# The No-Three-in-Line Problem

In 1917 Henry Dudeney asked: What’s the maximum number of lattice points that can be placed on an n × n grid so that no three points are collinear?

The answer can’t be more than 2n, since if we place one point more than this, we’re forced to put three into the same row or column. (The 10 × 10 grid above contains 20 points.)

For a grid of each size up to 52 × 52, it’s possible to place 2n points without making a triple. For larger grids it’s conjectured that fewer than 2n points are possible, but today, more than a century after Dudeney posed the question, a final answer has yet to be found.

# The Nimm0 Property

In the 17th century the French mathematician Bernard Frénicle de Bessy described all 880 possible order-4 magic squares — that is, all the ways in which the numbers 1 to 16 can be arranged in a 4 × 4 array so that the long diagonals and all the rows and columns have the same sum.

These squares share a curious property: If we subtract 1 from each cell, to get a square of the numbers 0-15, then each of the rows and columns has a nim sum of 0. A nim sum is a binary sum in which 1 + 1 is evaluated as 0 rather than “0, carry 1.” For example, here’s one of Frénicle’s squares:

$\displaystyle \begin{matrix} 0 & 5 & 10 & 15\\ 14 & 11 & 4 & 1\\ 13 & 8 & 7 & 2\\ 3 & 6 & 9 & 12 \end{matrix}$

Translating each of these numbers into binary we get

$\displaystyle \begin{bmatrix} 0000 & 0101 & 1010 & 1111\\ 1110 & 1011 & 0100 & 0001\\ 1101 & 1000 & 0111 & 0010\\ 0011 & 0110 & 1001 & 1100 \end{bmatrix}$

And the binary sums of the four rows, evaluated without carry, are

0000 + 0101 + 1010 + 1111 = 0000
1110 + 1011 + 0100 + 0001 = 0000
1101 + 1000 + 0111 + 0010 = 0000
0011 + 0110 + 1001 + 1100 = 0000

The same is true of the columns. (The diagonals won’t necessarily sum to zero, but they will equal one another. And note that the property described above won’t necessarily work in a “submagic” square in which the diagonals don’t add to the magic constant … but it does work in all 880 of Frénicle’s “true” 4 × 4 squares.)

(John Conway, Simon Norton, and Alex Ryba, “Frenicle’s 880 Magic Squares,” in Jennifer Beineke and Jason Rosenhouse, eds., The Mathematics of Various Entertaining Subjects, Vol. 2, 2017.)

# Podcast Episode 204: Mary Anning’s Fossils

In 1804, when she was 5 years old, Mary Anning began to dig in the cliffs that flanked her English seaside town. What she found amazed the scientists of her time and challenged the established view of world history. In this week’s episode of the Futility Closet podcast we’ll tell the story of “the greatest fossilist the world ever knew.”

We’ll also try to identify a Norwegian commando and puzzle over some further string pulling.

See full show notes …

# Five Up

A card curiosity via Martin Gardner: Deal 10 cards from an ordinary deck and hold this packet face down in your left hand. Turn the top two cards face up and then cut the packet anywhere you like. Again, turn the top two cards and cut. Continue doing this for as long as you like, turning over the top two cards and cutting the packet.

When you’ve finished, deal the cards in a row on the table and turn over the cards at even positions in the row: the second, fourth, sixth, eighth, and tenth cards.

This will always leave five cards face up.

(Martin Gardner, “Curious Counts,” Math Horizons 10:3 [February 2003], 20-22.)

# The Cremona–Richmond Configuration

This figure contains 15 lines and 15 points, with three points on each line and three lines through each point, yet no three points are connected by three lines to form a triangle.

It’s named after mathematicians Luigi Cremona and Herbert William Richmond, who studied it in the late 19th century.

# Good Boy

As Washington State University anthropologist Grover Krantz was dying of pancreatic cancer, he told his colleague David Hunt of the Smithsonian:

“I’ve been a teacher all my life and I think I might as well be a teacher after I’m dead, so why don’t I just give you my body.”

When Hunt agreed, Krantz added, “But there’s one catch: You have to keep my dogs with me.”

Accordingly, in 2003, Krantz’s skeleton was laid to rest in a green cabinet at the National Museum of Natural History alongside the bones of his Irish wolfhounds Clyde, Icky, and Yahoo.

Krantz’s bones have been used to teach forensics and advanced osteology to students at George Washington University.

And in 2009 his skeleton was articulated and, along with Clyde’s, displayed in the exhibition “Written in Bone: Forensic Files of the 17th Century Chesapeake.”

