Coming and Going

https://pixabay.com/en/autumn-dog-running-dog-forest-665149/

In 1978 the Chronicle of Higher Education mentioned an old exam question:

Q. How far can a dog run into the woods?

A. Halfway. The rest of the time he is running out.

Harvard’s Richard E. Baym wrote in to take issue with the answer:

The correct answer is ‘All the way’. Certainly we understand that the dog is running ‘in’ only until he reaches the middle of the forest, but this is in fact, all the way in. If the dog ran only half ‘in’, he would not yet be at the middle. Indeed if the dog ran halfway in and then ran halfway out, he would still be in the woods.

The editors noted, “It occurs to us that the dog’s continued presence there would be useful, in case something happens to that tree that we’ve been hearing about since high school physics — the one that falls when no one is in the forest and since there is no eardum to register sound waves, makes no noise. You know what a fine sense of hearing a dog has. Let him run halfway in (or as Mr. Baym argues, all the way), settle there, and keep an ear cocked for that tree.”

(from Robert L. Weber, ed., Science With a Smile, 1992.)

| Puzzles

Technicalities

In presenting the rules of chess, some writers carelessly say that a pawn that reaches the eighth rank can be promoted to any piece that the player chooses. That’s a bit too generous, as a couple of puzzle composers have noted. In 1941 Leonid Kubbel presented this problem — White is to mate in two moves:

kubbel promotion puzzle

It’s not immediately clear how to release Black from his stalemate and still mate him on the next move. The solution is to promote the e7 pawn to a black king!

kubbel promotion puzzle - solution

Now it’s Black’s move — he has to play 1. … Kd8, and White can mate both kings with 2. Qd7#!

The Polish master Johannes Zukertort offered this one: White is to mate on the move:

zukertort promotion puzzle

Here White promotes the pawn to a black knight, ending the game. (Note that it must be a knight — crazy as it seems, this is the only black piece that produces mate.)

| Puzzles

Divide and Conquer

Facing dental surgery one day, mathematician Matt Parker asked Twitter for a math puzzle to distract him. A friend challenged him to put the digits 1-9 in order so that the first two digits formed a number that was a multiple of 2, the first three digits were a multiple of 3, and so on.

Leaving the digits in the conventional order 1234356789 doesn’t work: 12 is divisible by 2 and 123 by 3, but 1234 isn’t evenly divisible by 4. “By the end of my dental procedure, I had some but not all of the digits worked out, but, apparently, you’re not allowed to stay in the dentist’s chair after they’re finished.” At home he finished working out the solution, which is unique. What is it?

Click for Answer
| Puzzles

Last Words

https://commons.wikimedia.org/wiki/File:After_the_Battle_of_Flers-Courcelette.jpg

Note found in the pocketbook of an English corporal killed during the battle of the Somme:

Dear Mother

I am writing these few lines severely wounded. We have done well our Batt. advanced about 3 quarters of a mile. I am laid in a shell hole with 2 wounds in my hip and through my back. I cannot move or crawl. I have been here for 24 hours and never seen a living soul. I hope you will receive these few lines as I don’t expect anyone will come to take me away, but you know I have done my duty out here now for 1 year and 8 months and you will always have the consolation that I died quite happy doing my duty.

Must give my best of love to all the cousins who [have] been so kind to me since I have been out here and the Best of love to Arthur and Harry and all at Swinefleet. xxx

He was identified as John Duesbery of Bradford, and the note and his other belongings were sent home to his family. In The Quick and the Dead, Richard van Emden writes, “His grave would have been marked in a rudimentary way, perhaps with a piece of wood or an upturned rifle, but whatever was placed there it was subsequently destroyed and John’s body lost.”

| History

The One-Seventh Ellipse

The decimal expansion of 1/7 is 0.142857142857 …, a repeating decimal. Arrange the six repeating digits into overlapping ordered pairs, like so:

(1, 4), (4, 2), (2, 8), (8, 5), (5, 7) (7, 1),

and, remarkably, all six lie on an ellipse:

19x2 + 36yx + 41y2 – 333x – 531y + 1638 = 0

one-seventh ellipse #1

Even more remarkably, if we take the digits two at a time:

(14, 28), (42, 85), (28, 57), (85, 71), (57, 14), (71, 42),

these points also lie on an ellipse:

-165104x2 + 160804yx + 8385498x – 41651y2 – 3836349y – 7999600 = 0

one-seventh ellipse #2

That’s from David Wells, The Penguin Dictionary of Curious and Interesting Numbers (1986). Victor Hugo wrote, “Mankind is not a circle with a single center but an ellipse with two focal points, of which facts are one and ideas the other.”

| Science & Math

Overdone Bacon

https://en.wikipedia.org/wiki/Orville_Ward_Owen

Exponents of the theory that Francis Bacon wrote the plays attributed to Shakespeare have gone to sometimes elaborate lengths to find messages hidden in the plays. American physician Orville Ward Owen even invented a “cipher wheel” that could pass the texts under his eyes at various speeds as he looked for hidden meanings.

