The Four Travelers Problem

four travelers problem

Four straight roads cross a plain. No two are parallel, and no three meet in a point. On each road is a traveler who moves at some constant speed. If Blue and Red meet each other at their crossroad, and each of them meets Yellow and Green at their respective crossroads, will Yellow and Green necessarily meet at their own crossroad?

Suprisingly, the answer is yes. Rob Fatland offers a beautiful solution at the CTK Exchange:

Imagine the whole scenario unfolding from Blue’s rest frame; that is, regard Blue as unmoving and consider the movements of the other travelers relative to him. What does he see? Objectively we know that each traveler moves along a straight line at a constant speed, eventually encounters Blue, and moves on, so from Blue’s perspective each of them moves directly toward him on a straight line, passes through his position, and continues.

Very well, let Red do that. But we know that Red also encounters Yellow and Green without deviating from his own path. And we know that (from Blue’s perspective) Yellow and Green are also traveling straight lines that intersect Blue’s position. This can only mean that Red, Yellow, and Green are all traveling along the same straight line from Blue’s point of view. And this means that Yellow and Green must meet one another.

(It might be objected that two points traveling the same line needn’t meet if they’re going in the same direction at the same speed. But here this would mean that two of the roads are parallel, and that possibility is excluded by the conditions of the problem.)

05/27/2016 UPDATE: Reader Derek Christie recalls a three-dimensional solution: “Make a time axis perpendicular to the plane. Then each traveller moves in a straight line trajectory through this 3D space. Blue and Red meet at a particular place and time, so their two trajectories must meet at a point, and these two trajectories define a plane. Both Yellow and Green trajectories meet the Blue and Red trajectories and so also lie in this same plane. So Yellow and Green must also meet somewhere in that plane.”

Small World

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

A striking example of the strength of the British Empire in the early 20th century: In 1911 Britain completed the “All Red Line,” a network of telegraphs that linked its possessions. The system was so redundant that an enemy would have had to cut 49 cables to isolate the United Kingdom, 15 to isolate Canada, or 5 to isolate South Africa. As a result, British communications remained uninterrupted throughout World War I.

Sir Sandford Fleming described the network as “the cerebro-spinal axis of our political system … through which would freely pass the sensory impressions and the motor impulses of the British people in every longitude.”

Related: In Air Facts and Problems (1927), Secretary of State for Air Christopher Thomson noted that the whole empire might be visited by an aircraft capable of “long hops”:

For the purposes of the immediate future a ‘long hop’ may be taken as 1,500 miles in length. One such hop would cover the distance from the south coast of England to Malta, a second would reach Egypt, a third Bushire (on the Persian Gulf), a fourth India, at Karachi or Bombay, a fifth Ceylon, a sixth the Straits Settlements, a seventh Port Darwin in Northern Australia; three more would reach New Zealand.

“Thus, in ten ‘long hops,’ or ten days and nights, the traveller and the mailbag out of England would arrive in the most distant of our dominions without landing at an air station which was not either British or under British control.”

Wallpaper

futility closet wallpaper

Thanks again to the world-bestriding Von Glitschka for our new logo. At Von’s suggestion we’re calling the penguin Mr. Curious.

Von has also thoughtfully created wallpaper files for iPhone, iPad, and desktops, which are free for the downloading:

iPhone 4, iPhone 5, iPhone 6, iPhone 6 Plus

iPad Air, Mini, Pro

Desktop: 1920 × 1080, 2560 × 1440, 3410 × 1918

I’ll add these links to the About page as well.

As we get organized we’ll announce other offerings here and in the Patreon feeds, and Von will be creating custom promo graphics to download, print out, and collect. A million thanks again to him for all his contributions, and to you for your kind words about the redesign.

Not So Fast

José Paluzie offered this chess poser in 1910. White is to move and mate in 1:

paluzie chess puzzle

The key is to notice that the position is illegal: There’s no legal way for the black king to have arrived at a2. Black, desperate to avoid mate, must have put it there when White wasn’t looking.

Where did it come from? It doesn’t matter: White can place the black king on any legal square and mate in 1.

