Schwenk Dice

Western Michigan University mathematician Allen J. Schwenk discovered this oddity in 2000: Consider three fair six-sided dice of different colors, marked with the following numbers:

  • Red: 2, 2, 2, 11, 11, 14
  • Blue: 0, 3, 3, 12, 12, 12
  • Green: 1, 1, 1, 13, 13, 13

Now:

  • The red die beats the green die 7/12 of the time.
  • The blue die beats the red die 7/12 of the time.
  • The green die beats the blue die 7/12 of the time.

We’ve seen that before. But look at this:

  • A pair of green dice beats a pair of red dice 693/1296 of the time.
  • A pair of red dice beats a pair of blue dice 675/1296 of the time.
  • A pair of blue dice beats a pair of green dice 693/1296 of the time.

The favored color in each pairing has changed! Schwenk writes, “I call this a perverse reversal.”

(And a bonus: It turns out that a pair of Schwenk dice of any one color is an even match against a mixed pair of the other two colors.)

(Allen J. Schwenk, “Beware of Geeks Bearing Grifts,” Math Horizons 7:4 [April 2000], 10-13, via Jennifer Beineke and Lowell Beineke, “Some ABCs of Graphs and Games,” in Jennifer Beineke and Jason Rosenhouse, eds., The Mathematics of Various Entertaining Subjects, 2016.)

Unquote

Omne ignotum pro magnifico est. Everything unknown is assumed to be grand.” — Tacitus

“As a rule, what is out of sight disturbs men’s minds more seriously than what they see.” — Julius Caesar

“Ignorance is the parent of fear.” — Herman Melville

Dedication

https://commons.wikimedia.org/wiki/File:James_Pullen%27s_model_of_SS_Great_Eastern_at_Langdon_Down_Museum.JPG
Image: Wikimedia Commons

Confined since age 15 in Surrey’s Earlswood Asylum, autistic savant James Henry Pullen spent seven years building a 10-foot replica of the iron steamship Great Eastern. Completed in 1877, it included brass anchors, copper paddles, 13 lifeboats, hundreds of individually molded planks, 5,585 rivets, and more than 1 million wooden pins made in a specially constructed pin mill. The upper deck could be hoisted to reveal state cabins and furniture inside. It’s now on display at the Museum at the Langdon Down Centre in Teddington.

Below is the sectional plan of the actual 692-foot steamship, for comparison.

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

(Thanks, Charlie.)

Clergymen and Chickens

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

Why, let me ask, should a hen lay an egg which egg can become a chicken in about three weeks and a full-grown hen in less than a twelvemonth, while a clergyman and his wife lay no eggs but give birth to a baby which will take three-and-twenty years before it can become another clergyman? Why should not chickens be born and clergymen be laid and hatched? Or why, at any rate, should not the clergyman be born full grown and in Holy Orders, not to say already beneficed? The present arrangement is not convenient, it is not cheap, it is not free from danger, it is not only not perfect but is so much the reverse that we could hardly find words to express our sense of its awkwardness if we could look upon it with new eyes, or as the cuckoo perhaps observes it.

— Samuel Butler, “On Memory as a Key to the Phenomena of Heredity,” Working Men’s College, London, Dec. 2, 1882

Podcast Episode 206: The Sky and the Sea

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Image: Wikimedia Commons

Swiss physicist Auguste Piccard opened two new worlds in the 20th century. He was the first person to fly 10 miles above the earth and the first to travel 2 miles beneath the sea, using inventions that opened the doors to these new frontiers. In this week’s episode of the Futility Closet podcast we’ll follow Piccard on his historic journeys into the sky and the sea.

We’ll also admire some beekeeping serendipity and puzzle over a sudden need for locksmiths.

See full show notes …

The Slothouber-Graatsma Puzzle

A deceptively simple packing problem by Dutch architects Jan Slothouber and William Graatsma: How can you assemble six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into a 3 × 3 × 3 cube? There’s no trick to it, but it can be quite difficult to solve — the solution is unique, not counting mirror reflections and rotations.

