Vital Signs

In 1921 someone began stealing money, jewelry, and clothing from a girls’ dormitory at the University of California. When the residents themselves were unable to identify the culprit, student Margaret Taylor made a formal complaint to the police.

The police elected to try something new. One of their number, 23-year-old John A. Larson, had been experimenting with a lie-detecting device that measured a subject’s respiration, blood pressure, and other physical reactions as she responded to a series of questions. He asked the women’s consent to use the device in his investigation, and they agreed.

He started with Margaret Taylor, the student who had first reported the thefts. Part of his technique was to engage in preliminary small talk with the subject, to put her at ease. He found Taylor intelligent and witty, and she said she found his work fascinating and admired his ambition. She passed the test easily, showing no response to key terms such as crime, locker, or purse among Larson’s questions.

In reviewing the results, Larson realized that he might not have eliminated all the extraneous factors that could have affected the young women’s responses — he had tried to make them comfortable with the machine, but some of them also “might have been reacting to the questioner, not the questions.” So he called back Margaret Taylor to test this proposition. He connected her to the machine again and asked her to lie to him. Then he asked her out.

The two were married a year later. One of Larson’s assistants said, “It was an odd way to begin a romance.” The dorm thief was discovered among the other women, and Margaret recovered a $400 diamond ring she had lost. Today Larson is remembered as the father of the polygraph.

In recounting this story in his 2009 book The Lie Detectors, Ken Alder writes:

“Years later, he still had the record of their first meeting in his files, the zigzag trace of her heart as he asked her, ‘Are you interested in this test?'”

Seeking Stability

You’re standing in a room with an uneven floor. Before you is a square table with four legs. The table wobbles, but by turning it gradually you manage to find a position in which all four feet are supported, eliminating the wobble (though now the tabletop isn’t level).

You wonder: Is this always possible? Assuming that the four legs are of equal length and that the surface of the floor varies smoothly, is it always possible to position a four-legged table so that all four legs are supported?

Click for solution …

Crime and Punishment

In the eyes of the law, a corporation is a person. But it’s a strangely bodiless person, which makes it tricky to punish with laws designed for human offenders.

Tired of this, federal judge Robert Doumar in 1988 sentenced Allegheny Bottling Company to three years in prison and a fine of $1 million. The company had been caught in a price-fixing scheme that cheated consumers out of at least $10 million. “Congress has not said a corporation could not be imprisoned,” Doumar said. “This court will deal with any individual who similarly disregards the law.”

How? The court decided that the essence of imprisonment is restraint, or a deprivation of liberty, and that it could restrain or immobilize a corporation — by closing the physical plant and guarding it, for example. Doumar suspended the sentence but said that if the company violated its probation he would send a U.S. Marshal to “padlock every facility Allegheny owns.”

“Some 200 years ago, the Lord Chancellor of England said, ‘You cannot expect a corporation to have a conscience when it has no soul to be damned and no body to be kicked,'” he said. “Obviously, Allegheny Bottling Company did not have a conscience.”

More recently, California resident Jonathan Frieman put a charity’s incorporation papers in his passenger seat and drove in the carpool lane, arguing that his car now had multiple occupants.

“After I explained the reason I was citing him, he explained to me that he was exempt because he was in essence a corporation,” CHP Officer Troy Dorn testified. “I explained to him I was not sure about his standing as a corporation but he could explain it later in a Marin County court.”

Jurist Frank Drago admired the novelty of Frieman’s argument but said, “I look at it a little differently. … Common sense says carrying a sheath of papers in the front seat does not relieve traffic congestion. And so I’m finding you guilty.”

The Conway Immobilizer

Three positions, “left,” “middle,” and “right,” are marked on a table. Three cards, an ace, a king, and a queen, lie face up in some or all three of the positions. If more than one card occupies a given position then only the top card is visible, and a hidden card is completely hidden; that is, if only two cards are visible then you don’t know which of them conceals the missing card.

Your goal is to have the cards stacked in the left position with the ace on top, the king in the middle, and the queen on the bottom. To do this you can move one card at a time from the top of one stack to the top of another stack (which may be empty).

The problem is that you have no short-term memory, so you must design an algorithm that tells you what to do based only on what is currently visible. You can’t recall what you’ve done in the past, and you can’t count moves. An observer will tell you when you’ve succeeded. Can you devise a policy that will meet the goal in a bounded number of steps, regardless of the initial position?

