A puzzle by N. Avilov, from the Russian science magazine Kvant. Prove that the shaded region of this pentagram takes up half its area.
Author: Greg Ross
Unquote
“Politeness and a sense of honor have this advantage: we bestow them on others without losing a thing.” — Baltasar Gracián
“Be not niggardly of what costs thee nothing, as courtesy, counsel, & countenance.” — Ben Franklin
Some Enchanted Evening
For the 1968 exhibition Cybernetic Serendipity in London, inventor Gordon Pask created a society of mobiles, two “males” and three “females” that had to learn to communicate, cooperate, and compete in order to satisfy their drives. The males could project light beams, and the females could, if they chose, reflect a beam back to the sending male, which he desired. The males had to compete with one another to find cooperative females, and the females competed to find males projecting suitably colored light.
Male I sends out an intermittent directional visual signal which serves to identify it as ‘male I’ and its desire as ‘O [orange] satisfaction.’ … Should the directional signal fall on the receptor of a female who is trying to cooperate, she produces an identifying sound in synchrony with the intermittent light signal. Male I detects the correlation between the female and his light signal and stops his motion (unless he is prevented from doing so by male II). At this point he triggers off an autonomous energetic event which consists in shining an intense orange light for at least a minimum interval in the direction of the located female. The immediate result is an increase of the O drive. However, male I anticipates reinforcement (which he will achieve if the female behaves appropriately and if the moving part, C, is appropriately positioned during at least some of this behaviour). Reinforcement, which substantially reduces the O drive, is obtained if the O goal is satisfied; that is if orange light falls on receptor C. Supposing reinforcement occurs, male I emits an identifying sound signal which is received by the cooperating female, the autonomous energetic event is prolonged and the O drive is decreased.
“The cooperative encounter terminates after a short time if reinforcement does not occur, or if it is externally disrupted. Otherwise it continues until the drive state of male I is modified so that he aims for a different goal.”
Pask stressed that the mobiles needed to learn from experience in order to satisfy their drives. The females had to learn how to position their reflectors to attract males, but the system encouraged them to find different strategies so that not all males demanded stimulation of the same receptor. Pask said, “Some may like O light on D and P [puce] light on C. She can learn that trick also.”
(From Paul Brown, et al., eds., White Heat Cold Logic: British Computer Art 1960-1980, 2008.)
Outsiders
After several delusional episodes, seamstress Agnes Richter was institutionalized at the University of Heidelberg Psychiatric Clinic in 1893, at age 49. While performing the needlework expected of female patients, she sewed a diary of sorts into a remarkable jacket pieced together of wool and linen. “Writing” in a now-obsolete German script, she recorded brief, enigmatic expressions reflecting life in a psychiatric hospital: I wish to read, I am not big, I plunge headlong into disaster. Her laundry number, 583, appears several times, apparently to ensure that the jacket was not lost during cleaning.
Another patient, Mary Lieb, institutionalized periodically at Heidelberg for mania, would sometimes decorate the floors of various rooms with patterns of cloth strips. The warders found some of these remarkable enough to photograph (below). Physician Hans Prinzhorn included some of the photographs in his collection of the art of the insane, and the images have survived to the present day as strangely vivid marks of an inscrutable self-expression.
“The patterns are extraordinary, comprising rows of starbursts (or perhaps flowers), letters, crosses, geometric patterns, and sometimes intricate curved figures,” writes Lyle Rexer in How to Look at Outsider Art. “Their purpose and organization are unclear, but like much outsider art, the work appears to be a combination of decoration and communication, an attempt to reorder the space ‘from the ground up,’ visually transform it, and invest it with new significance.” What it means only Lieb knew.
Signature
In the 13th century, in England’s Worcester Priory, an anonymous scribe worked at inserting interlinear notations into Old English manuscripts. Though his identity has been lost, his shaky, leftward-sloping handwriting is so distinctive that he’s noted among scholars more than 700 years later. He’s known as the Tremulous Hand of Worcester.