Click for Answer

No Rest

https://commons.wikimedia.org/wiki/File:Prohibition_of_death_around_the_world.jpg
Image: Wikimedia Commons

In the 5th century BC, Athens forbade anyone to die or to give birth on the island of Delos, to render it fit for the proper worship of the gods.

In 2005 Roberto Pereira, mayor of the Brazilian town of Biritiba Mirim, proposed a ban on death because the local cemetery had reached its capacity.

The French settlements of Le Lavandou (in 2000), Cugnaux (in 2007), and Sarpourenx (in 2008) have all outlawed death because of limited capacity in local cemeteries. The Sarpourenx ordinance added: “Offenders will be severely punished.”

Since 1878, no births or deaths have been permitted near Japan’s Itsukushima Shrine, a sacred site in Shinto belief.

In 1999 the mayor of the Spanish town of Lanjarón outlawed death, again because of an overcrowded cemetery. His edict ordered residents “to take utmost care of their health so they do not die until town hall takes the necessary steps to acquire land suitable for our deceased to rest in glory.”

“I don’t want to achieve immortality through my work,” said Woody Allen. “I want to achieve immortality through not dying.”

06/24/2018 UPDATE: It’s illegal to die in Longyearbyen, Norway, because digging in the cemetery might unleash the century-old Spanish flu virus buried in the permafrost. (Thanks, Michael.)

An Apparition

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Shortly after conquering the Matterhorn on July 14, 1865, Edward Whymper watched four of his companions fall to their deaths down the mountain’s precipitous north face. Afterward he and his two Swiss guides, the Taugwalders, beheld a remarkable figure in the sky:

A mighty arch appeared, rising above the Lyskamm, high into the sky. Pale, colourless, and noiseless, but perfectly sharp and defined, except where it was lost in the clouds, this unearthly apparition seemed like a vision from another world; and, almost appalled, we watched with amazement the gradual development of two vast crosses, one on either side. If the Taugwalders had not been the first to perceive it, I should have doubted my senses. They thought it might have some connection with the accident, and I, after a while, that it might bear some relation to ourselves. But our movements had no effect on it. It was a fearful and wonderful sight; unique in my experience, and impressive beyond description, coming at such a moment.

What was this? Whymper later called it a fog bow, a bow that forms in fog rather than rain. Unfortunately, we have no photograph, only a sketch and a woodcut. In a 2002 simulation C.J. Hardwick tried to account for the features as Whymper had described them. “A fogbow and ice crystal arcs could have produced a circle and crosses in a direction consistent with the apparition,” he concluded in 2005. “However, while this simulation used a crystal type that can occur, it required an unusual alignment that would be very rare.”

(C.J. Hardwick, “Simulation of the Whymper Apparition,” Weather 57:12 [December 2002], 457-463; Cedric John Hardwick and Jason C. Knievel, “Speculations on the Possible Causes of the Whymper Apparition,” Applied Optics 44:27 [Sept. 20, 2005], 5637-5643.)

One-Man Band

In the early days of silent movies, large theaters would engage orchestras to play the accompanying music. But starting around 1910, small venues that lacked the money or the space could use a machine instead. In Film Music, music editor Roy M. Pendergast writes, “In addition to music, these machines were capable of providing a battery of sound effects, and they ranged in size from what was essentially a player piano with small percussion setup to elaborate instruments nearly equaling a twenty-piece pit orchestra.” He quotes Samuel A. Peeples in Films in Review:

The crowning achievement of the American Photo Player Company was their Fotoplayer Style 50, only one of which is presently known to survive in operating condition. Among the most splendid automatic musical instruments ever built, it was 21 feet long, 5 feet wide, and 5 feet 2 inches tall. It was capable of recreating the volume of a 20-piece pit orchestra, plus a full-scale theatre pipe-organ, with an incredible range of effects, such as the lowing of cattle, the drumming of hoofs in assorted gaits, several varieties of klaxons, street traffic noises, crackling flames, breaking wood and brush, rifle, pistol and machine gun shots, even the sound of a French 75MM cannon!

Pendergast adds, “One wonders about the quality of genius it must have taken to operate one of these devices.”

Black and White

korolikov chess problem

A curiously ambiguous problem by V.A. Korolikov. Mate in 1. Either side can fulfill this easily — but which has the move?

Click for Answer