Time Saver

Ogden Nash invented a streamlined limerick he called the “limick”:

An old person of Troy
In the bath is so coy
That it doesn’t know yet
If it’s a girl or a boy.

Two nudists of Dover,
When purple all over,
Were munched by a cow,
When mistaken for clover.

A cook called McMurray
Got a raise in a hurry
From his Hindu employer,
By flavouring curry.

A young flirt of Ceylon,
Who led the boys on,
Playing “Follow the Leda,”
Succumbed to a swan.

For What It’s Worth


Based on an ancient Hindu game, Snakes and Ladders (Chutes and Ladders in ophidiophobic America) is at heart a morality lesson: As you progress by die roll from square 1 to square 100 and spiritual enlightenment, your way is complicated by virtues and vices. Landing on a snake (or chute) will send you back to an earlier square, and landing on a ladder will send you ahead to a later one. Each of these shortcuts is associated with a precept — “Carelessness” leads to “Injury,” “Study” leads to “Knowledge,” and so on.

In 1993 University of Michigan mathematician S.C. Althoen and his colleagues considered the game as a 101-state absorbing Markov chain. The shortest possible game lasts seven moves, the longest is infinite, and according to their calculations the expected number of moves in the Milton Bradley version of Chutes and Ladders is

\displaystyle  \frac{225837582538403273407117496273279920181931269186581786048583}{5757472998140039232950575874628786131130999406013041613400},

which is about 39.2.

Troublingly, the average length of a game without snakes or ladders (just the 100-square board) is almost exactly 33 moves: “Apparently the snakes lengthen the game more than the ladders shorten it.” And, while adding a ladder will generally shorten the game and adding a snake will lengthen it, this isn’t always the case: In the original game, adding a ladder from square 79 to square 81 lengthens the expected playing time by more than two moves (to about 41.9), since it increases the chance of missing the important ladder leading from square 80 to square 100. And adding a snake from square 29 to square 27 shortens the game by more than a move (to about 38.0), since it offers a second chance at the long ladder from 28 to 84.

So, arguably, we might advance more quickly through life with more vice and less virtue.

(S.C. Althoen, L. King, and K. Schilling, “How Long Is a Game of Snakes and Ladders?”, Mathematical Gazette 77:478 [March 1993], 71-76.)

Black and White


I just ran across this in Benjamin Glover Laws’ The Two-Move Chess Problem, from 1890. It’s by G. Chocholous. White is to mate in two moves.

Click for Answer



On its 35th anniversary, THX, the sound quality assurance company founded by George Lucas, released the original score of “Deep Note,” its audio trademark, which debuted at the premiere of Return of the Jedi in 1983 and is now familiar from countless films. Essentially it’s a stupendous D chord; the U.S. trademark registration reads:

The THX logo theme consists of 30 voices over seven measures, starting in a narrow range, 200 to 400 Hz, and slowly diverting to preselected pitches encompassing three octaves. The 30 voices begin at pitches between 200 Hz and 400 Hz and arrive at pre-selected pitches spanning three octaves by the fourth measure. The highest pitch is slightly detuned while there are double the number of voices of the lowest two pitches.

“I like to say that the THX sound is the most widely-recognized piece of computer-generated music in the world,” says James A. Moorer, who wrote it. “This may or may not be true, but it sounds cool.” And now that we have the score you can do this:

Cube Route

Created by Franz Armbruster in 1967, “Instant Insanity” was the Rubik’s Cube of its day, a simple configuration task with a dismaying number of combinations. You’re given four cubes whose faces are colored red, blue, green, and yellow:

Image: Wikimedia Commons

The task is to arrange them into a stack so that each of the four colors appears on each side of the stack. This is difficult to achieve by trial and error, as the cubes can be arranged in 41,472 ways, and only 8 of these give a valid solution.

One approach is to use graph theory — draw points of the four face colors and connect them to show which pairs of colors fall on opposite faces of each cube:

Image: Wikimedia Commons

Then, using certain criteria (explained here), we can derive two directed subgraphs that describe the solution:

Image: Wikimedia Commons

The first graph shows which colors appear on the front and back of each cube, the second which colors appear on the left and right. Each arrow represents one of the four cubes and the position of each of the two colors it indicates. So, for example, the black arrow at the top of the first graph indicates that the first cube will have yellow on the front face and blue on the rear.

This solution isn’t unique, of course — once you’ve compiled a winning stack you can rotate it or rearrange the order of the cubes without affecting its validity. B.L. Schwartz gives an alternative method, through inspection of a table, as well as tips for solving by trial and error using physical cubes, in “An Improved Solution to ‘Instant Insanity,'” Mathematics Magazine 43:1 (January 1970), 20-23.


Image: Flickr

“If you have an apple and I have an apple and we exchange apples then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, each of us will have two ideas.” — George Bernard Shaw

“He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me. That ideas should freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition, seems to have been peculiarly and benevolently designed by nature, when she made them, like fire, expansible over all space, without lessening their density in any point.” — Thomas Jefferson

Podcast Episode 217: The Bone Wars


The end of the Civil War opened a new era of fossil hunting in the American West — and a bitter feud between two rival paleontologists, who spent 20 years sabotaging one another in a constant struggle for supremacy. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Bone Wars, the greatest scientific feud of the 19th century.

We’ll also sympathize with Scunthorpe and puzzle over why a driver can’t drive.


Nepal’s constitution contains instructions for drawing its flag.

The tombstone of Constanze Mozart’s second husband calls him “the husband of Mozart’s widow.”


Othniel Charles Marsh and Edward Drinker Cope.

Sources for our feature on the Bone Wars:

David Rains Wallace, The Bonehunters’ Revenge, 1999.