(From Burt Hochberg’s Chess Braintwisters, 1999.)

Podcast Episode 106: The Popgun War

https://commons.wikimedia.org/wiki/File:USArmy_M114_howitzer.jpg

During wargames in Louisiana in September 1941, the U.S. Army found itself drawn into a tense firefight with an unseen enemy across the Cane River. The attacker turned out to be three boys with a toy cannon. In this week’s episode of the Futility Closet podcast we’ll revisit the Battle of Bermuda Bridge and the Prudhomme brothers’ account of their historic engagement.

We’ll also rhapsodize on guinea pigs and puzzle over some praiseworthy incompetence.

Sources for our feature on the “Battle of Bermuda Bridge”:

Elizabeth M. Collins, “Patton ‘Bested’ at the Battle of Bermuda Bridge,” Soldiers 64:9 (September 2009), 10-12.

Terry Isbell, “The Battle of the Bayous: The Louisiana Maneuvers,” Old Natchitoches Parish Magazine 2 (1997), 2-7.

Special thanks to the staff at the University of North Carolina’s Wilson Library for access to the Prudhomme family records.

Listener mail:

Alastair Bland, “From Pets To Plates: Why More People Are Eating Guinea Pigs,” The Salt, National Public Radio, April 2, 2013.

Christine Dell’Amore, “Guinea Pigs Were Widespread as Elizabethan Pets,” National Geographic, Feb. 9, 2012.

Wikipedia, “Guinea Pig” (accessed May 20, 2016).

David Adam, “Why Use Guinea Pigs in Animal Testing?”, Guardian, Aug. 25, 2005.

Maev Kennedy, “Elizabethan Portraits Offer Snapshot of Fashion for Exotic Pets,” Guardian, Aug. 20, 2013.

“How Did the Guinea Pig Get Its Name?”, Grammarphobia, Dec. 22, 2009.

This week’s lateral thinking puzzle was contributed by listener Tommy Honton, who sent these corroborating links (warning: these spoil the puzzle).

You can listen using the player above, download this episode directly, or subscribe on iTunes or Google Play Music or via the RSS feed at http://feedpress.me/futilitycloset.

Enter code CLOSET to get $5 off your first purchase of high-quality razor blades at Harry’s.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

Followup

During World War I, cable censors would sometimes change a word here and there in a telegram, preserving the meaning but hoping to interfere with any enemy codes the messages might contain.

‘Father is dead,’ ran a cablegram from Sweden to New York which passed through the British censorship.

For some inexplicable reason the censor didn’t like the word ‘dead.’ He changed it to ‘deceased.’

Within a short time this question, sent from New York to Sweden, passed through the hands of the same censor: ‘Is father dead or deceased?’

“What did that word ‘dead’ mean? It might have covered a whole volume of enemy news; it might have provoked a disaster on land or sea. And yet the censor had no better reason for cutting it out than a certain ‘hunch’ which came over him that the word ought to be changed.”

(“Our Dear Friend, the Censor,” American Printer, June 5, 1917.)

Balance

balance puzzle

Point P lies within acute angle XOY. How can we find a point A on OX and a point B on OY such that P is the midpoint of a segment drawn between them?

Click for Answer

Elements

https://www.pexels.com/photo/sky-blue-sun-cloud-9236/

In James Joyce’s Ulysses, the events of Bloomsday are so carefully worked out that even incidents of weather can be recognized across the various episodes. Episode 1, at the tower:

A cloud began to cover the sun slowly, wholly, shadowing the bay in deeper green.

In Episode 4, Bloom notices the same thing as he walks home from Dlugacz’s shop:

A cloud began to cover the sun slowly, wholly. Grey. Far.

Four paragraphs later the cloud has passed:

Quick warm sunlight came running from Berkeley road, swiftly, in slim sandals, along the brightening footpath.

And back at the tower it passes as well:

Stephen, still trembling at his soul’s cry, heard warm running sunlight and in the air behind him friendly words.

“The breeze is therefore approximately from the west, that being the prevailing direction of winds in the British Isles,” observes Ian Gunn in James Joyce’s Dublin (2004).