Psychologist Abraham S. Luchins discovered a discouraging phenomenon in 1942: When people find a problem-solving strategy that works successfully in multiple trials, they’ll tend to adopt the same strategy even in situations when more efficient solutions are available.

Suppose you’re given water jugs with capacities of 21 units (A), 127 units (B), and 3 units (C) of water. Then you’re asked to use these to measure out 100 units of water. You find that you can do this by filling jug B and using that to fill jug A once and jug C twice, leaving 100 units in jug B. In several similar problems you find that this strategy, B – A – 2C, works.

But then you’re given an “extinction problem” in which this strategy doesn’t work. And in the next problem you’re given jugs of capacity 15, 39, and 3 and asked to measure 18 units of water. The old strategy, B – A – 2C, works here, but there’s a simpler solution: A + C.

In Luchins’ experiments, a control group that had not been primed with the early B – A – 2C successes hit immediately on the A + C solution in the last test. But subjects who did have those early successes tended to revert to the original strategy, reflecting a “mechanized” state of mind that prevented them from seeking more efficient solutions. When Luchins told them “Don’t be blind,” more than half of them found the A + C solution.

The reasons for this are still being studied in different populations and with different types of tasks. Luchins acknowledges that habit can be useful, but notes that when, “instead of the individual mastering the habit, the habit masters the individual — then mechanization is indeed a dangerous thing.”

(Abraham S. Luchins, “Mechanization in Problem Solving: The Effect of Einstellung,” Psychological Monographs 54:6 [1942]: i.)

Lime juice is phototoxic — on contact it sensitizes the skin to ultraviolet light, so that exposure to sunlight can then produce redness, itching, burning, and even blisters.

The reaction is called phytophotodermatitis, or margarita photodermatitis. It can also be caused by celery, parsnips, parsley, and figs.

A dominant male long-tailed manakin acquires a team of subordinate males to help him woo females. “It’s the only example of cooperative male-male displays ever discovered in the entire animal kingdom,” writes Noah Strycker in The Thing With Feathers.

It’s common for male animals to cooperate to impress females, but typically each of those males is hoping to mate. Among manakins the eldest male gets this right, and the others defer until they succeed him.

Strycker writes, “A pair of male long-tailed manakins may work together like this for five years, building up their jungle reputation as hot dancers, before the alpha male dies and the backup singer takes his place with a new apprentice.”

Conducting a workshop on paper folding and geometry for a group of gifted 10-year-olds in 1977, Santa Clara University mathematician Jean Pedersen passed around a collection of polyhedra and asked the students which shapes they’d classify as “regular.” To her surprise, the only one who chose the five platonic solids was Peter Wilson, a blind student.

The others immediately responded, “That’s not fair, Peter’s blind!” So Pedersen agreed to let them try again, this time feeling the models with their eyes closed. Now every student chose the five platonic solids.

“I’m not sure what all the ramifications of these events are,” Pedersen wrote in a letter to the Mathematical Intelligencer, “but begin with this: we can perceive things with just our hands that we miss when we use both our eyes and our hands. Sometimes less really is more.”

(Jean Pedersen, “Seeing the Idea,” Mathematical Intelligencer 20:4 [Fall 1998], 6.)

In 1822, when Europeans were still searching for an explanation for the annual disappearance of some bird species, a white stork appeared bearing a central African arrow in its neck. This helped to show that some birds migrate long distances for the winter.

The stuffed stork can be seen today at the University of Rostock, where it bears the magnificent name Rostocker Pfeilstorch (“arrow stork from Rostock”).

A curious puzzle from Pi Mu Epsilon Journal, Fall 1968 [Volume 4, Issue 9]:

Where must a man stand so as to hear simultaneously the report of a rifle and the impact of the bullet on the target?

On the second day of Apollo 16’s trip to the moon in 1972, command module pilot Ken Mattingly lost his wedding ring. “It just floated off somewhere, and none of us could find it,” lunar module pilot Charlie Duke told Wired in 2016.

Mattingly looked for it intermittently over the ensuing week, with no luck. By the eighth day, Duke and Commander John Young had visited the moon and rejoined him, but there was still no sign of the ring.

But during a spacewalk the following day, Mattingly was just heading back toward the open hatch when Duke said, “Look at that!” The ring was floating just outside the hatch. “I grabbed it,” he said, “and we put it in the pocket. We had the chances of a gazillion to one.”

Duke said later, “You plan and plan and plan but the unexpected always jumps up and bites you.”

(From Ben Evans, Foothold in the Heavens: The Seventies, 2010.)

A stunning geometric alphamagic square by Lee Sallows. The 3 × 3 grid is a familiar magic square in which each number is spelled out: The first cell contains the number 25, the second 2, and so on. Interpreted in this way, each row, column, and long diagonal sums to 45.

But there’s more: The English name of the number in each cell has been arranged onto a distinctive tile, such that the three tiles in any row, column, or long diagonal can be combined to form the same 21-cell figure, as shown. (Shapes with dotted outlines have been turned over.)

And yet more: Count the number of letters in each of the number names (or, equivalently, count the number of cells that make up each tile). So, for example, TWENTY-FIVE has 10 letters, so replace the TWENTYFIVE tile with the number 10. Similarly, replace TWO with 3, EIGHTEEN with 8, and so on. This produces another magic square:

10  3  8
 5  7  9
 6 11  4

Each row, column, and long diagonal totals 21.

I’m just sharing this because I think it’s pretty — it’s the smallest arrangement of identical non-crossing matchsticks that one can make on a tabletop in which each match-end touches three others.

Presented by German mathematician Heiko Harborth in 1986, it’s known as the Harborth graph.