Left or Right?


A curious physics puzzle from Mark Levi’s excellent Why Cats Land on Their Feet: Suppose two astronauts, Al and Bob, are strapped to opposite ends of a space capsule’s interior, Al on the left and Bob on the right. Al is holding a large helium balloon, and everything is at rest. If Al pushes the balloon toward Bob, which way will the capsule drift?

It would be reasonable to guess that the capsule will drift to the left. Newton’s third law says that action equals reaction, so as Al pushes the balloon to the right, the balloon pushes Al to the left, and since he’s strapped to the capsule, he and it should drift left.

In fact the capsule will drift right as well. Because there are no external forces, the center of mass of the whole system is fixed. The helium balloon has less mass than the air it displaces, so from Al’s point of view the center of mass moves left. But the center of mass of the whole system is fixed in space, so the capsule must move right from the point of view of an external observer.

One way to make this intuitive is to imagine that the capsule is full of water rather than air. The mass of water essentially stays in place while we transfer a bubble of helium from the water’s left to its right. To accommodate this, the shell (whose mass we neglect) must move to the right.


In May 1936 a publisher invited Albert Einstein to contribute a message to be sealed in a metal box in the cornerstone of a new library wing in his country home, to be opened a thousand years hence. He sent this:

Dear Posterity,

If you have not become more just, more peaceful, and generally more rational than we are (or were) — why then, the Devil take you.

(From Helen Dukas and Banesh Hoffmann, eds., Albert Einstein, the Human Side, 1979.)

An Odd Bond


This arrangement of rubber bands is “Brunnian”: Though the bands are entangled, no two are directly linked; and though no band can be extricated from the mass, cutting any one of them will free all the others.

By following the pattern here, any number of bands can be joined in a configuration with the same properties.


During World War I, British physicist G.I. Taylor was asked to design a dart to be dropped onto enemy troops from the air. He and a colleague dropped a bundle of darts as a trial and then “went over the field and pushed a square of paper over every dart we could find sticking out of the ground.”

When we had gone over the field in this way and were looking at the distribution, a cavalry officer came up and asked us what we were doing. When we explained that the darts had been dropped from an airplane, he looked at them and, seeing a dart piercing every sheet remarked: ‘If I had not seen it with my own eyes I would never have believed it possible to make such good shooting from the air.’

(The darts were never used — “we were told they were regarded as inhuman weapons and could not be used by gentlemen.”)

(From T.W. Körner, The Pleasures of Counting, 1996.)


Artist Patrick Hughes calls this illusion “reverspective” — the “end” of each gallery hallway is actually nearest the viewer.

“Hughes acknowledges that these types of paintings have been his most successful and they continue to intrigue him,” writes Brad Honeycutt in The Art of Deception. “As such, he says that he could very well concentrate on creating paradoxical perspective pictures for the rest of his days.”

More examples.

The Martians

In the first half of the 20th century, a considerable number of famous scientists emigrated from Hungary to the United States, including physicists Eugene Wigner, Edward Teller, and Dennis Gabor and mathematicians Theodore von Kármán, John von Neumann, Paul Halmos, George Pólya, and Paul Erdős. Most were Jewish refugees from Nazi Germany, but they had surprising further similarities — many had been born near Budapest, had shown an early interest in chemistry, and had studied physics at German universities before emigrating to America.

One of their number, Leo Szilard, joked that he knew the reason: They were all descended from a Martian scout force that had landed on Earth in that period. The Martians had left eventually, but not before impregnating some Earth women.

The “Martians” adopted Szilard’s name because in many ways they felt themselves to be outsiders in America: All were brilliant, spoke English with a strong accent, and came from a small little-known country.

When Enrico Fermi posed his famous paradox — if intelligent aliens are as common as we believe, why haven’t we encountered one? — Szilard answered, “They are among us — but they call themselves Hungarians.”

(Thanks, Rini.)

The Octave Illusion

University of California psychologist Diana Deutsch discovered this illusion in 1973. Play the file using stereo headphones. If you hear a high tone in one ear and a low tone in the other, decide which ear is hearing the high tone. Then reverse the headphones and play the file again.

“Despite its simplicity, this pattern is almost never heard correctly, and instead produces a number of illusions,” Deutsch writes. Some people hear a single moving tone; some hear silence; some notice no change when the headphones are reversed. Some impressions even seem to vary with the handedness of the subject!

What you’re hearing is simply an octave interval, with the high note played in one ear and the low in the other, the two regularly switching places. Seen on paper it’s remarkably simple, which makes the confusion all the more striking. Deutsch suspects that two different decision mechanisms are being invoked at once — one determines what pitch we hear, and the other determines where it’s coming from. More info here.

Buttoned Up

Divide a pile of 14 buttons into two smaller piles, say of 9 and 5 buttons. Then write on a piece of paper: 9 × 5 = 45. Divide the pile of 9 into two smaller piles, say of 6 and 3, and write 6 × 3 = 18 on the paper. Keeping doing this, splitting each pile into two and recording the pair of numbers you get, until you have 14 separate piles of one button each. An example might run like this:

9 × 5 = 45

6 × 3 = 18
1 × 4 = 4

4 × 2 = 8
2 × 1 = 2
2 × 2 = 4

3 × 1 = 3
1 × 1 = 1
1 × 1 = 1
1 × 1 = 1
1 × 1 = 1

1 × 2 = 2

1 × 1 = 1

No matter how you proceed, if you start with a pile of 14 buttons, the products in the right column will always sum to 91.

(James Tanton, “A Dozen Questions About Pile Splitting,” Math Horizons 12:1 [September 2004], 28-31.)

“Coal Is Decayed Vegetarians”

Memorable excerpts from student geology examinations, from W.D. Ian Rolfe’s 1980 collection Geological Howlers:

  • The average person does not have to dig a deep hole to remind himself of the past.
  • Dust is mud with the juice squeezed out.
  • Articulate brachiopods have teeth and socks.
  • A skeleton is a man with his inside out and his outside off.
  • There are three kinds of rocks, ingenious, sedentary and metaphoric.
  • The term Caledonian Orogeny is brandished about by many people.
  • Nine-eighths of an iceberg is beneath the sea.
  • It has been found by a gentle man that organic remains can be converted to petroleum by the processes of metabolism.
  • Sedimentation is a rather lengthy affair.

“A dinosaur is an extinct animal still found in Australia,” one student contended. “It was sometimes so large that its feet are found in the Precambrian and its head in the Silurian because it was too big to lie down where it died.”