Threes and Fours
Image: Wikimedia Commons

A problem from the Tenth International Mathematical Olympiad, 1968:

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

Click for Answer


  • The negative space in the eight of diamonds forms an 8.
  • William Brewster, leader of the Plymouth Colony, named his children Jonathan, Patience, Fear, Love, and Wrestling.
  • Wilfred Owen’s mother received the news of his death on Armistice Day.
  • “I never had any philosophic instruction, the first lecture on psychology I ever heard being the first I ever gave.” — William James

Good Boy

In 1919, engineer James Cowan Smith bequeathed £55,000 to the National Gallery of Scotland.

He set two conditions. One was that the gallery provide for his dog Fury.

The other was that this picture of his previous dog Callum, by painter John Emms, always be hung in the gallery.

Both conditions were fulfilled.

Image: Wikimedia Commons

In 1940 Bertrand Russell was invited to teach logic at the City College of New York.

A Mrs. Kay of Brooklyn opposed the appointment, citing Russell’s agnosticism and his alleged practice of sexual immorality.

In the lawsuit his works were described as “lecherous, libidinous, lustful, venerous, erotomaniac, aphrodisiac, irreverent, narrowminded, untruthful, and bereft of moral fiber.”

“Although he lost the case, the aging Russell was delighted to have been described as ‘aphrodisiac,'” writes Betsy Devine in Absolute Zero Gravity. “‘I cannot think of any predecessors,’ he claimed, ‘except Apuleius and Othello.'”

Dance Lessons

The quicksort computer sorting algorithm demonstrated with Hungarian folk dance, from Romania’s Sapientia University.


The four queens puzzle solved using ballet.

Binary search through flamenco dance.

Merge sort via Transylvanian-Saxon folk dance.

Selection sort using Gypsy folk dance.


(Via MetaFilter.)

01/19/2019 UPDATE: When Gavin Taylor showed these algorithms to his students at the United States Naval Academy, they asked whether they themselves could dance for extra credit. He said yes. So here are the U.S. Naval Academy midshipmen dancing the InsertionSort algorithm:

(Thanks, Gavin.)

Tempting Fate

What remained of the Tenth [Massachusetts] departed from City Point, on the James River, on June 21 [1864], for the return to Springfield and Northampton. But before leaving Virginia, on June 20, Sgt. Maj. George F. Polley, who was originally in Brewster’s company and had just reenlisted, carved his name and the inscription ‘Killed June –, 1864’ on a piece of board torn from a cracker box. After participating in the ‘goodbye’ rituals with his comrades and sharing an awkward amusement with them about his carving, Polley was struck flush by an artillery shell and killed. In his diary, brigade member Elisha Hunt Rhodes recorded this incident in his matter-of-fact style. Polley ‘showed me a board on which he had carved his name, date of birth and had left a place for the date of his death,’ reported Rhodes. ‘I asked him if he expected to be killed and he said no, and that he had made his head board only for fun. To day he was killed by a shell from a Rebel Battery.’ The last act of the Tenth before boarding the mailboat for Washington, D.C., was to bury Polley.

— David W. Blight, When This Cruel War Is Over, 2009

Area Matters

area matters 1

If you know the vertices of a polygon, here’s an interesting way to find its area:

  1. Arrange the vertices in a vertical list, repeating the first vertex at the end (see below).
  2. Multiply diagonally downward both ways as shown.
  3. Add the products on each side.
  4. Find the difference of these sums.
  5. Halve that difference to get the area.

area matters 2

This works for any polygon, no matter the number of points, so long as it doesn’t intersect itself. It’s a slight restatement of the shoelace formula.

(Thanks, Derek, Dan, and Kyle.)

The Vowel Triangle

Chris McManus discovered this oddity. If W and Y are accepted as vowels, that gives us AEIOUWY. Starting with O, number these according to their positions on a circular alphabet without starting the count over for A (that is, O is the 15th letter of the alphabet, so it’s assigned number 15; beyond Z we’d reach A as the “27th” letter; and so on). Now write these numbers into a triangle, again starting with O:

   O                15
 U W Y           21 23 25
A  E  I         27  31  35

Each of the five lines in the figure gives a different arithmetic progression:

UWY: difference of 2
AEI: difference of 4
OUA: difference of 6
OWE: difference of 8
OYI: difference of 10

(David Morice, “Kickshaws,” Word Ways 34:4 [November 2001], 292-305.)

Podcast Episode 232: The Indomitable Spirit of Douglas Bader,_CO_of_No._242_Squadron,_seated_on_his_Hawker_Hurricane_at_Duxford,_September_1940._CH1406.jpg

Douglas Bader was beginning a promising career as a British fighter pilot when he lost both legs in a crash. But that didn’t stop him — he learned to use artificial legs and went on to become a top flying ace in World War II. In this week’s episode of the Futility Closet podcast we’ll review Bader’s inspiring story and the personal philosophy underlay it.

