life is of necessity fallacious; they can only represent what is unusual. However democratic they may be, they are only concerned with the minority.

— G.K. Chesterton, The Ball and the Cross, 1909