P. Anning noted this curiosity in Scripta Mathematica in 1956 — if the middle digit 1 in both the numerator and denominator of 101010101/110010011 is replaced with any odd number of 1s, then the proportion remains the same. And all of these numbers are palindromes!

Waclaw Sierpinski gives a proof in 250 Problems in Elementary Number Theory (1970).

In 1983, University of British Columbia physicist Lorne Whitehead noted “a simple and dramatic demonstration of exponential growth, as in a nuclear chain reaction.” He determined that one domino can knock down another that’s about half again as large in all dimensions; since the gravitational potential energy of an upright domino is proportional to the fourth power of its size, this means that one tiny domino can set off a graduated chain reaction with impressively thunderous results.

Whitehead’s first domino was less than 10 mm high; he nudged it with a piece of cotton. The resulting chain reaction brought down a 13th domino that was 64 times as tall; an investment of 0.024 microjoules at one end had released 51 joules of energy at the other, an amplification factor of about 2 billion.

Of course, it’s possible to construct impressive chains of graduated dominoes even if they grow less dramatically than this one. Here’s a world record set in the Netherlands in 2009:

(Lorne A. Whitehead, “Domino ‘Chain Reaction,'” American Journal of Physics 51:2 [February 1983], 183.)

To multiply this 58-digit number by 6, just move its last digit to its head.

Surprisingly, no smaller number displays this behavior when multiplied by 6. It’s called a parasitic number.

In a 2009 experiment, Keele University researchers Richard Stephens, John Atkins, and Andrew Kingston asked two groups of subjects to hold their hands in ice water. One group was asked to swear while they did so, and the other was asked to say neutral words. The swearers were able to hold their hands in the water twice as long, and these subjects reported feeling less pain.

No one’s quite sure why this works — perhaps swearing activates the amygdala, which leads to a release of adrenaline, producing natural pain relief. Stephens said, “I would advise people, if they hurt themselves, to swear.”

Interestingly, Stephens later found that people who reported swearing every day reported a lesser pain-deadening effect than those who swore less often. Perhaps people who seldom swear place a higher emotional value on these words, which triggers a stronger chemical response.

“Swearing is a very emotive form of language and our findings suggest that over-use of swear words can water down their emotional effect,” Stephens said. “Used in moderation, swearing can be an effective and readily available short-term pain reliever if, for example, you are in a situation where there is no access to medical care or painkillers. However, if you’re used to swearing all the time, our research suggests you won’t get the same effect.”

Clusters of soap bubbles obey some pleasingly simple rules: They arrange themselves into constant-mean-curvature surfaces (such as pieces of spheres) that meet in threes at 120° along seams, which in turn meet in fours at about 109° angles at points.

“Nothing more complicated ever happens,” writes mathematician Frank Morgan, even in complicated clusters with thousands of bubbles.

Belgian physicist Joseph Plateau observed and recorded this fact in the 19th century, but he offered no proof. More than 100 years would go by before Rutgers University mathematician Jean Taylor produced a complete explanation. Her demonstration required no physics or chemistry, just the principle of area minimization.

Morgan writes, “Many pages of complicated mathematics later came the conclusion: Plateau’s laws, 120° angles, 109° angles, and all.”

(Frank Morgan, “Mathematicians, Including Undergraduates, Look at Soap Bubbles,” American Mathematical Monthly 101:4 [April 1994], 343-351.)

In 1959 chemist William J. Buehler of the Naval Ordnance Laboratory was trying to devise a missile nose cone that could withstand extraordinary heat and fatigue. He found a promising alloy of nickel and titanium and passed around a sample at a 1961 laboratory management meeting. The sample had been folded like an accordion, but in examining it Buehler’s colleagues flexed and twisted it out of shape. When of them idly held it over his pipe lighter, they got a surprise: The sample sorted itself back into its accordion shape.

Buehler’s alloy is now known as nitinol (for “nickel titanium Naval Ordnance Laboratory”), and this property is known as “shape memory.” In Nature’s Building Blocks, John Emsley notes, “Spectacle frames made from nitinol can be bent and twisted into remarkable shapes and, when released, will jump back to their original shape.”

Here’s an odd little animal: Get two rigid disks, cut a notch in each one, fit them together as shown, and try to send them rolling across a table. If the notches are too deep, marrying the discs too closely together, then the object will pretty quickly slow to a stop with each disc standing at a 45° angle to the table. If the notches are too shallow, it will stop with one disc standing up at right angles to the table. But if the notches are about the right length, ideally 29.2893 percent of the radius, then the contraption will roll along quite happily for a surprisingly long distance.

The reason is that in that configuration the object’s center of mass remains level as it rolls along. (It does move from side to side, which is why it’s called the wobbler.)

Apparently this was originally discovered by A.T. Stewart, who dubbed his creation the “two-circle roller” in a 1966 note in the American Journal of Physics. I found it described in Matt Parker’s 2014 book Things to Make and Do in the Fourth Dimension, which includes a simple proof of the principle involved. There’s a more rigorous discussion here.