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.)

If you erect equilateral triangles on two adjacent sides of a square and then connect the triangle vertices distant from the square to the square vertex distant from the triangles, you get a third equilateral triangle.

Pleasingly, this works whether the triangles are erected inside or outside the square. It was discovered by French mathematician Victor Thébault.