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 Double Day

Lyndon Johnson averaged only 3 to 4 hours of sleep a night and worked most of the rest; his wife once said, “Lyndon acts as if there is never going to be a tomorrow.” He arranged his time in a curious pattern:

Johnson began every day with a bedroom conference at 6:30 a.m., then worked straight through until 2:00 p.m., when he had lunch, relaxed, sometimes with a swim, and took a quick nap. By 4:00 p.m. he was ready to go again. ‘It’s like starting a new day,’ Johnson observed, and he would then proceed to work straight through to one or two in the morning. This Johnsonian ‘double day’ amazed the press and exhausted and frustrated his over-worked aides. His assistant Jack Valenti opined that Johnson had ‘extra glands’ that gave him energy that ordinary men did not possess: ‘He goes to bed late, rises early, and the words I have never heard him say are “I’m tired.”‘

He once called a congressman at 3 a.m. to discuss a piece of pending legislation. When Johnson asked, “Were you asleep?” the congressman thought quickly and said, “No, Mr. President, I was just lying here hoping you’d call.”

(From Larry F. Vrzalik and Michael Minor, From the President’s Pen, 1991.)


jealousy glass

This is sneaky: Operagoers in the 18th century could spy on their neighbors using a “jealousy glass” — you’d appear to be watching the stage but a mirror would direct your view to the side, like a horizontal periscope. Marc Thomin, optician to the queen of France, wrote in 1749:

It is sufficient to turn this opening in the direction of whatever one wishes to observe and the curiosity is immediately satisfied. Its usefulness is confined to letting us see surreptitiously a person we seem not to be observing. This lorgnette may have been called a decorum glass because there is nothing more rude than to use an ordinary opera glass for looking at some one face to face.

Hanneke Grootenboer writes in Treasuring the Gaze, “Apparently, it was very convenient in allowing one to keep track of latecomers entering the opera without having to turn one’s head.”

(From J. William Rosenthal, From Spectacles and Other Vision Aids: A History and Guide to Collecting, 1996.)

Lost Voices

In 2009 three historians engaged forensic lip reader Jessica Rees to analyze silent film shot at the Battle of the Somme in 1916.

Soldiers of the Essex Regiment washing at a pool shout “Hi Mum!” and “Hello Mum, it’s me.” A soldier with a wounded foot repeats, “Jesus, Jesus, Jesus, Jesus.” And another soldier tells the crew, “Stop filming, this is awful.”

“What struck me the most was the optimism of the soldiers and their bravery,” Rees said. “They all seemed very positive, full of team spirit and jocular. Yet, as I was stunned to learn, many of them did not even survive the day of filming. I came away feeling a bit humble.”

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.)

Needs Analysis

I do not believe in freedom of will. Schopenhauer’s words, ‘Man can indeed do what he wants, but he cannot want what he wants,’ accompany me in all life situations and console me in my dealings with people, even those that are really painful to me. This recognition of the unfreedom of the will protects me from taking myself and my fellow men too seriously as acting and judging individuals and losing good humor.

— Albert Einstein, Mein Glaubensbekenntnis, August 1932

Absent Friends

If existence is taken as betokening thisness and thereness, then nonexistence is going to have, speaking informally, this problem: It obliges us to speak of a nothing. If a nonexistent object were always like a footprint in the sand, we might refer to it by its mold, its negative place. But usually the world closes up without much trace around things that have passed their time and ceased to exist, and often there is not even a world left to hold the mold — think of extinct dodos and never existent unicorns; there is no empty niche left in our ‘real’ world for the former and there never was — some say — one for the latter. What kind of focus allows us then to speak of things that are definitely and determinately nowhere and not now and not ever? What, if anything, is it we are referring to when we say: This does not exist?

— Eva Brann, The Ways of Naysaying, 2001

False or True?

A Russian coin-weighing puzzle:

You have 101 coins, and you know that 50 of them are counterfeit. Every true coin has the same weight, an unknown integer, and every false coin has the same weight, which differs from that of a true coin by 1 gram. You also have a two-pan pointer scale that will show you the difference in weight between the contents of each pan. You choose one coin. Can you tell in a single weighing whether it’s true or false?

Click for Answer

Making Cases,_Leonardo_da_-_Crossbow_sketch_-_1500.jpg

Testimony is like an arrow shot from a longbow; the force of it depends on the strength of the hand that draws it.

Argument is like an arrow from a crossbow, which has equal force though shot by a child.

— Robert Boyle, paraphrased by Samuel Johnson