“Problem for Poultry Farmers”

Eureka, the honored journal in recreational mathematics published at Cambridge University, has placed its archives online under a Creative Commons license. Here’s a problem from May 1940 (no solution is given):

The chicken was twice as old when when the day before yesterday was to-morrow to-day was as far from Sunday as to-day will be when the day after to-morrow is yesterday as it was when when to-morrow will be to-day when the day before yesterday is to-morrow yesterday will be as far from Thursday as yesterday was when to-morrow was to-day when the day after to-morrow was yesterday. On what day was the chicken hatched out?

01/11/2021 UPDATE: The solution is here. (Thanks, Bruce.)

Child’s Play

https://patents.google.com/patent/US6368227B1/en

In 2002 a 7-year-old boy, Steven Olson, patented a “method of swinging on a swing”:

The method comprises the steps of: a) positioning a user on the seat; and b) having the user pull alternately on one chain to induce movement of the user and the swing toward one side, and then on the other chain to induce movement of the user and the swing toward the other side, to create side-to-side motion.

Steven’s father, Peter, a patent attorney, wanted to show him how the system works. Steven’s submission was approved at first (the patent office said that its technical definition of obviousness “is not necessarily the conventional definition”) but later reconsidered and invalidated, perhaps due to criticism.

A year earlier, to test the workability of a new national patent system, an Australian man had patented the wheel.

Vanishing Act

There is often peculiar humour about self-frustration. Consider, for example, a train of events which started outside the old Clarendon Laboratory, Oxford. I came across a dirty beaker full of water just when I happened to have a pistol in my hand. Almost without thinking I fired, and was surprised at the spectacular way in which the beaker disappeared. I had, of course, fired at beakers before; but they had merely broken, and not shattered into small fragments. Following Rutherford’s precept I repeated the experiment and obtained the same result: it was the presence of the water which caused the difference in behavior. Years later, after the War, I found myself having to lecture to a large elementary class at Aberdeen, teaching hydrostatics ab initio. Right at the beginning came the definitions — a gas having little resistance to change of volume but a liquid having great resistance. I thought that I would drive the definitions home by repeating for the class my experiments with the pistol, for one can look at them from the point of view of the beaker, thus suddenly challenged to accommodate not only the liquid that it held before the bullet entered it, but also the bullet. It cannot accommodate the extra volume with the speed demanded, and so it shatters.

— R.V. Jones, “Impotence and Achievement in Physics and Technology,” Nature 207:4993 (1965), 120-125

(When the Royal Engineers tried to use this trick to demolish a tall chimney, filling its base with 6 feet of water and firing an explosive charge into the water, “it succeeded so well that it failed completely”: The incompressible water flung the surrounding ring of bricks outward, leaving a foreshortened chimney suspended above in midair. This dropped down neatly onto the old foundation, upright and intact, “presenting the Sappers with an exquisite problem.”)

A Special Year

Reader Alan Jackson points out that 2021 is the product of consecutive primes: 2021 = 43 × 47.

It’s the first time this has happened since 1763 (= 41 × 43), and it won’t happen again until 2491 (= 47 × 53).

A quick way to find the factors is to note that 2021 is the difference of two squares, 452 – 22, so 2021 = (45 – 2) × (45 + 2).

(Thanks, Alan.)

Small World

https://commons.wikimedia.org/wiki/File:Statue_of_Little_Prince,_Kyiv.jpg
Image: Wikimedia Commons

In Antoine de Saint-Exupéry’s 1943 novella The Little Prince, the narrator encounters “a most extraordinary small person” whose planet is “scarcely any larger than a house.”

This led University of Ljubljana physicist Janez Strnad to consider the implications. If the radius of the prince’s planet were 64 meters and it had Earth’s density, then the weight of a prince with a mass of 30 kg would amount to 0.003 newtons, corresponding on Earth to the weight of a mass of 0.3 g. (If the planet had the density of an asteroid, his weight would be lower still.)

The planet cannot have an atmosphere, because the mean velocity of gas molecules is greater than the escape velocity.

If the prince moved faster than 80 millimeters per second he’d be sent into orbit around the planet; if faster than 11 centimeters per second he’d leave it altogether.

“He could overcome the limitations concerning his velocity by either binding himself with a rope to his planet or building a spherical shell around it,” Strnad concluded. “The human body adapts to weightlessness and astronauts have to perform special gymnastic exercises not to suffer on returning to the Earth. For the little prince, coming to Earth would be a serious adventure, were he not a fictitious character.”

(Janez Strnad, “The Planet of the Little Prince,” Physics Education 23:4 [1988], 224.)

The Flettner Rotor

https://commons.wikimedia.org/wiki/File:Buckau_Flettner_Rotor_Ship_LOC_37764u.jpg

Canvas sails had been used for thousands of years when German engineer Anton Flettner realized that rotating cylinders might work as well: The spin produces a difference in pressure on opposite sides of the cylinder, and this can propel a ship through the water.

The idea made a splash when it was introduced in the 1920s, and an experimental vessel crossed the Atlantic in 1926, but the rotating drums consumed a discouraging amount of power and Flettner moved on to other projects. The principle has been revived lately, though, in hopes of increasing the fuel efficiency of conventionally powered ships.

More Bad Verse

For years I’ve been hearing about an immortally bad volume of poetry, The Captain of the ‘Dolphin’ and Other Poems of the Sea, by Frederick J. Johnston-Smith. The glimpses I’ve seen look fantastic:

We piled more wood upon the blazing hearth —
More broken planks from off the mould’ring wreck;
The billets, all composed of Norway pine,
Were evidently portions of the deck.

The morning came the tempest’s trail impatient to elute;
The merry birds assistance gave — played each his fife or flute.

A balminess the darkened hours had brought from out the south.
Each breaker doffed its cap of white and shut its blatant mouth.

Strike, strike your flag, Sidonia,
And lessen death and pain!
“Strike!” “Fight!” are but synonyma
For misery to Spain.

(Where the Goddess Aurora sits shaking her fan
In the face of a vapourless moon —
Where the sun circles round for the half of the year
And is cold — like a yellow balloon)

(to a lapwing:)

I thank you for cutting the thread of my thought
With a snip of your scissors-like bill,
For why should my mind with a thinking be fraught
Of men’s indefectible ill?

I’ve just discovered that the whole thing is on the Internet Archive — including a Glossary of Nautical Terms in which the poet informs us that the “wheel” is “that with which the helmsman steers the ship.”

Prime Circles

Is it always possible to arrange the numbers from 1 to 2n in a circle so that each adjacent pair sums to a prime?

  4  1
7      2
6      3
  5  8

As of 2016, solutions have been found for every case up to n = 106 — but no one has yet proven that it’s always possible.