Dependent Claws

The first motion picture to feature a live cat is believed to be this 1894 short in which French physiologist Étienne-Jules Marey drops an inverted feline to watch it land on its feet.

When the experiment was published in Nature in 1894, the editors wrote, “The expression of offended dignity shown by the cat at the end of the first series indicates a want of interest in scientific investigation.”

The Backward Club

A self-working card curiosity by Shippensburg University mathematician Douglas E. Ensley:

I give you the four aces from a deck of cards and turn my back. Then I ask you to stack the four cards face up with the heart at the bottom, then the club, the diamond, and the spade. Now turn the uppermost card, the spade, face down.

Now you’re invited to perform any of these operations as many times and in any order that you wish:

  • Cut any number of cards from the top of the stack to the bottom.
  • Turn the top two cards over as one.
  • Turn the entire stack over.

When you’ve finished, I ask you to turn the topmost card over, then turn the top two cards over as one, then turn the top three cards over as one. I predict that the club is the only card facing the opposite way from the others, and as long as you’ve followed the directions above, it always will be.

The answer is explained by group theory — see the article below for the details.

(Douglas E. Ensley, “Invariants Under Group Actions to Amaze Your Friends,” Mathematics Magazine 72:5 [December 1999], 383-387.)


  • Thomas Jefferson, John Adams, and James Monroe all died on July 4.
  • Australia is wider than the moon.
  • NoNRePReSeNTaTiONaLiSm can be assembled from chemical symbols.
  • 1 × 56 – 1 – 7 = 15617
  • “‘Needless to say’ is, needless to say, needless to say.” — Enoch Haga

His Image
Images: PLOS One

In 2018 a team of researchers at the University of North Carolina presented 511 American Christians with randomly paired pictures of faces and asked them to identify which of the pair more closely resembled the face of God. By combining the selected faces, the psychologists could produce a composite image of the Creator as envisioned by various groups. (Here, the image on the left is God as young participants imagine him; the one on the right is how he’s seen by older participants.)

Liberals tend to imagine that God is younger, more feminine, and more loving than conservatives, and African-Americans picture a God who looks more African-American than Caucasians do, but the traditional image of the powerful older man with the flowing beard is nowhere to be seen.

“People’s tendency to believe in a God that looks like them is consistent with an egocentric bias,” said senior author Kurt Gray. “People often project their beliefs and traits onto others, and our study shows that God’s appearance is no different — people believe in a God who not only thinks like them, but also looks like them.”

One exception, though: Men and women believed in an equally masculine-looking God.

(Joshua Conrad Jackson, Neil Hester, and Kurt Gray, “The Faces of God in America: Revealing Religious Diversity Across People and Politics,” PLOS One, June 11, 2018.)


In 2006, Math Horizons challenged its readers to pose a problem in such a way that it contained its own answer. Rheta Rubenstein of the University of Michigan-Dearborn offered a pair of questions that answer one another:

  1. What fraction of the letters in three-eighths are vowels?
  2. What fraction of the letters in one-third are vowels?

(“Self-Answering Problems,” Math Horizons 13:4 [April 2006], 19.)

When in Rome …
Image: Wikimedia Commons

Oxford zoologists Tim Guilford and Dora Biro discovered a surprise in 2004: Homing pigeons sometimes just follow roads like the rest of us. Although the birds have inbuilt magnetic compasses, they fall back on the known landscape when they’re in familiar territory, following the lines of motorways and trunk roads.

Guilford and Biro strapped cameras and GPS devices to pigeons’ backs and watched them follow the A34 Oxford Bypass, turning at traffic lights and curving around roundabouts. They write, “One dominant linear feature, the A34 Oxford Bypass, appears to be associated with low entropy for much of its length, even where individual birds fly along or over it for a relatively short distance.”

“In fact, you don’t need a mini-GPS to find the circumstantial evidence” of this phenomenon, writes Joe Moran in On Roads. “You will often see seagulls in landlocked Birmingham because they have flown up the Bristol Channel and followed the M5, mistaking it for a river.”

