Rising Masses

A writer in The Builder has cleverly suggested that bridges might be erected in the crowded thoroughfares of London for the convenience of foot passengers, who lose so much valuable time in crossing. As the stairs would occupy a considerable space, and occasion much fatigue, I beg to propose an amendment: Might not the ascending pedestrians be raised up by the descending? The bridge would then resemble the letter H, and occupy but little room. Three or four at a time, stepping into an iron framework, would be gently elevated, walk across, and perform by their weight the same friendly office for others rising on the opposite side. Surely no obstacles can arise which might not be surmounted by ingenuity. If a temporary bridge were erected in one of the parks the experiment might be tried at little cost, and, at any rate, some amusement would be afforded. C.T.

Notes and Queries, July 17, 1852

The Four Points, Two Distances Problem

winkler distances problem

Alex Bellos set a pleasingly simple puzzle in Monday’s Guardian: How many ways are there to arrange four points in the plane so that only two distances occur between any two points? He gives one solution, which helps to illustrate the problem: In a square, any two vertices are separated by either the length of a side or the length of a diagonal — no matter which two points are chosen, the distance between them will be one of two values. Besides the square, how many other configurations have this property?

The puzzle comes originally from Dartmouth mathematician Peter Winkler, who writes, “Nearly everyone misses at least one [solution], and for each possible solution, it’s been missed by at least one person.”

The answer is here.


In On the Origin of Species, Darwin wrote that bumblebees are the only pollinators of red clover.

In 1862 he discovered that this is wrong — honeybees do it as well.

He wrote to his friend John Lubbock, “I hate myself, I hate clover, and I hate bees.”

Black and White

laws chess problem

“A fairly good two-mover” from Benjamin Glover Laws’ The Two-Move Chess Problem, 1890. What’s the key move?

Click for Answer


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Podcast Episode 269: The Sack of Baltimore


One night in 1631, pirates from the Barbary coast stole ashore at the little Irish village of Baltimore and abducted 107 people to a life of slavery in Algiers — a rare instance of African raiders seizing white slaves from the British Isles. In this week’s episode of the Futility Closet podcast we’ll describe the sack of Baltimore and the new life that awaited the captives in North Africa.

We’ll also save the Tower of London and puzzle over a controversial number.


In 1999, inventor Allison Andrews proposed dividing all our pants in half.

In 1955, test pilot Alvin Johnston put an airliner through a barrel roll.

Sources for our feature on the sack of Baltimore:

Des Ekin, The Stolen Village: Baltimore and the Barbary Pirates, 2012.

Nabil Matar, British Captives From the Mediterranean to the Atlantic, 2014.

David D. Hebb, Piracy and the English Government 1616–1642: Policy-Making Under the Early Stuarts, 2016.

Sir Robert Lambert Playfair, The Scourge of Christendom: Annals of British Relations With Algiers Prior to the French Conquest, 1884.

Theresa D. Murray, “From Baltimore to Barbary: The 1631 Sack of Baltimore,” History Ireland 14:4 (July/August 2006).

Nabil Matar, “The Barbary Corsairs, King Charles I and the Civil War,” Seventeenth Century 16:2 (October 2001), 239-258.

Nabil I. Matar, “Wives, Captive Husbands, and Turks: The First Women Petitioners in Caroline England,” Explorations in Renaissance Culture 40:1-2 (Summer-Winter 2014), 125+.

Paul Baepler, “The Barbary Captivity Narrative in American Culture,” Early American Literature 39:2 (January 2004), 217-246.

Robert C. Davis, “Counting European Slaves on the Barbary Coast,” Past & Present 172 (August 2001), 87-124.

Paul Baepler, “White Slaves, African Masters,” Annals of the American Academy of Political and Social Science 588 (July 2003), 90-111.

Erik Göbel, “The Danish Algerian Sea Passes, 1747-1838: An Example of Extraterritorial Production of Human Security,” Historical Social Research 35:4, 164-189.

Des Ekin, “The Irish Village That Was Kidnapped by Islamist Extremists,” Irish Independent, Sept. 2, 2006, 1.

“Cork Village to Recall Historical Event in Summer Festival,” Irish Times, June 20, 2000, 2.

Frank McNally, “An Irishman’s Diary,” Irish Times, June 20, 2007, 17.

“Algerian Pirates Who Took Devon Settlers as Slaves,” [Exeter] Express & Echo, June 20, 2007, 15.

“Pirates of the Cork Coast,” Irish Times, Sept. 16, 2006, 9.

“Village Stolen for a Life of Slavery,” [Plymouth, U.K.] Western Morning News, Feb. 17, 2007, 10.

“Islamic Pirates’ Raid on Baltimore,” Nationalist, Nov. 16, 2010.

Lara Marlowe, “Secrets of Barbary Corsair Life Uncovered: Historian Debunks Myth of ‘Savage’ Attack by Barbary Corsairs on Baltimore,” Irish Times, Feb. 18, 1998, 13.

“Anniversary of Baltimore Pirate Raid,” Irish Examiner, June 20, 2013.

“The Irish Slaves in North Africa,” Irish Independent, Nov. 11, 2006, 1.

