“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?



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.


Image: Wikimedia Commons

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.

Image: Wikimedia Commons

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.

Image: Wikimedia Commons

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.)

A Splitting Headache

Image: Wikimedia Commons

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?'”

(From Oppositions and Paradoxes, 2016.)



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.



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.)

Foreign Relations


“Such a question as, Why does everything in Spain look Spanish? is one of those apparently idiotic questions that opens up quite unexpected areas of vision.” — Northrop Frye, “Spirit and Symbol,” Words With Power, 2008

Podcast Episode 127: Rowing Across the Atlantic


In 1896 two New Jersey clam diggers made a bold bid for fame: They set out to cross the North Atlantic in a rowboat, a feat that had never been accomplished before. In this week’s episode of the Futility Closet podcast we’ll follow the adventure of George Harbo and Frank Samuelsen, which one newspaper called “the most remarkable event in the way of ocean navigation that ever transpired.”

We’ll also meet some military mammals and puzzle over a thwarted burglar.

See full show notes …

A Premonition


First-Sergeant Thomas Innes Woods, of Company B, was killed on May 8th [1864]. The first time that Sergeant Woods was ever known to ask permission to leave his post on march or in battle occurred this day, after the Regiment’s all-night march to reach Spottsylvania ahead of Lee. When it became evident that a battle was imminent, Sergeant Woods asked Captain H.W. Grubbs for a pass to go to the rear. On his declaring that he was not sick, he was advised by the Captain that under the circumstances he could not be excused, and Sergeant Woods resumed his post at the head of the Company. Shortly after, during a halt by the roadside, Sergeant Woods wrote in his diary the following, addressed to his friend, Sergeant James A. McMillen: ‘I am going to fall to-day. If you find my body, I desire you to bury it and mark my grave so that if my friends desire to take it home they can find it. Please read the Ninetieth Psalm at my burial.’ He was killed early in the battle. His body was found by Sergeant McMillen and others of Company B, the diary being found in his pocket. His request for the Ninetieth Psalm to be read at the grave was complied with.

— Charles F. McKenna, ed., Under the Maltese Cross, Antietam to Appomattox: The Loyal Uprising in Western Pennsylvania, 1861-1865: Campaigns 155th Pennsylvania Regiment, 1910

Hope and Change


Canada’s “victory nickel,” struck from 1943 to 1945, included a special message to stimulate the war effort: Engraved around the rim were the words WE WIN WHEN WE WORK WILLINGLY in Morse Code.

The coin was reissued in 2005 to commemorate the 60th anniversary of V-E Day.