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

October 26, 2023October 25, 2023 | Quotations · Society

Round Numbers

halmos map

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.

October 26, 2023October 25, 2023 | Science & Math

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?

	Select Click for Answer
This solution is by Mandy Rodgers and Josh Trejo. C accuses B of lying, so they can’t both be knights or both knaves. Assume first that B is a knight and C is a knave. That would mean that A really did tell the visitor that there was one knight among the three islanders. But if A’s statement were true then both A and B would be knights, which is a contradiction; and if A’s statement were false then B really would be the only knight, which again is a contradiction. So B can’t be a knight. That means that B is a knave, which makes C a knight. Note that we don’t know whether A is a knight or a knave. It’s possible that he’s a knight and said that the trio includes two knights, which would be true (in this case A/B/C are knight/knave/knight). And it’s possible that he’s a knave and said that all three were knights or all knaves, both of which statements we know are false. These possibilities are both self-consistent, so we can’t say anything about A’s identity, only that B lies and C tells the truth. Obligatory John Finnemore sketch:

Click for Answer

October 25, 2023October 24, 2023 | Puzzles

The Region Beta Paradox

https://commons.wikimedia.org/wiki/File:Region-beta_paradox_cropped.png — Image: Wikimedia Commons

Suppose you decide that you’ll walk (at 3 mph) to any destination that’s within a mile of your house, and bike (at 15 mph) to any destination that’s farther away. That’s a reasonable choice, but it has a surprising result: You’ll actually arrive more quickly at moderately distant points (1 to 5 miles away) than at most points closer to home (less than 1 mile away).

Psychologist Daniel Gilbert uses this example to illustrate a phenomenon in our reactions to stressful events: Sometimes we’ll recover more quickly from particularly distressing experiences because they’re strong enough to invoke defense processes that attentuate stress.

October 25, 2023October 24, 2023 | Science & Math

The Six Submarines

A puzzle by Henry Dudeney:

If five submarines, sunk on the same day, all went down at the same spot where another had previously been sunk, how might they all lie at rest so that every one of the six U-boats should touch every other one? To simplify we will say, place six ordinary wooden matches so that every match shall touch every other match. No bending or breaking allowed.

	Select Click for Answer
It will be seen from the illustration that this puzzle is absurdly easy — when you know how to do it! And yet I have not the slightest doubt that many readers found it a hard nut to crack. It will be seen that every match undoubtedly touches every other match. From Dudeney’s Modern Puzzles and How to Solve Them, 1926. Related: Suppose we’re coloring a map and we decide that no two adjoining regions should have the same color. What’s the smallest number of colors we’ll need to be sure of doing the job? On a flat map the answer is famously four. But what if we extend the question to three dimensions and try to color regions in space? In that case we might need any number of colors, as shown in this “proof without words” by CMG Lee: Image: Wikimedia Commons

Click for Answer

October 24, 2023October 24, 2023 | Puzzles · Science & Math

Boo!

J.H. Brown’s 1864 book Spectropia: Or, Surprising Spectral Illusions promises to show “ghosts everywhere, and of any colour.” It accomplishes this by relying on two simple principles: persistence of vision and complementary colors. Readers are directed to stare at any of the figures for 15 seconds and then turn their eyes to a white surface (or the sky); “the spectre will soon begin to make its appearance, increasing in intensity, and then gradually vanishing,” in the color complementary to that of the stimulus.

Try it yourself.

October 24, 2023October 23, 2023 | Entertainment · Science & Math

All Together Now

Kurt Vonnegut wrote, “The function of the artist is to make people like life better than before.”

Asked whether he’d ever seen this done, he said, “Yes, the Beatles did it.”

(From Dan Wakefield’s introduction to Vonnegut’s If This Isn’t Nice, What Is?, 2013.)

October 23, 2023October 22, 2023 | Entertainment

Renewal

After Archimedes’ death in 212 B.C., his tomb in Sicily fell into obscurity and was eventually lost. It was rediscovered by, of all people, Cicero, who had been sent to the island in 75 B.C. to administer corn production:

When I was Quaestor, I tracked down his grave; the Syracusans not only had no idea where it was, they denied it even existed. I found it surrounded and covered by brambles and thickets. I remembered that some lines of doggerel I had heard were inscribed on his tomb to the effect that a sphere and a cylinder had been placed on its top. So I took a good look around (for there are a lot of graves at the Agrigentine Gate cemetery) and noticed a small column rising a little way above some bushes, on which stood a sphere and a cylinder. I immediately told the Syracusans (some of their leading men were with me) that I thought I had found what I was looking for. Slaves were sent in with scythes to clear the ground and once a path had been opened up we approached the pedestal. About half the lines of the epigram were still legible although the rest had worn away.

“So, you see, one of the most celebrated cities of Greece, once upon a time a great seat of learning too, would have been ignorant of the grave of one of its most intellectually gifted citizens — had it not been for a man from Arpinum who pointed it out to them.”

(From Anthony Everitt, Cicero, 2003.)

October 23, 2023October 22, 2023 | History · Science & Math

Poser

From reader Éric Angelini:

What is the smallest number that doesn’t have a 1 and yet has one?

	Select Click for Answer
2,000,008. It doesn’t contain the digit 1, but its English name contains the string ONE (two milliON Eight). (Thanks, Éric.) 10/22/2023 UPDATE: Full credit for the answer none! Thanks to the readers who pointed this out.

October 22, 2023October 22, 2023 | Puzzles