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.

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.

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.

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.)

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.”

Webster’s New World Dictionary of the American Language defines paradox as “1. a statement that seems contradictory, etc. but may be true in fact. 2. a statement that is self-contradictory and, hence, false.”

(Noted by David Morice in the August 1993 issue of Word Ways.)