Mark Jaffe, The Gilded Dinosaur, 2000.

Elizabeth Noble Shor, The Fossil Feud, 1974.

Hal Hellman, Great Feuds in Science, 1998.

Tom Huntington, “The Great Feud,” American History 33:3 (August 1998), 14.

Richard A. Kissel, “The Sauropod Chronicles,” Natural History 116:3 (April 2007), 34-38.

Keith Stewart Thomson, “Marginalia: Dinosaurs as a Cultural Phenomenon,” American Scientist 93:3 (May-June 2005), 212-214.

Genevieve Rajewski, “Where Dinosaurs Roamed,” Smithsonian 39:2 (May 2008), 20-24.

James Penick Jr., “Professor Cope vs. Professor Marsh,” American Heritage 22:5 (August 1971).

Alfred S. Romer, “Cope versus Marsh,” Systematic Zoology 13:4 (December 1964), 201-207.

Renee Clary, James Wandersee, and Amy Carpinelli, “The Great Dinosaur Feud: Science Against All Odds,” Science Scope 32:2 (October 2008), 34-40.

Susan West, “Dinosaur Head Hunt,” Science News 116:18 (Nov. 3, 1979), 314-315.

P.D. Brinkman, “Edward Drinker Cope’s Final Feud,” Archives of Natural History 43:2 (October 2016), 305-320.

Eric J. Hilton, Joseph C. Mitchell and David G. Smith, “Edward Drinker Cope (1840–1897): Naturalist, Namesake, Icon,” Copeia 2014:4 (December 2014), 747-761.

John Koster, “Good to the Old Bones: Dreaming of Dinosaurs, Digging for Dollars,” Wild West 25:2 (August 2012), 26-27.

Daniel Engber, “Bone Thugs-N-Disharmony,” Slate, Aug. 7, 2013.

Walter H. Wheeler, “The Uintatheres and the Cope-Marsh War,” Science, New Series 131:3408 (April 22, 1960), 1171-1176.

Lukas Rieppel, “Prospecting for Dinosaurs on the Mining Frontier: The Value of Information in America’s Gilded Age,” Social Studies of Science 45:2 (2015), 161-186.

Michael J. Benton, “Naming Dinosaur Species: The Performance of Prolific Authors,” Journal of Vertebrate Paleontology 30:5 (2010), 1478-1485.

Cary Woodruff and John R. Foster, “The Fragile Legacy of Amphicoelias fragillimus (Dinosauria: Sauropoda; Morrison Formation-Latest Jurassic),” PeerJ PrePrints 3 (2014), e838v1.

Paul Semonin, “Empire and Extinction: The Dinosaur as a Metaphor for Dominance in Prehistoric Nature,” Leonardo 30:3 (1997), 171-182.

Jennie Erin Smith, “When Fossil-Finding Was a Contact Sport,” Wall Street Journal Asia, June 10, 2016, A.11.

Adam Lusher, “The Brontosaurus Is Back After 150 Million Years… At Least in Name,” Independent, April 8, 2015, 10.

Will Bagley, “Rivals Fought Tooth and Nail Over Dinosaurs,” Salt Lake Tribune, March 25, 2001, B1.

Clive Coy, “Skeletons in the Closet,” Ontario National Post, Jan. 22, 2000, 10.

Rose DeWolf, “Philly Is Facile With Fossils,” Philadelphia Daily News, March 27, 1998, D.6.

Mark Jaffe, “Phila. and Fossils Go Way Back,” Philadelphia Inquirer, March 22, 1998, 2.

Malcolm W. Browne, “Dinosaurs Still Star in Many Human Dramas and Dreams,” New York Times, Oct. 14, 1997.

John Noble Wilford, “Horses, Mollusks and the Evolution of Bigness,” New York Times, Jan. 21, 1997.

Jerry E. Bishop, “Bones of Contention: Should Dr. Cope’s Be The Human Model?” Wall Street Journal, Nov. 1, 1994, A1.

“Dinosaur Book Has Museum Aide Losing His Head,” Baltimore Sun, Oct. 17, 1994, 6A.

“The Bricks of Scholarship,” New York Times, Jan. 21, 1988.

Dick Pothier, “Fossil Factions: Dinosaur Exhibit Points Out a Battle in Science,” Philadelphia Inquirer, Feb. 9, 1986, B.14.

Rose DeWolf, “Dinosaurs: Bone in the USA,” Philadelphia Daily News, Jan. 24, 1986, 52.

William Harper Davis, “Cope, a Master Pioneer of American Paleontology,” New York Times, July 5, 1931.

George Gaylord Simpson, “Mammals Were Humble When Dinosaurs Roved,” New York Times, Oct. 18, 1925.

“A Prehistoric Monster,” Hartford Republican, Sept. 1, 1905.

“The Scientists’ New President,” Topeka State Journal, Oct. 9, 1895.

Listener mail:

David Mack, “This Woman With a ‘Rude’ Last Name Started the Best Thread on Twitter,” BuzzFeed News, Aug. 29, 2018.

Natalie Weiner, Twitter, Sept. 6, 2018.

Wikipedia, “Scunthorpe Problem” (accessed Sept. 6, 2018).

Declan McCullagh, “Google’s Chastity Belt Too Tight,” CNET, April 23, 2004.

Daniel Oberhaus, “Life on the Internet Is Hard When Your Last Name is ‘Butts,'” Motherboard, Aug. 29, 2018.

Matthew Moore, “The Clbuttic Mistake: When Obscenity Filters Go Wrong,” Telegraph, Sept. 2, 2008.

This week’s lateral thinking puzzle was contributed by listener David Malki.

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.

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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 podcast@futilitycloset.com. Thanks for listening!