Moving Pictures

In 1864 a photographer employed by Mathew Brady used a four-lens camera to record activity at a Union Army wharf along Potomac Creek in Virginia. The four images were taken in quick succession, so staggering them produces a crude time lapse of the events they record:

In effect they present a four-frame film, perhaps the closest we’ll come to a contemporary movie of life during the Civil War. Here are a few more, all taken in Virginia in 1864:

Union cavalry crossing a pontoon bridge over the James River:

Traffic in front of the Marshall House in Alexandria:

Union soldiers working on a bridge over the Pamunkey River near White House Landing:

There’s more information at the National Park Service’s Fredericksburg & Spotsylvania National Military Park blog.

April 26, 2016April 25, 2016 | History · Technology

Podcast Episode 103: Legislating Pi

In 1897, confused physician Edward J. Goodwin submitted a bill to the Indiana General Assembly declaring that he’d squared the circle — a mathematical feat that was known to be impossible. In today’s show we’ll examine the Indiana pi bill, its colorful and eccentric sponsor, and its celebrated course through a bewildered legislature and into mathematical history.

We’ll also marvel at the confusion wrought by turkeys and puzzle over a perplexing baseball game.

See full show notes …

April 24, 2016October 10, 2017 | Podcast · Science & Math

Pagan Island

Twenty-six villages are ranged around the coastline of an island. Their names, in order, are A, B, C, …, Z. At various times in its history, the island has been visited by 26 missionaries, who names are also A, B, C, …, Z. Each missionary landed first at the village that bore his name and began his work there. Each village was pagan to begin with but became converted when visited by a missionary. Whenever a missionary converted a village he would move along the coastline to the next village in the cycle A–B–C-…-Z–A. If a missionary arrived at an uncoverted village he’d convert it and continue along the cycle, but there was never more than one missionary in a village at a time. If a missionary arrived at a village that had already been converted, the villagers, feeling oppressed, would kill him and revert to a state of paganism; they would do this even to a missionary who had converted them himself and then traveled all the way around the island. There’s no restriction as to how many missionaries can be on the island at any given time. After all 26 missionaries have come and gone, how many villages remain converted?

	Select Click for Answer
None. Each village is converted by its namesake missionary, unless it has already been converted by someone else, so every village gets converted at some point. But then each becomes a deathtrap for the next missionary who visits it. As each missionary works his way around the island, he leaves a string of such traps behind him, the first of which will spring on him when he comes around again unless he’s killed even sooner by stumbling on a village converted by someone else. So all 26 missionaries must die. Since a murderous village reverts to a pagan state, over time the 26 deaths provide 26 unconverted villages, but these are not necessarily different. If we could show that once a village has killed a missionary it is never visited again, and is thus effectively removed from the story, then the 26 deaths leave behind 26 different unconverted villages. So suppose village A has just murdered its first victim. That means it has been visited twice, once by the missionary who converted it from paganism and then by the missionary whom it murdered. What is the possibility that it will receive a third visit? Well, one of the two known visitors might be missionary A on his first visit to the island. But any other visitor to A must come from village Z. So if A were to get 3 visitors, then at least 2 of them must have come from Z; that is, at least 2 missionaries must have got into and out of village Z. But Z kills every other missionary who visits it. So if 2 missionaries left Z, then at least 3 must have entered it. Thus in order for A to receive a third visitor, Z must already have had at least 3 visitors. By the same reasoning, Y must have had at least 3 visitors, and so on around the island, until we reach the impossible condition that in order to have a third visitor, village A must already have had 3 visitors. This shows that A cannot receive a third visitor, and the conclusion follows. (From Crux Mathematicorum, via Ross Honsberger’s More Mathematical Morsels, 1991.)

Click for Answer

None. Each village is converted by its namesake missionary, unless it has already been converted by someone else, so every village gets converted at some point. But then each becomes a deathtrap for the next missionary who visits it. As each missionary works his way around the island, he leaves a string of such traps behind him, the first of which will spring on him when he comes around again unless he’s killed even sooner by stumbling on a village converted by someone else. So all 26 missionaries must die. Since a murderous village reverts to a pagan state, over time the 26 deaths provide 26 unconverted villages, but these are not necessarily different. If we could show that once a village has killed a missionary it is never visited again, and is thus effectively removed from the story, then the 26 deaths leave behind 26 different unconverted villages.

So suppose village A has just murdered its first victim. That means it has been visited twice, once by the missionary who converted it from paganism and then by the missionary whom it murdered. What is the possibility that it will receive a third visit? Well, one of the two known visitors might be missionary A on his first visit to the island. But any other visitor to A must come from village Z. So if A were to get 3 visitors, then at least 2 of them must have come from Z; that is, at least 2 missionaries must have got into and out of village Z. But Z kills every other missionary who visits it. So if 2 missionaries left Z, then at least 3 must have entered it. Thus in order for A to receive a third visitor, Z must already have had at least 3 visitors. By the same reasoning, Y must have had at least 3 visitors, and so on around the island, until we reach the impossible condition that in order to have a third visitor, village A must already have had 3 visitors. This shows that A cannot receive a third visitor, and the conclusion follows.

(From Crux Mathematicorum, via Ross Honsberger’s More Mathematical Morsels, 1991.)

April 24, 2016May 1, 2016 | Puzzles

“A Dialogue Between the Head and Heart”

Of course, she’s only a digestive tube, like all of us.
Yes, but look what it’s attached to!