We’ll also revisit the year 536 and puzzle over the fate of a suitcase.


In 1872 Celia Thaxter published an unsettling poem about an iceberg.

In 193 the Praetorian Guard auctioned off the Roman empire.

Sources for our story on Douglas Bader:

Paul Brickhill, Reach for the Sky, 1954.

S.P. Mackenzie, Bader’s War, 2008.

Andy Saunders, Bader’s Last Fight, 2007.

Joel Ralph, “Their Finest Hour,” Canada’s History 95:6 (December 2015/January 2016), 22-31.

Paul Laib, “Bader, Sir Douglas Robert Steuart,” Oxford Dictionary of National Biography, May 19, 2011.

A.W.G. English, “Psychology of Limb Loss,” BMJ: British Medical Journal 299:6710 (Nov. 18, 1989), 1287.

“Obituary,” Journal of the Royal Society of Arts 130:5315 (October 1982), 750-751.

The Douglas Bader Foundation.

Neil Tweedie, “Tribute to a Very British Hero,” Daily Telegraph, Aug. 10, 2001, 10.

“Reaching for the Sky: Lady Bader Unveils Statue in Honour of Sir Douglas,” Birmingham Post, Aug. 10, 2001, 6.

“Who Really Shot Down Douglas Bader?” Daily Telegraph, Aug. 9, 2001, 23.

Arifa Akbar, “In Memory of a Legendary Hero,” [Darlington, UK] Northern Echo, Aug. 8, 2001, 8.

“Sir Douglas Bader, Legless RAF Ace Who Shot Down 22 German Planes,” Associated Press, Sept. 6, 1982, 1.

“Sir Douglas Bader, World War II Ace,” Associated Press, Sept. 5, 1982.

Herbert Mitgang, “He Fought Sitting Down,” New York Times, Nov. 17, 1957.

“Legless British Pilot to Aid Veterans Here,” New York Times, May 7, 1947.

“Legless Air Hero Enters British Title Golf Event,” New York Times, April 5, 1946.

“Legless RAF Ace Honored,” New York Times, Nov. 28, 1945.

“Bader, Legless RAF Flier, Freed by Yanks in Reich,” New York Times, April 19, 1945.

“Germans Recapture Flier Bader As He Tries Out Those New Legs; Bader Is Caught Trying to Escape,” New York Times, Sept. 29, 1941.

“Bader Gets New Artificial Leg, But Escape Attempt Fails,” [Washington D.C.] Evening Star, Sept. 29, 1941 A-4.

“Legless Pilot Honored; Bader, Now War Prisoner, Gets Bar to Flying Cross,” New York Times, Sept. 5, 1941.

“Epic of Bader’s Leg,” New York Times, Aug. 21, 1941.

“R.A.F., on Sweep, Drops Artificial Leg for Bader,” New York Times, Aug. 20, 1941.

“Bader Is Nazi Prisoner; Legless R.A.F. Ace Safe After Parachuting in France,” New York Times, Aug. 15, 1941.

“Bader, Legless R.A.F. Ace, Reported Missing,” New York Times, Aug. 13, 1941.

“Two British Air Force Aces, One Legless, Reported Missing,” [Washington D.C.] Evening Star, Aug. 12, 1941, A-18.

“10 Leading R.A.F. Aces Listed for Exploits,” New York Times, Jan. 10, 1941.,_Flight_Lieutenant_Eric_Ball_and_Pilot_Officer_Willie_McKnight,_admire_the_nose_art_on_Bader%27s_Hawker_Hurricane_at_Duxford,_October_1940._CH1412.jpg

Bader with Flight Lieutenant Eric Ball and Pilot Officer Willie McKnight of No. 242 Squadron, Duxford, October 1940. Bader himself designed the squadron’s emblem, a boot kicking Hitler in the breeches.

Listener mail:

Wikipedia, “Settlement of Iceland” (accessed Jan. 4, 2019).

Wikipedia, “History of Iceland” (accessed Jan. 4, 2019).

Wikipedia, “Papar” (accessed Jan. 4, 2019)., “The Discovery and Settlement of Iceland” (accessed Jan. 4, 2019).

Neil Schlager, Science and Its Times: Understanding the Social Significance of Scientific Discovery, 2001.

Wikipedia, “Thule” (accessed Jan. 4, 2019).

Wikipedia, “(486958) 2014 MU69” (accessed Jan. 4, 2019).

NASA, “New Horizons Chooses Nickname for ‘Ultimate’ Flyby Target,” March 13, 2018.

“Is This the Reason Ireland Converted to Christianity?,” Smithsonian Channel, June 26, 2014.

Mike Wall, “How Halley’s Comet Is Linked to a Famine 1,500 Years Ago,” NBC News, Dec. 19 2013.

Colin Barras, “The Year of Darkness,” New Scientist 221:2952 (2014), 34-38.

This week’s lateral thinking puzzle was contributed by listener Jeff King.

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

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

If you have any questions or comments you can reach us at Thanks for listening!