(Tim Guilford et al., “Positional Entropy During Pigeon Homing Ii: Navigational Interpretation of Bayesian Latent State Models,” Journal of Theoretical Biology 227:1 [2004], 25-38.)

Double Magic
Image: Wikimedia Commons

This style of compound magic square was first devised by Kenneth Kelsey of Great Britain. The numbers 70-94 appear in the blue boxes, making a pandiagonal magic square. The numbers 95-110 appear in the yellow circles, making a pandiagonal magic square of their own. And embedding one in the other produces a compound square — the numbers in the circles can be added to the numbers in the squares in either of the adjoining sections. So, for example, the second row, 87 + 79 + 91 + 83 + 70 = 410, can include the row of circles above it (87 + 108 + 79 + 105 + 91 + 99 + 83 + 98 + 70 = 820) or below it (87 + 95 + 79 + 102 + 91 + 104 + 83 + 109 + 70 = 820).

In the finished figure, every number from 70 to 110 appears once, and the blue square and the yellow square have the same magic constant — 410!

The Bicycle Puzzle

Stand a bicycle so that one pedal is in its lowest position and one in its highest. Now if we pull backward on the low pedal, will the bicycle move forward or backward?

Intuition suggests it will move forward: We’re turning the pedals in the same direction that a rider would, and normally this motion drives the rear wheel to propel the bike forward.

Surprisingly, though, in the experiment the bike moves backward. That’s because (in most bikes, in most gears) each pedal is constantly moving forward with respect to the ground. So pulling backward on the pedal doesn’t produce the intuitive result — it moves the pedal backward relative to the ground, and so produces the opposite result to the one we expect.

An exception: If the bike is in a sufficiently low gear, pulling backward on the pedal will drive the bike forward. But the sprocket ratio must be so low that really we’re betraying intuition rather than the reverse (see the video).

Tursiops Economicus

In the 1970s, dolphin trainer Jim Mullen sought to encourage the dolphins at Marine World in Redwood City, California, to tidy up their pool at the end of the day. Each dolphin received a reward of fish for each piece of litter it brought to him.

“It worked very well,” Mullen told psychologist Diana Reiss. “The pool was kept neat and clean, and the dolphins seemed to enjoy the game.”

One day in the summer of 1978, a dolphin named Spock seemed unusually diligent, bringing one piece after another of brown paper to Mullen and receiving a reward each time. Eventually Mullen grew suspicious and asked an assistant to go below and look through the pool windows.

“It turned out that there was a brown paper bag lodged behind an inlet pipe,” Mullen said. “Spock went to the paper bag, tore a piece off, and brought it to me. I then gave Spock a fish, as per our arrangement, and back he went. The second time my assistant saw Spock go to the paper bag, Spock pulled at it to remove a piece, but the whole bag came out. Spock promptly shoved the bag back into place, tore a small piece off, and brought it to me. He knew what he was doing, I’m sure. He completely had me.”

Spock hadn’t been trained to tear debris to pieces, and in doing so he was certainly maximizing his reward, Reiss writes. “And when he pushed the bag back behind the pipe when it came out in one piece, that certainly had the ring of deliberate action. Whether you can call it deliberate deception is a tough call.”

(Diana Reiss, The Dolphin in the Mirror, 2011.)

The Chameleon Vine
Image: Wikimedia Commons

Native to Chile and Argentina, Boquila trifoliolata has a remarkable ability: Once it’s wrapped its vines around a host plant, it can alter its leaves to mimic those of the host, a phenomenon called mimetic polymorphism.

“It modifies its size, shape, color, orientation, and even the pattern of its veins in such a way that it fuses perfectly with the foliage of the tree that bears it,” writes botanist Francis Hallé in his 2018 Atlas of Poetic Botany. “If, in the course of growing, it changes its support, the same stalk can even display leaves that are completely different, corresponding to the new tree — even if these leaves are much bigger.”

This helps it to avoid predators. If the plant grew along the forest floor it would be eaten by weevils, snails, and leaf beetles, but these tend to leave it alone when it disguises itself with “tree leaves.” But how it accomplishes the mimicry remains unclear.