# Shifting Areas

This square of 8 × 8 = 64 square units can apparently be reassembled into a rectangle of 5 × 13 = 65 square units:

This paradox is described in W.W. Rouse Ball’s 1892 Mathematical Recreations and Essays; it seems to have been published first in 1868 in Zeitschrift für Mathematik und Physik.

In 1938 the Rockefeller Foundation’s Warren Weaver discovered an old trove of papers from the 1890s in which Lewis Carroll puzzled out the dimensions of all possible squares in which this illusion is possible (the other sizes include squares of 21 and 55 units on a side).

Regardless of publication, it’s not clear who first came up with the idea. Sam Loyd claimed to have presented it to the American Chess Congress in 1858. That would be interesting, as it was his son who later discovered that the four pieces can be assembled into a figure of 63 squares:

(Warren Weaver, “Lewis Carroll and a Geometrical Paradox,” American Mathematical Monthly 45:4 [April 1938], 234-236.)

Australia was named before it was discovered. Ancient geographers had supposed that land in the north must be balanced by land in the south — Aristotle had written, “there must be a region bearing the same relation to the southern pole as the place we live in bears to our pole” — and Romans told legends of a Terra Australis Incognita, an “unknown land of the South,” more than a millennium before Europeans first sighted the continent.

In 1814 the British explorer Matthew Flinders suggested applying the speculative name, Terra Australis, to the actual place — and in a footnote he wrote, “Had I permitted myself any innovation on the original term, it would have been to convert it to AUSTRALIA; as being more agreeable to the ear, and an assimilation to the names of the other great portions of the earth.”

# Podcast Episode 129: The Voynich Manuscript

In 1912, bookseller Wilfrid Voynich discovered an illustrated manuscript that was written in a mysterious alphabet that had never been seen before. The text bears the hallmarks of natural language, but no one has ever been able to determine its meaning. In this week’s episode of the Futility Closet podcast we’ll learn about the Voynich manuscript, which has been bewildering scholars for more than a century.

We’ll also ponder some parliamentary hostages and puzzle over a tormenting acquisition.

Intro:

In 1851, George Merryweather invented the Tempest Prognosticator, a rack of bottled leeches who would ring a bell when a storm approached.

Between 1884 and 1896, visitors to Coney Island could stay in a 31-room hotel shaped like an elephant.

Sources for our feature on the Voynich manuscript:

Gerry Kennedy and Rob Churchill, The Voynich Manuscript, 2004.

“Voynich Manuscript,” Beinecke Rare Book & Manuscript Library, Yale University.

Klaus Schmeh, “The Voynich Manuscript: The Book Nobody Can Read,” Skeptical Inquirer 35:1 (January/February 2011).

Diego R. Amancio et al., “Probing the Statistical Properties of Unknown Texts: Application to the Voynich Manuscript,” PLoS One, July 2, 2013.

Andreas Schinner, “The Voynich Manuscript: Evidence of the Hoax Hypothesis,” Cryptologia 31:2 (March 2007).

Marcelo A. Montemurro and Damián H. Zanette, “Keywords and Co-Occurrence Patterns in the Voynich Manuscript: An Information-Theoretic Analysis,” PLoS One, June 21, 2013.

Bec Crew, “Researcher Finds Evidence That the ‘World’s Most Mysterious Book’ Is an Elaborate Hoax,” Science Alert, Sept. 23, 2016.

Melissa Hogenboom, “Mysterious Voynich Manuscript Has ‘Genuine Message’,” BBC News, June 22, 2013.

Reed Johnson, “The Unread: The Mystery of the Voynich Manuscript,” New Yorker, July 9, 2013.

Rich McCormick, “Decrypting the Most Mysterious Book in the World,” The Verge, Feb. 28, 2014.

Wikipedia has scans of the entire manuscript, sortable by page, folio, or topic.

Listener mail:

Wikipedia, “Hostage MP” (accessed Nov. 12, 2016).

Wikipedia, “State Opening of Parliament” (accessed Nov. 12, 2016).

Matt Field, “Queen’s Speech: Your Guide to All the Parliamentary Pomp and Pageantry,” Guardian, May 27, 2015.

“Intertwined Love Story: Twins Who Married Twins,” Morning Edition, National Public Radio, May 28, 2010.

“Identical Twins Marry, Give Birth to Identical Twins,” Telegraph, July 22, 2008.

Danielle Centoni, “The Secret Life of Pears (in Brandy),” Oregon Live, September 2011.

This week’s lateral thinking puzzle was contributed by listener Jake Koethler.

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.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, 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!

# The Margate Shell Grotto

In the chalky soil under the English seaside town of Margate, someone has hewn an artificial cave and lined it with millions of seashells. No one knows who, when, or why — the popular story is that a laborer was digging in a field in 1835 when his spade disappeared into a void. Alerted to this mystery, James Newlove, the master of the nearby Dane House School, lowered his son Joshua into the darkness bearing a candle. Joshua would have found himself in a domed rotunda lined with shells, beyond which a winding passageway leads to a rectangular chamber of uncertain purpose. Newlove later purchased the land, installed gas lighting, and opened them to the public.