James McConnachie, “Scourge of the Waves: Mediterranean Pirates Were Easily as Exotic and Terrifying as Their More Famous Caribbean Counterparts,” Sunday Times, March 28, 2010, 42.

Barry Roche, “Pirate Raid That Stunned Nation,” Sun, Jan. 10, 2003, 8.

Thomas Osborne Davis, “The Sack of Baltimore,” in Edmund Clarence Stedman, ed., A Victorian Anthology, 1895.

Listener mail:

Encyclopaedia Britannica, “Victoria Day” (accessed Oct. 10, 2019).

Canadian Encyclopedia, “Victoria Day,” Feb. 7, 2006.

“Why Do We Celebrate Victoria Day?”, CBC Kids (accessed Oct. 10, 2019).

Wikipedia, “Victoria Day (Scotland)” (accessed Oct. 10, 2019).

No Such Thing as a Fish, Episode 111, “No Such Thing as Snappedy Chat,” originally aired April 28, 2016.

Wikipedia, “Queen’s Official Birthday” (accessed Oct. 9, 2019).

Wikipedia, “Ravens of the Tower of London” (accessed Oct. 10, 2019).

William Booth, “The Secrets of the Tower of London’s Royal Ravenmaster,” Washington Post, Oct. 20, 2018.

“How the Ravenmaster of London Protects the Kingdom With Birds,” CBC Radio, Oct. 30, 2018.

PD Smith, “The Ravenmaster by Christopher Skaife Review — My Life at the Tower of London,” Guardian, Nov. 3, 2018.

“Tower of London Welcomes First Raven Chicks in 30 Years,” BBC News, May 17, 2019.

Meilan Solly, “Tower of London Welcomes Baby Ravens for the First Time in 30 Years,” Smithsonian.com, May 21, 2019.

This week’s lateral thinking puzzle was contributed by listener Mendel Smith, who sent this corroborating link (warning — this spoils the puzzle).

You can listen using the player above, download this episode directly, or subscribe on Google Podcasts, on Apple Podcasts, or via the RSS feed at https://futilitycloset.libsyn.com/rss.

Please consider becoming a patron of Futility Closet — you can choose the amount you want to pledge, and we’ve set up some rewards to help thank you for your support. You can also make a one-time donation on the Support Us page of the Futility Closet website.

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!

A Wine Slide Rule

Image: Science Museum Group

Revenue agents in 18th-century London faced a curious challenge: how to calculate the excise tax on a barrel that was partially full of liquor. The answer was an “ullage slide rule” — this gauging rod was dipped into the barrel, some brass sliding pieces were adjusted to reflect the height of the surface, and a mathematical calculation would tell how much liquid the barrel contained.

The Science Museum says, “The calculations were very complicated.” A correspondent to the Mathematical Gazette wrote in 1990, “I well remember puzzling, unsuccessfully, over graphs and calculations of measurements until I wrote to the makers whose name was stamped on the rule and who still existed [in 1966] at the same address in London Bridge. At that time they were still making a modern equivalent for the same use by revenue officers.” More at the link below.

(Tom Martin, “Gauging: The Art Behind the Slide Rule,” Brewery History 133 [2009], 69-86.)

Narrators and Film


Ishmael narrates Moby-Dick, just as Gulliver narrates his travels and John Watson narrates the Sherlock Holmes stories. In each case we can assume that all the information presented in the literary story is imparted to us by its fictional narrator.

But the filmed version of each story contains thousands of details that are apparent to us but clearly never observed directly by the narrator. Yet it’s still the narrator who’s ostensibly telling us the story. If the narrator isn’t supplying these details, then … who is?

Malfatti Circles


What’s the best way to squeeze three circles into a triangle so that the area of the circles is maximized? In 1803 Italian mathematician Gian Francesco Malfatti decided that the best course was to place each circle tangent to the other two and to two sides of the triangle (left) — he thought that some instance of this arrangement would give the best solution.

But that’s not actually so: In an equilateral triangle, Malfatti’s circles occupy less area than the solution on the right, found by Lob and Richmond in 1930 — their suggestion is to inscribe the largest possible circle in the triangle, then fit the second circle into one of the triangle’s three corners, and then fit the third circle into one of the five spaces now available, taking the largest available option in each case.

In the case of an equilateral triangle, Lob and Richmond’s solution is only about 1% larger than Malfatti’s. But in 1946 Howard Eves pointed out that for a long, narrow isosceles triangle (below), simply stacking three circles can cover nearly twice the area of the Malfatti circles.

Subsequent studies have borne this out — it turns out that Malfatti’s plan is never best. We now know that Lob and Richmond’s procedure will always find three area-maximizing circles — but whether their approach will work for more than three circles is an open question.

(Thanks, Larry.)

Image: Wikimedia Commons

“Let Liking Last”

Inscriptions found in 17th-century English wedding rings, from William Jones’ Finger-Ring Lore, 1898:

  • Let reason rule affection.
  • A token of good-will.
  • Live in Loue.
  • As I expect so let me find, A faithfull ❤ and constant mind.
  • Time lesseneth not my love.
  • Love the truth.
  • In loving wife spend all thy life.

A diamond ring bore the inscription “This sparke will grow.”