# De Bruijn’s Theorem

At age 7, F.W. de Bruijn found himself unable to pack a box measuring 6 × 6 × 6 quite completely with bricks measuring 1 × 2 × 4. The box had volume 216, so it might be expected to accommodate exactly 27 bricks, but he found there was no way to pack more than 26.

He mentioned this to his father, who happened to be mathematician Nicolaas Govert de Bruijn, and Nicolaas found that a “harmonic brick” (one in which the length of each side is a multiple of the next smaller side length) can be packed efficiently only into a box whose dimensions are multiples of the brick’s dimensions.

This can seen intuitively by imagining the 6 × 6 × 6 box filled with small colored cubes as shown here. No matter where it’s placed, each 1 × 2 × 4 brick must now displace an equal number of white and black cubes. But the box contains 112 white cubes and 104 black ones. So the task is impossible.

(Nicholas G. de Bruijn, “Filling Boxes With Bricks,” American Mathematical Monthly 76:1 [1969], 37-40.)

# Fortunate Numbers

Multiply the first n prime numbers:

2 × 3 × 5 × 7 × 11 × 13 = 30030

Now find the smallest integer greater than 1 that will produce a prime number when it’s added to that product. In this example it’s 17:

30030 + 17 = 30047,

which is prime. This makes 17 a Fortunate number, named for Reo Fortune, the social anthropologist who first studied this. The first few Fortunate numbers are

3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151 …

Are all Fortunate numbers prime? Fortune conjectured so, but whether it’s true remains an open problem.

# The Clever Way

When I give talks on factoring, I often repeat an incident that happened to me long ago in high school. I was involved in a math contest, and one of the problems was to factor the number 8051. A time limit of five minutes was given. It is not that we were not allowed to use pocket calculators; they did not exist in 1960, around when this event occurred! Well, I was fairly good at arithmetic, and I was sure I could trial divide up to the square root of 8051 (about 90) in the time allowed. But on any test, especially a contest, many students try to get into the mind of the person who made it up. Surely they would not give a problem where the only reasonable approach was to try possible divisors frantically until one was found. There must be a clever alternate route to the answer. So I spent a couple of minutes looking for the clever way, but grew worried that I was wasting too much time. I then belatedly started trial division, but I had wasted too much time, and I missed the problem. …

The trick is to write 8051 as 8100 – 49, which is 902 – 72, so we may use algebra, namely, factoring a difference of squares, to factor 8051. It is 83 × 97.

— Carl Pomerance, “A Tale of Two Sieves,” Notices of the AMS 43:12 (December 1996), 1473-1485

# Taylor–Couette Flow

This is surprising: A laminar flow induced in a viscous fluid confined in the gap between two rotating cylinders can be (to a large extent) reversible. The dyes here appear to mix, but in fact they’re being stretched into distinct spirals that can then be “unmade” by reversing the direction of the rotation.

# Application

“A Polish girl, desperate and vengeful after failing her examination, took a gun, concealed herself in [Alfred] Werner’s garden, and awaited his return. He arrived home. She fired and missed. Werner calmly turned to her and remarked, ‘Your aim is no better than your knowledge of chemistry.'”

— George B. Kauffman, Alfred Werner: Founder of Coordination Chemistry, 2013

(“The Polizeiinspektorat der Stadt Zürich reports that neither the Stadtpolizei nor the Kantonspolizei have any record of the incident. Se non è vero, è ben trovato!”)