Even then the origins of the grotto were a mystery — and, as no scientific dating has been undertaken, we still don’t know when it was created. R.F. LeGear, who made an assessment for the Kent Archaeological Society, wrote, “Whoever commissioned and/or planned the elaborate designs for the shell panels must have been a well educated person who managed to entwine many different themes into the intricate patterns of literally millions of shells.” He suspects that a medieval denehole, or chalk-mining shaft, was reworked and expanded in the 17th or 18th century.

But “[a]s to the purpose of this enigmatic structure the writer can make no useful comment except that it is highly likely that the Shell Grotto’s original designer, whoever and whenever that was, has accomplished exactly what he set out to achieve i.e., speculation, controversy and conjecture which started with the discovery in 1835 and continues to the present day.”

(Thanks, Ron.)

# “A Postal Problem”

Browsing the Post Office Guide in June 1891, Lewis Carroll discovered an ambiguity that produces “a very curious verbal puzzle” — he sent this pamphlet to friends and interested parties:

The Rule, for Commissions chargeable on overdue Postal Orders, is given in the ‘Post Office Guide’ in these words, (it is here divided, for convenience of reference, into 3 clauses)—

(a) After the expiration of 3 months from the last day of the month of issue, a Postal Order will be payable only on payment of a Commission, equal to the amount of the original poundage;

(b) with the addition (if more than 3 months have elapsed since the said expiration) of the amount of the original poundage for every further period of 3 months which has so elapsed;

(c) and for every portion of any such period of 3 months over and above every complete period.

You are requested to answer the following questions, in reference to a Postal Order for 10/- (on which the ‘original poundage’ would be 1d.) issued during the month of January, so that the 1st ‘period’ would consist of the months February, March, April; the 2nd would consist of the months May, June, July; and the 3rd would consist of the months August, September, October.

(1) Supposing the Rule to consist of clause (a) only, on what day would a ‘Commission’ begin to be chargeable?

(2) What would be its amount?

(3) Supposing the Rule to consist of clauses (a) and (b), on what day would the lowest ‘Commission’ begin to be chargeable?

(4) What would be its amount?

(5) On what day would a larger ‘Commission’ (being the sum of 2 ‘Commissions’) begin to be chargeable?

(6) What would be its amount?

(7) On what day would a yet larger ‘Commission’ begin to be chargeable?

(8) What would be its amount?

(9) Taking the Rule as consisting of all 3 clauses, in which of the above-named 3 ‘periods’ does clause (c) first begin to take effect?

(10) Which day, of any ‘period,’ is the earliest on which it can be said that a ‘portion’ of the ‘period’ has elapsed?

(11) On what day would the lowest ‘Commission’ begin to be chargeable?

(12) What would be its amount?

(13) On what day would a larger ‘Commission’ begin to be chargeable?

(14) What would be its amount?

(15) On what day would a yet larger ‘Commission’ begin to be chargeable?

(16) What would be its amount?

Signature:

Date:

He followed up with this supplement later that month:

The trouble, as I read it, is that clause (c) is ambiguous. Presumably the postal authorities intended the general rule to be that a patron had three months to redeem a postal order, and beyond this would be charged a commission (here, 1 penny) for every three months that had elapsed since the deadline — including the last such period, which would not be prorated. Unfortunately, the phrase “every complete period” means exactly that — it refers to every completed period on the calendar. This sets the clock going twice as fast as intended. Our patron should owe 1d on May 1, 2d on August 1, and 3d on November 1. But with clause (c) worded as it is, she’ll owe 1d on May 1, 4d on August 1, and 6d on November 1. The final effect is that, beyond the first period, postal patrons who follow these rules will pay twice the intended commission.

I don’t know whether the post office ever learned about this. I imagine most patrons trusted them to do the math, and no one but Carroll recognized the ambiguity.

# Vacancies

Between 1937 and 1939, Nazi Germany built a colossal beach resort on the island of Rügen in the Baltic Sea. Its scale was enormous: Meant to host 20,000 holiday-makers at a time in shifts of 10 days, the six-story edifice of 10,000 double rooms stretches for 4.5 kilometers, requiring almost an hour to walk its length. At the end of the war seven of a planned eight blocks and part of a main square had been completed. Since then it’s housed small-scale projects, including a youth and a family hostel, a skating hall, a theatre, workshops, museums, art galleries, secondhand shops, and a disco. Today five of the blocks have been developed as apartments and a new hostel, while the remaining three lie in ruins.

Until four years ago, fully 1 percent of Greenland’s population was housed in a single building, Blok P. Erected in the 1960s, it was five stories high and stretched 200 meters, the largest construction project in the Kingdom of Denmark, with one end boasting the world’s largest flag of Greenland. But its poor design made the building a difficult and depressing home for its residents, and it was demolished in 2012.

