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?

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.

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.

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.

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.)

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.)