# Franklin’s Magickest Square

When a friend showed him a 16 × 16 magic square devised by Michel Stifelius, Ben Franklin went home and composed the square above, “not willing to be outdone.” An admirer describes its properties:

1. The sum of the sixteen numbers in each column or row, vertical or horizontal, is 2,056. — 2. Every half column, vertical or horizontal, makes 1,028, or just one half of the same sum, 2,056. — 3. Any half vertical row added to any half horizontal, makes 2,056. — 4. Half a diagonal ascending added to half a diagonal descending, makes 2,056, taking these half diagonals from the ends of any side of the square to the middle of it, and so reckoning them either upward, or downward, or sideways. — 5. The same with all the parallels to the half diagonals, as many as can be drawn in the great square: for any two of them being directed upward and downward, from the place where they begin to that where they end, make the sum 2,056; thus, for example, from 64 up to 52, then 77 down to 65, or from 194 up to 204, and from 181 down to 191; nine of these bent rows may be made from each side. — 6. The four corner numbers in the great square added to the four central ones, make 1,028, the half of any column. — 7. If the great square be divided into four, the diagonals of the little squares united, make, each, 2,056. — 8. The same number arises from the diagonals of an eight sided square taken from any part of the great square. — 9. If a square hole, equal in breadth to four of the little squares or cells, be cut in a paper, through which any of the sixteen little cells may be seen, and the paper be laid on the great square, the sum of all the sixteen numbers seen through the hole is always equal to 2,056.

Franklin wrote, “This I sent to our friend the next morning, who, after some days, sent it back in a letter with these words: ‘I return to thee thy astonishing or most stupendous piece of the magical square, in which’ — but the compliment is too extravagant, and therefore, for his sake as well as my own, I ought not to repeat it. Nor is it necessary; for I make no question but you will readily allow this square of 16 to be the most magically magical of any magic square ever made by any magician.”

(“Clavis,” “Magic Squares,” The Mirror of Literature, Amusement, and Instruction 4:109 [Oct. 23, 1824], 293-294.) (Thanks, Walker.)

12/21/2020 UPDATE: The square appeared originally in 1767 in James Ferguson’s Tables and Tracts, Relative to Several Arts and Sciences and was reprinted a year later in the Gentleman’s Magazine. Only the second publication credits Franklin. I don’t have a date for Franklin’s purported composition, so I don’t know what to make of this. (Thanks, Tom.)

# Hex

Invented independently by Piet Hein and John Nash, the game of Hex is both simple and deep. Each player is assigned two opposite sides of the board and tries to connect them with an unbroken chain of stones. Draws are impossible, and in principle it can be shown that the first player has a winning strategy (if the second player had such a strategy, the first player could “steal” it with a move in hand). But succeeding in practical play requires careful, subtle thought.

You can try it here.

# MIThenge

Around January 31 and November 11 each year, the setting sun illuminates the “Infinite Corridor,” the 251-meter hallway that runs through five main buildings at MIT.

The hallway isn’t optimally aligned, so the phenomenon lasts only about 2 minutes. MIT maintains a page with predictions, etiquette, background, and more photos.

# Neighbors

Astronomer Thomas Dick believed that every planet in the solar system was inhabited. In his 1838 book Celestial Scenery he worked out that they contain 21 trillion inhabitants in all — assuming “280 inhabitants to a square mile, which is the rate of population in England.”

# The Value of Disagreement

In 1907, Francis Galton famously found that when a crowd were asked to guess the weight of an ox, the average value of their responses was surprisingly accurate — in Galton’s experiment, it fell within 1 percent of the ox’s true weight. This is “the wisdom of crowds”: By canceling errors across individuals, the mean response often proves more accurate than individual estimates.

Interestingly, the same phenomenon can arise when we aggregate multiple estimates made by a single person (the “wisdom of the inner crowd”). And organizational behavior researchers Philippe van de Calseyde and Emir Efendić now find that the accuracy can be refined still further when people are asked to consider a question from the perspective of someone they often disagree with.

“In explaining its accuracy, we find that taking a disagreeing perspective prompts people to consider and adopt second estimates they normally would not consider as viable option, resulting in first and second estimates that are highly diverse (and by extension more accurate when aggregated),” the researchers write. “Our results suggest that disagreement, often highlighted for its negative impact, can be a powerful tool in producing accurate judgments.”

(Philippe van de Calseyde and Emir Efendić, “Taking a Disagreeing Perspective Improves the Accuracy of People’s Quantitative Estimates,” PsyArXiv, Nov. 15, 2019.)