Before the Turkish invasion of Cyprus in 1974, the tourist district of Varosha was the nation’s premier vacation destination, with high-rise hotels, shopping centers, restaurants, and nightclubs. With the invasion, the entire population of 39,000 fled, leaving behind an opulent ghost town. Since then it’s been fenced off, accessible only by the Turkish military and United Nations personnel. Negotiations continue, but after 40 years of mounting disrepair it’s not clear how much of it might still be salvaged.

(Thanks, Matthias and Steve.)

Tom has a crystal ball that shows him the future. One day it shows him a bomb going off in the city. He alerts the authorities, who disable the bomb, saving millions of people. Tom is glad, but he wonders: How can this outcome be logically consistent with the future that the crystal ball had shown him? In that future he saw millions of people die, but in this future they’re still alive. He realizes that when he contacted the authorities the timeline must have split in two. In the original timeline, the bomb went off just as the crystal ball had foretold, and the city’s population did die. But in this new timeline, the authorities defused the bomb and everyone lived.

This understanding seems to explain what has happened, but it leaves a worrying subjective question. If there are two timelines then there are two Toms, both sharing the same history and presumably each realizing that two instances of his identity now exist. “We are familiar with physical things splitting into two, and can accept in principle that they could even be duplicated,” writes Western University philosopher John L. Bell. “But it is extremely difficult to make sense of the idea that an individual consciousness can be so split.”

The doomed Tom might ask himself, “Why am I the doomed Tom?” Objectively the answer is that he’s the Tom who failed to alert the authorities to the coming catastrophe. “But from a subjective point of view he can ask: why was I the Tom who failed to act? Why couldn’t I have been the saved Tom? There seems to be no satisfactory answer to this question.”

In his Philosophy of Mathematics and Natural Science, mathematician Hermann Weyl notes that Leibniz thought he had resolved the tension between freedom and predestination by letting God consider the infinite number of possible universes and assign existence to one of them. “This solution may objectively be sufficient,” Weyl wrote, “but it is shattered by the desperate outcry of Judas, ‘Why did I have to be Judas?'”

# Smoothly

This is an excerpt from Kaikhosru Sorabji’s Opus Clavicembalisticum of 1930. The snaky line running through it is a slur (!) encompassing the whole complex passage.

Indiana University information scientist Donald Byrd observes, “It has a total of 10(!) inflection points; it spans three systems, repeatedly crosses three staves (this is also the most staves within a system for any slur I know of), and goes slightly backwards — i.e., from right to left — several times.”

More at Byrd’s Gallery of Interesting Music Notation.

# Inspiration

A poor artist is visited by a time traveler from the future. The traveler is an art critic who has seen the artist’s work and is convinced that he’s one of the greatest painters of his time. In looking at the artist’s current paintings, the critic realizes that the artist hasn’t yet reached the zenith of his ability. He gives him some reproductions of his later work and then returns to the future. The artist spends the rest of his life copying these reproductions onto canvas, securing his reputation.

What is the problem here? Kurt Gödel showed in 1949 that time travel might be physically possible, and there’s no contradiction involved in the critic arriving in the artist’s garret, giving him the reproductions, and later admiring the painter’s copies of them — that loop might simply exist in the fabric of time.

What’s missing is the source of the artistic creativity that produces the paintings. “No one doubts the aesthetic value of the artist’s paintings, nor the sense in which the critic’s reproductions reflect this value,” writes philospher Storrs McCall. “What is incomprehensible is: who or what creates the works that future generations value? Where is the artistic creativity to be found? Unlike the traditional ‘paradoxes of time travel’, this problem has no solution.”

(Storrs McCall, “An Insoluble Problem,” Analysis 70:4 [October 2010], 647-648.)

# Out of Bounds

If a game is anything, it’s a set of rules. And playing a game requires following these rules. If we take this definition seriously, then a cheater, one who breaks the rules, not only doesn’t deserve to win — he literally isn’t playing the game. University of Waterloo philosopher Bernard Suits writes:

The end in poker is not to gain money, nor in golf simply to get a ball into a hole, but to do these things in prescribed (or, perhaps more accurately, not to do them in proscribed) ways: that is, to do them only in accordance with rules. Rules in games thus seem to be in some sense inseparable from ends. … If the rules are broken, the original end becomes impossible of attainment, since one cannot (really) win the game unless he plays it, and one cannot (really) play the game unless he obeys the rules of the game.

So, strictly speaking, it’s impossible for a cheater to win a game — he can win only by following the rules. “In a game I cannot disjoin the end, winning, from the rules in terms of which winning possesses its meaning. I of course can decide to cheat in order to gain the pot, but then I have changed my end from winning a game to gaining money.”

(Bernard Suits, “What Is a Game?”, Philosophy of Science 34:2 [June 1967], 148-156.)