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.

A perplexing story from logician Raymond Smullyan:

Oh, one other thing. I must tell you of a certain great Sage in the East who was reputed to be the wisest man in the world. A philosopher heard about him and was anxious to meet him. It took him fifteen years to find him, but when he finally did, he asked him: ‘What is the best question that can be asked, and what is the best answer that can be given?’ The great Sage replied: ‘The best question that can be asked is the question you have asked, and the best answer that can be given is the answer I am now giving.’

It’s at the very end of his last book, A Mixed Bag, from 2016.

A normal die is painted so that it has four green faces and two red. Then it’s shaken in a cup and thrown repeatedly onto a table. You’re invited to guess which of these three sequences results. If you guess wrong you lose $10; and if you guess right you win $30.

1. RGRRR
2. GRGRRR
3. GRRRRR

Most people express the preferences 2, 1, 3, in that order. Red is less likely than green, but it predominates in all three sequences, so many subjects explain that sequence 2 is more “balanced,” and therefore more probable. In fact 65 percent of all subjects (excluding expert statisticians and people whose business is probability) show a strong propensity to vote for sequence 2, even when it’s pointed out explicitly that sequence 1 is just sequence 2 minus the first throw — so sequence 2 cannot be more likely!

“The longer the sequence, the less probable it is, independently of its being ‘balanced’ or ‘unbalanced,'” writes Massimo Piattelli-Palmarini in Inevitable Illusions. “This shows how resistant certain cognitive illusions are. Many other more complex examples have been advanced, and these show that even professional statisticians are sometimes subject to the same illusion.”