“It’s tricky to design an algorithm that makes progress, avoids cycling, and doesn’t do something stupid when it’s about to win,” wrote Dartmouth mathematician Peter Winkler in sharing this puzzle in his book Mathematical Puzzles: A Connoisseur’s Collection (2003). It’s called “The Conway Immobilizer” because it originated with legendary Princeton mathematician John H. Conway and because it’s said to have immobilized one solver in his chair for six hours.

Click for solution …

Still Life

http://www.todayandtomorrow.net/page/101/

In 2009 British artist Roger Hiorns won the Turner Prize by melting a passenger aircraft engine, pouring it through a funnel, and spraying it with water to break it into fine granules.

Its dimensions are listed as “variable.”

Likewise

A charming little scene from mathematical history — in 1615 Gresham College geometry professor Henry Briggs rode the 300 miles from London to Edinburgh to meet John Napier, the discoverer of logarithms. A contemporary witnessed their meeting:

He brings Mr. Briggs up into My Lord’s chamber, where almost one quarter of an hour was spent, each beholding the other with admiration, before one word was spoke: at last Mr. Briggs began. ‘My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto Astronomy, viz. the Logarithms: but my Lord, being by you found out, I wonder nobody else found it before, when now being known it appears so easy.’

Their friendship was fast but short-lived: The first tables were published in 1614, and Napier died in 1617, perhaps due to overwork. In his last writings he notes that “owing to our bodily weakness we leave the actual computation of the new canon to others skilled in this kind of work, more particularly to that very learned scholar, my dear friend, Henry Briggs, public Professor of Geometry in London.”

The Amazing Sand Counter

A man presents himself as the The Amazing Sand Counter. He claims that if you put some quantity of sand into a bucket, he will know at a glance how many grains there are, but he won’t tell you the number. Can you devise a test that can verify this ability without telling you anything that you don’t already know? You can ask the Sand Counter to leave the room or turn away, for example, and you can ask him questions. How can you convince yourself that he knows how many grains of sand are in the bucket when he won’t actually tell you the number?

Click for solution …

Self-Test

leder patent

Xavier Henry Leder, who declares himself “by profession a seaman,” patented this “foul breath indicator” in 1902, perhaps after inventing it for his own use.

“It is an appliance in the shape of a tube, made of any non-absorbent material and curved so as to transmit without any obstruction the breath from the mouth to the nostrils. … By breathing from the mouth through the tube any foulness or unpleasant state of the breath may be readily detected by the sense of smell.”

The hard part is exhaling and inhaling at the same time.

Unquote

“The greatest deception men suffer is from their own opinions.” — Leonardo

“Convictions are more dangerous enemies of truth than lies.” — Nietzsche

“Opinions have caused more ills than the plague or earthquakes on this little globe of ours.” — Voltaire

“The most difficult subjects can be explained to the most slow-witted man if he has not formed any idea of them already; but the simplest thing cannot be made clear to the most intelligent man if he is firmly persuaded that he knows already, without a shadow of doubt, what is laid before him.” — Tolstoy

“Nothing is so firmly believed as that which we least know.” — Montaigne

“Opinions are made to be changed, or how is truth to be got at?” — Lord Byron

“Nothing is more conducive to peace of mind than not having any opinions at all.” — G.C. Lichtenberg

Encounter

https://commons.wikimedia.org/wiki/File:Paul_Klee_Eine_Art_Katze_1937.jpg

We had a dark grey cat (Norfolk bred, very Norfolk in character) called Tom. He was reserved, domineering, voluptuous — much as I imagine Tiber to be. When he was middle-aged he gave up nocturnal prowlings and slept on my bed, against my feet. One evening I was reading in bed when I became aware that Tom was staring at me. I put down my book, said nothing, watched. Slowly, with a look of intense concentration, he got up and advanced on me, like Tarquin with ravishing strides, poised himself, put out a front paw, and stroked my cheek as I used to stroke his chops. A human caress from a cat. I felt very meagre and ill-educated that I could not purr. It had never occurred to me that their furry love develops from what was shown them as kittens.

— Sylvia Townsend Warner, letter to David Garnett, June 18, 1973, quoted in The Oxford Book of Friendship

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