# Puffery

From a letter of Charles Darwin to Charles Lyell, April 1860:

I must say one more word about our quasi-theological controversy about natural selection, and let me have your opinion when we meet in London. Do you consider that the successive variations in the size of the crop of the Pouter Pigeon, which man has accumulated to please his caprice, have been due to ‘the creative and sustaining powers of Brahma?’ In the sense that an omnipotent and omniscient Deity must order and know everything, this must be admitted; yet, in honest truth, I can hardly admit it. It seems preposterous that a maker of a universe should care about the crop of a pigeon solely to please man’s silly fancies. But if you agree with me in thinking such an interposition of the Deity uncalled for, I can see no reason whatever for believing in such interpositions in the case of natural beings, in which strange and admirable peculiarities have been naturally selected for the creature’s own benefit. Imagine a Pouter in a state of nature wading into the water and then, being buoyed up by its inflated crop, sailing about in search of food. What admiration this would have excited — adaptation to the laws of hydrostatic pressure, &c &c For the life of me I cannot see any difficulty in natural selection producing the most exquisite structure, if such structure can be arrived at by gradation, and I know from experience how hard it is to name any structure towards which at least some gradations are not known.

Ever yours,

C. Darwin.

# “A Terrific Banquet in an Iguanodon”

In 1852, British artist Benjamin Waterhouse Hawkins engaged to make 33 life-size concrete models of extinct dinosaurs, to be arranged in a park in southern London around the relocated Crystal Palace. Throughout the work he conferred with a team of leading British scientists, and on New Year’s Eve 1853 they celebrated their accomplishment with a dinner party held inside one of the sculptures:

Twenty-one of the guests were accommodated with seats ranged on each side of the table, within the sides of the iguanodon. Professor Owen, one of the most eminent geologists of the day, occupied a seat at the head of the table, and within the skull of the monster. Mr. Francis Fuller, the Managing Director, and Professor Forbes, were seated on commodious benches placed in the rear of the beast. An awning of pink and white drapery was raised above the novel banqueting-hall, and small banners bearing the names of Conybeare, Buckland, Forbes, Owen, Mantell, and other well-known geologists, gave character and interest to the scene. When the more substantial viands were disposed of, Professor Owen proposed that the company should drink in silence ‘The memory of Mantell, the discoverer of the iguanodon,’ the monster in whose bowels they had just dined.

They concluded with a “roaring chorus” in praise of the “antediluvian dragon”:

A thousand ages underground
His skeleton had lain;
But now his body’s big and round,
And he’s himself again!
His bones, like Adam’s, wrapped in clay,
His ribs of iron stout,
Where is the brute alive to-day
That dares with him turn out?
Beneath his hide he’s got inside
The souls of living men,
Who dare our Saurian now deride
With life in him again?

(Chorus) The jolly old beast
Is not deceased,
There’s life in him again. (A roar.)

In fairy land are fountains gay,
With dragons for their guard:
To keep the people from the sight,
The brutes hold watch and ward!
But far more gay our founts shall play,
Our dragons, far more true,
Will bid the nations enter in
And see what skill can do!
For monsters wise our saurians are,
And wisely shall they reign,
To spread sound knowledge near and far
They’ve come to life again!

Though savage war her teeth may gnash,
And human blood may flow,
And foul ambition, fierce and rash,
Would plunge the world in woe,
Each column of this palace fair
That heavenward soars on high,
A flag of hope shall on it bear,
Proclaiming strife must die!
And art and science far shall spread
Around this fair domain,
The People’s Palace rears its head
With life in it again.

# Total Victory

The familiar posture of victory — raising the arms, tilting the head back, and expanding the chest — appears to be hard-wired into the human brain, probably because it was a universal sign of dominance in our ape ancestors.

In 2008, psychologists Jessica Tracy and David Matsumoto compared the expressions and body language of sighted, blind, and congenitally blind judo competitors representing more than 30 countries in the 2004 Olympic and Paralympic Games. They found that the blind athletes used the same gestures as their sighted peers, even though they’d never seen anyone else use them.

“Since congenitally blind individuals could not have learned pride and shame behaviours from watching others, these displays of victory or defeat are likely to be an innate biological propensity,” Tracy told the Telegraph.

The same victory gesture is seen in children as young as 3. Tracy said she was studying similar behaviors in chimps and that “anecdotal evidence mentioned in the paper suggests that, yes, the human pride and shame displays are very similar to non-human displays of dominance and submission, seen in a wide range of animals.”

(Jessica L. Tracy and David Matsumoto, “The Spontaneous Expression of Pride and Shame: Evidence for Biologically Innate Nonverbal Displays,” Proceedings of the National Academy of Sciences 105:33 [August 19, 2008], 11655-11660.)

# Podcast Episode 202: The Rosenhan Experiment

In the 1970s psychologist David Rosenhan sent healthy volunteers to 12 psychiatric hospitals, where they claimed to be hearing voices. Once they were admitted, they behaved normally, but the hospitals diagnosed all of them as seriously mentally ill. In this week’s episode of the Futility Closet podcast we’ll describe the Rosenhan experiment, which challenged the validity of psychiatric diagnosis and set off a furor in the field.

We’ll also spot hawks at Wimbledon and puzzle over a finicky payment processor.

See full show notes …