The cause of the tremor is uncertain, but its identifiable character has shed light on the evolution of the language and on the ability to read Old English in this period. “For us at least,” writes literary scholar Christine Franzen, “his infirmity was fortuitous — if his hand had remained steady and unchanged throughout his glossing career, it might have been impossible to distinguish the layers of glossing, but as it is, we can watch his methods and knowledge develop along with his tremble.”
(Christine Franzen, The Tremulous Hand of Worcester, 1991.)
“The Unlucky Hatter”
From The Book of 500 Curious Puzzles, 1859:
A blackleg passing through a town in Ohio, bought a hat for $8 and gave in payment a $50 bill. The hatter called on a merchant near by, who changed the note for him, and the blackleg having received his $42 change went his way. The next day the merchant discovered the note to be a counterfeit, and called upon the hatter, who was compelled forthwith to borrow $50 of another friend to redeem it with; but on turning to search for the blackleg he had left town, so that the note was useless on the hatter’s hands. The question is, what did he lose — was it $50 besides the hat, or was it $50 including the hat?
This is not so much a puzzle as a perplexity. “[I]n almost every case the first impression is, that the hatter lost $50 besides the hat, though it is evident he was paid for the hat, and had he kept the $8 he needed only to have borrowed $42 additional to redeem the note.”
Private Collection
In November 1985, a couple walked into an art museum in Tucson, Arizona. While the woman chatted with a security guard, the man disappeared briefly upstairs, and then the pair departed. Then the guard discovered that Willem de Kooning’s painting Woman-Ochre was missing — it had been cut out of its canvas.
More than 30 years later, in 2017, retired New York speech pathologist Rita Alter passed away in the little town of Cliff, N.M., five years after her husband, Jerry, a former schoolteacher. In their bedroom was the missing de Kooning, in a position that was visible only when the door was closed. The painting appeared to have been reframed only once in the 31 years it had been missing, suggesting that it had had only one owner in that time.
Had the Alters stolen the painting? They were admirers of de Kooning and had been in Tucson the day before the theft. But such a crime seems vastly out of character for the retiring couple. “[They wouldn’t] risk something as wild and crazy as grand larceny — risk the possibility of winding up in prison, for God’s sake — they wouldn’t do that,” Rita’s sister told the New York Times.
Had the pair then bought the painting from a third party? That seems impossible too — it was worth an estimated $160 million. Perhaps the painting’s authenticity had been forgotten by the time of the transaction, so that both buyer and seller thought it was a copy? How could that have come about?
Jerry Alter once published a story in which a woman and her granddaughter steal an emerald from a museum and keep it on private display, “where two pairs of eyes, exclusively, are there to see.” Is that a coincidence? A veiled admission?
We may never know. The FBI’s case remains open.
(Thanks, Daniel.)
Skill
Letter to the Times, Oct. 23, 2001:
Sir, As a schoolboy in the 1940s I heard the late Sir Robert Wood, Principal of the (then) University College of Southampton, proclaim at a school speech day:
‘The advantage of a classical education is that it teaches you to do without the money it makes you unable to acquire.’
Yours faithfully,
Bill Kirkman
Willingham, Cambridge
Round Numbers
A bit more on map coloring: Suppose a map consists of a number of overlapping circles, like this, so that the borders of each “country” are all arcs of circles. How many colors would we need to color this map, again with the proviso that no two countries that share a border will receive the same color?
Here we need only two. Each country occupies the interior of some number of circles. If that number is even, color the country white; if odd, black. Crossing a border always changes the number by 1, so each border will divide countries of opposite colors.
From Paul R. Halmos, Problems for Mathematicians, Young and Old, 1991.
The Unheard Islander
A puzzle by Edward J. Barbeau, from the February 2007 issue of Crux Mathematicorum:
A certain familiar island is inhabited by knights, who can only speak the truth, and knaves, who can only lie. One day a visitor meets three inhabitants, A, B, and C. The visitor asked, “How many knights are there among you three?”
A gave an answer, which the visitor didn’t hear. When the visitor asked B what A had said, B replied, “A said that there is one knight among us.” At this C said, “Don’t believe B. He is lying.”
What are B and C?