He didn’t find many supporters. Even Owen’s friend Frederick Mann wrote, “We are asked to believe that such peerless creations as Hamlet, The Tempest, and Romeo and Juliet were not prime productions of the transcendent genius who wrote them, but were subsidiary devices which Bacon designed for the purpose of concealing the cipher therein.”

In his 1910 book Bacon Is Shake-speare, Sir Edwin Durning-Lawrence argues that the long word honorificabilitudinitatibus in Love’s Labour’s Lost is really an anagram:

HI LUDI F. BACONIS NATI TUITI ORBIS
These plays, F. Bacon’s offspring, are preserved for the world.

“It surpasses the wit of man,” he wrote, to produce another sensible anagram from the long word, and he offered a hundred guineas to anyone who could do it. A Mr. Beevor of St. Albans rather promptly sent him this:

ABI INIVIT F. BACON HISTRIO LUDIT
Be off, F. Bacon, the actor has entered and is playing.

Durning-Lawrence was taken aback, but he was a good sport: He paid Beevor his money.

(From John Michell, Who Wrote Shakespeare?, 1996.)

| Language · Literature

In a Word

https://commons.wikimedia.org/wiki/File:David_Livingstone_attacked_by_a_lion_in_Africa._Lithograph._Wellcome_V0018847.jpg

succussion
n. violent shaking

squassation
n. a severe shaking

A lion attacks David Livingstone, 1843:

Starting, and looking half round, I saw the lion just in the act of springing upon me. I was upon a little height; he caught my shoulder as he sprang, and we both came to the ground below together. Growling horribly close to my ear, he shook me as a terrier dog does a rat. The shock produces a stupor similar to that which seems to be felt by a mouse after the first shake of the cat. It caused a sort of dreaminess, in which there was no sense of pain nor feeling of terror, though quite conscious of all that was happening. It was like what patients partially under the influence of chloroform describe, who see all the operation, but feel not the knife. This singular condition was not the result of any mental process. The shake annihilated fear, and allowed no sense of horror in looking round at the beast. This peculiar state is probably produced in all animals killed by carnivora; and if so, is a merciful provision by our benevolent Creator for lessening the pain of death.

The lion left him to attack his companions, who eventually dispatched it. Livingstone could never afterward raise his left arm above his shoulder; when asked by a group of sympathetic friends what he had been thinking during the attack, he said, “I was thinking, with a feeling of disinterested curiosity, which part of me the lion would eat first.”

| Language

Podcast Episode 108: The Greenwich Time Lady

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

As recently as 1939, a London woman made her living by setting her watch precisely at the Greenwich observatory and “carrying the time” to her customers in the city. In this week’s episode of the Futility Closet podcast we’ll meet Ruth Belville, London’s last time carrier, who conducted her strange occupation for 50 years.

We’ll also sample the colorful history of bicycle races and puzzle over a stymied prizewinner.

See full show notes …

| Podcast

A Guest Appearance

The Fibonacci numbers are the ones in this sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Each number is the sum of the two that precede it. But now, interestingly:

\displaystyle \mathrm{arctan} \left ( \frac{1}{1} \right ) = \mathrm{arctan} \left ( \frac{1}{2} \right ) + \mathrm{arctan} \left ( \frac{1}{3} \right )\\ \mathrm{arctan} \left ( \frac{1}{3} \right ) = \mathrm{arctan} \left ( \frac{1}{5} \right ) + \mathrm{arctan} \left ( \frac{1}{8} \right )\\ \mathrm{arctan} \left ( \frac{1}{8} \right ) = \mathrm{arctan} \left ( \frac{1}{13} \right ) + \mathrm{arctan} \left ( \frac{1}{21} \right )\\ \mathrm{arctan} \left ( \frac{1}{21} \right ) = \mathrm{arctan} \left ( \frac{1}{34} \right ) + \mathrm{arctan} \left ( \frac{1}{55} \right )\\

“And so on!” writes James Tanton in Mathematics Galore! (2012). “The first relation, for instance, states that a line of slope 1/2 stacked with a line of slope 1/3 gives a line of slope 1. (Can you prove the relations?)”

(Ko Hayashi, “Fibonacci Numbers and the Arctangent Function,” Mathematics Magazine 76:3 [June 2003], 215.)

| Science & Math