— Gavin Ewart

April 23, 2016 | Poems

“Sweet-Seasoned Showers”

Today marks the 400th anniversary of William Shakespeare’s death. To commemorate it, Craig Knecht has devised a 44 × 44 magic square (click to enlarge). Like the squares we featured in 2013, this one is topographical — if the number in each cell is taken to represent its altitude, and if water runs “downhill,” then a fall of rain will produce the pools shown in blue, recalling the words of Griffith in Henry VIII:

Noble madam,
Men’s evil manners live in brass; their virtues
We write in water.

The square includes cells (in light blue) that reflect the number of Shakespeare’s plays (38) and sonnets (154) and the year of his death (1616).

(Thanks, Craig.)

April 23, 2016April 25, 2016 | Literature · Science & Math

The Public Figure

In 1956 Macedonian poet Venko Markovski was imprisoned under a fictitious name for circulating a poem critical of Marshal Tito.

Among the guards were individuals who were taking correspondence courses in an attempt to earn a degree. One of these guards, knowing I was a writer, came up to me one day and said: ‘I was told you are a writer. You have knowledge of literature. I have a request …’

‘Please, what do you want to know about literature?’

‘Tell me about Macedonian literature.’

‘Whom are you interested in?’

‘Venko Markovski.’

‘Is it possible you don’t recognize Venko Markovski?’

‘I don’t know him.’

There was an unpleasant pause. I felt sorry for this man who was ordered to guard someone without knowing whom he was guarding. I spoke to him as follows:

‘The best way for you to learn about Venko Markovski is to read his poetry written in Croatian. In this way you will understand Markovski the poet, the Partisan, the public figure, and you will pass your exam easily. But if you rely on me to tell you about Venko Markovski, you will find yourself — after you fail your exam — in the very place where Markovski now finds himself.’

‘What do you mean by that?’ the guard asked. ‘Where is he in fact?’

‘Right in front of you, here on Goli Otok.’

‘Can it be that you are really he?’

‘Yes, I am here under another name.’

The guard walked away silent and confused.

“The warden obviously thought that since he had physical possession of his prisoners he disposed of their minds and souls as well,” Markovski wrote after gaining his freedom in 1961. “But he was mistaken; the body is one thing and the soul is another. There is no way to bribe the human conscience once it has committed itself to the struggle for the rights of its people.”

(From Geoffrey Bould, ed., Conscience Be My Guide: An Anthology of Prison Writings, 2005.)

April 22, 2016April 21, 2016 | Literature · Society

Origins

A snob asked James McNeill Whistler, “Whatever possessed you to be born in a place like Lowell, Massachusetts?”

He said, “I wished to be near my mother.”

April 21, 2016 | Art

Leave-Taking

British infantry sergeant Harry Neale says goodbye to his 10-year-old daughter Lucy, April 4, 1917:

At about six o’clock in the evening, my father called me in and said he’d got to go back to Kidderminster, back to barracks. ‘Will you walk with me a little way, just up the hill, will you come with me?’ Of course I would. He said goodbye to my mother, who was crying, and we went off down the road and then up this long hill. It was a ten-minute walk, I suppose, but we didn’t hurry, we just walked slowly up the hill and I really can’t remember what we talked about. I held on to his hand so tight, and when we got to the top, he said, ‘I won’t take you any further, you must go back now, and I’ll stand here and watch you until you’re out of sight,’ and he put his arms round me and held me so close to him; I remember feeling how rough that khaki uniform was.

‘You must go now, wave to me at the bottom, won’t you?’ I went, I left him standing there and I went down the hill and I kept looking back and waving and he was still there, just standing there. I got to the bottom and then I’d got to turn off to go to where we lived, so I stopped and waved to him and he gestured as much as to say, ‘Go on, you must go home now,’ ever so gently gestured and then he waved and he was still waving when I went, and that was the last time I ever saw him.

Badly wounded in battle, he died of dysentery in East Africa that October.

(From Richard van Emden, The Quick and the Dead, 2011.)

April 21, 2016April 20, 2016 | History

In a Word

https://commons.wikimedia.org/wiki/File:EbniterStr04.JPG — Image: Wikimedia Commons

stirious
adj. like an icicle

April 20, 2016April 20, 2016 | Language

The Outer Limits

In January 2007, inspired by this article by computer scientist Scott Aaronson, philosophers Agustín Rayo of MIT and Adam Elga of Princeton joined in the “large number duel” to come up with the largest finite number ever written on an ordinary-sized chalkboard.

The rules were simple. The two would take turns writing down expressions denoting natural numbers, and whoever could name the largest number would win the duel. No primitive semantic vocabulary was allowed (so that it would be illegal simply to write the phrase “the smallest number bigger than any number named by a human so far”), and the two agreed not to build on one another’s contributions (so neither could simply write “the previous entry plus one”).

Elga went first, writing the number 1. Rayo countered with a string of 1s:

111111111111111111111111111111111111111111111111111111111111

and Elga erased a line through the base of half this string to produce a factorial:

1111111111111111111111111111!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

The two began defining their own functions, and toward the end Rayo wrote this phrase:

The smallest number bigger than any number that can be named by an expression in the language of first-order set theory with less than a googol (10¹⁰⁰) symbols.

With some tweaking, this became the winning entry, now enshrined as “Rayo’s number.”

“It was a great game,” Elga said after the match. “Heated at times, but nevertheless, a really great game.”

The use of philosophy was “crucial,” Rayo said. “The limit of math ability was reached at the end. Knowing a bit of philosophy, that was the key.”

Asked whether he thought his entry had set the Guinness world record, “It’s hard to be sure,” Rayo said, “but the number is bigger than any number I have ever seen.”

(Thanks, Erik.)

April 20, 2016April 19, 2016 | Science & Math