# Bird Brains

Crows are smart. In 2014, University of Auckland psychologist Sarah Jelbert and her colleagues assessed the causal understanding of water displacement in New Caledonian crows by presenting them with a narrow tube in which a reward floated out of reach. To get the reward, a bird had to drop objects into the tube to raise the water level.

“We found that crows preferentially dropped stones into a water-filled tube instead of a sand-filled tube; they dropped sinking objects rather than floating objects; solid objects rather than hollow objects, and they dropped objects into a tube with a high water level rather than a low one.”

Apparently crows read Aesop. And Aesop was right.

(Sarah A. Jelbert et al., “Using the Aesop’s Fable Paradigm to Investigate Causal Understanding of Water Displacement by New Caledonian Crows,” PloS One 9:3 [2014], e92895.)

# Magic

A “kinde of Divination” “to tell your friend how many pence or single peeces, reckoning them one with another, he hath in his purse, or should thinke in his minde,” from Robert Recorde’s The Ground of Arts, 1618:

[F]irst bid him double the peeces hee hath in his purse, or the number hee thinketh. … Now after hee hath doubled his number, bid him adde thereunto 5 more, which done, bid him multiply that his number by 5 also: which done bid him tell you the just sum of his last multiplication, which sum the giver thinking it nothing availeable, because it is so great above his pretended imagination: yet thereby shall you presently with the helpe of Subtraction tell his proposed number.

Apparently the section on “divers Sportes and Pastimes, done by Number” was contributed by Southwark schoolmaster John Mellis in 1582. “[T]he fact that this chapter on mathematical games was included in every subsequent edition of The Ground of Artes, save one, indicates that the idea of mathematical games found a receptive audience among arithmetic students.”

(Jessica Marie Otis, “‘Sportes and Pastimes, done by Number’: Mathematical Games in Early Modern England,” in Allison Levy, ed., Playthings in Early Modernity: Party Games, Word Games, Mind Games, 2017.)

# Dürer’s Unfolding Problem

Can every convex polyhedron be “unfolded” into a shape that doesn’t overlap itself, like this dodecahedron?

Surprisingly, no one knows.

The Renaissance artist Albrecht Dürer first wondered about this in the early 1500s, and in the ensuing five centuries no one has found a polyhedron that won’t permit it. But one may yet exist.

# Pursuit

A problem from Sam Loyd’s Cyclopedia of Puzzles, 1914:

Here is the puzzle of Tom the Piper’s Son, who, as told by ‘Mother Goose,’ stole the pig and away he run. It is known that Tom entered the far gate shown at the top on the right hand. The pig was rooting at the base of the tree 250 yards distant, and Tom captured it by always running directly towards it, while the pig made a bee-line towards the lower corner as shown. Now, assuming that Tom ran one-third faster than the pig, how far did the pig run before he was caught?

Intriguingly, Loyd adds, “The puzzle is a remarkable one on account of its apparent simplicity and yet the ordinary manner of handling problems of this character is so complicated that solvers are asked merely to submit approximately correct answers, based upon judgment and common sense, just to see who can make the best guess. The simple rule for solving it, however, which will doubtless be quite new to our puzzlists, is based upon elementary arithmetic.” What’s the answer?

# Root Words

Square roots:

EIGHTY-ONE has 9 letters.

ONE HUNDRED has 10 letters.

FIVE HUNDRED AND SEVENTY SIX has 24 letters.

Cube roots:

THIRTY-NINE THOUSAND THREE HUNDRED FOUR has 34 letters.

SIXTY-EIGHT THOUSAND NINE HUNDRED AND TWENTY-ONE has 41 letters.

ONE MILLION and ONE BILLION have 10 letters each, making them a sixth root and (in the United States) a ninth root word.

(Dave Morice, “Kickshaws,” Word Ways 30:2 [May 1997], 129-141.)

10/26/2020 UPDATE: Reader Hans Havermann has found many more, including this alarming specimen:

The name of that number contains 3,411 letters.

(Thanks, Hans.)

# Risk Analysis

The Society of Actuaries holds a regular speculative fiction contest. Here’s an excerpt from “The Temple of Screens,” by Nate Worrell, FSA, MAAA, recognized last year for describing the “most innovative actuarial career of the future”:

‘Ever since humans began to be aware of a future, we’ve wanted to explore it. We’ve cast stones, searched in tea leaves, held the entrails of animals in our hands to try to extract some knowledge of our fate. Some of our stories try to show us that like Oedipus, we can’t change our fate. In other stories, we find an escape, we have the power of choice, at least to some degree. But in either case, knowing our future changes how we act. Now that you’ve seen your possible futures, they are tainted. If you were to go back in, they’d all change, reflecting that you had some knowledge. The algorithm would reallocate a new set of weights to your tendencies, increasing some behaviors and decreasing others.’

A wave of anger flashes through me, and I stand and start pacing. ‘So what’s the use of this?’

‘To help you embrace what’s possible, to come to terms with it. You came here because you were afraid of a certain future, one you hoped to avoid somehow. We can’t fight or flee from the future, whatever one we fall into. But we can find serenity in any of our futures, if we so desire.’

MetaFilter has a guide to past contests.

# A Moving Target

The angle of Earth’s axis varies between 22.1 and 24.5 degrees over a 41,000-year period. This means that the Tropics of Cancer and Capricorn are moving slightly: Each is the most extreme circle of latitude in its hemisphere at which the Sun can be directly overhead. At the moment Cancer is drifting southward and Capricorn northward, each at about 15 meters a year.

In Mexico this movement is reflected precisely in a series of annual markers beside Federal Highway 83, from 2005 to 2010.

# The Longest Game

Chess includes a couple of rules intended to keep a game from running on forever. Specifically, a game is a draw (a) if the same position occurs five times or (b) if each player makes a series of 75 moves without a capture or a pawn move. (The more familiar “threefold repetition” and “50-move” rules describe circumstances in which a player can claim a draw but isn’t obliged to.)

At this year’s SIGBOVIK, the tongue-in-cheek scientific conference named after fictional student Harry C. Bovik, Carnegie Mellon’s Tom Murphy VII presented a legal game that carefully skirts these rules to run on as long as possible — 17,697 half-moves, enough to fill 6 pages of the conference proceedings even in small type.

“It can also be downloaded at tom7.org/chess/longest.pgn. Many chess programs fail to load the whole game, but this is because they decided not to implement the full glory of chess.”

(Tom Murphy VII, “Is This the Longest Chess Game?”, SIGBOVIK 2020, Carnegie Mellon University, April 1, 2020.) (Thanks, Noëlle.)

09/29/2020 UPDATE: Reader Alexander Bolton has set up a Longest Chess Game Bot on Twitter that’s playing through this game, tweeting out an image of the position after every halfmove. “It tweets every 4 hours so it should be finished in just over 8 years!”

# A Banana Split

Biologist Jonathan Eisen, who coined the term phylogenomics, called this “perhaps the best genomics Venn diagram ever.” The six-set diagram, published by Angélique D’Hont and her colleagues in Nature in 2012, presents the number of gene families that the banana shares with five other species.

“What the diagram says is that over time the 7,674 gene clusters shared by the six species did not change much in these lineages, as opposed to the 759 clusters specific to the banana (Musa acuminata), for example,” explains Anne Vézina at ProMusa. “Although the genes in these clusters probably share common ancestors with other species, they have since changed to the point that they haven taken on new functions.”

Here’s a similar (5-set) diagram relating to conifers.

(Angélique D’Hont et al., “The Banana (Musa acuminata) Genome and the Evolution of Monocotyledonous Plants,” Nature 488:7410 [2012], 213-217.) (Thanks, David.)

# Art and Artifice

Crockett Johnson, author of the 1955 children’s book Harold and the Purple Crayon, was trained as an engineer and produced more than 100 paintings based on diagrams used in the proofs of classical theorems. This one, Polar Line of a Point and a Circle (Apollonius), appears to have been inspired by a figure in Nathan A. Court’s 1966 College Geometry. The two circles are orthogonal: They cut one another at right angles. And as the square of the line connecting their centers equals the sum of the squares of their radii, these three segments form a right triangle.

Johnson was inspired to this work by his admiration of classical Greek architecture. Sitting in a restaurant in Syracuse in 1973, he managed to construct a heptagon using seven toothpicks and the edges of a menu and a wine list, a construction that had eluded the Greeks. (He found later that Archibald Finlay had illustrated similar constructions in 1959.)

(Stephanie Cawthorne and Judy Green, “Harold and the Purple Heptagon,” Math Horizons 17:1 [2009], 5-9.)