Les États Retrouvés

Here’s an outline map of one of the states of the U.S. Can you recognize it?


Here are the hidden states from the “À la Recherche des États Perdus” puzzle. (A little Proust allusion there.)

Hello! hi, Oscar! You knew me. X I couldn’t watch. You go gaga over Monty Python’s Flying Circus. I would hate to pain Diana’s fans. Never give a lab a machine. It might emit harmful radio waves.As final as kangaroo courts ever get. Fiat law and code law are at odds here. It’s all about a haven. You can win a new ham. P.S., Hire the handicapped. Don’t get exasperated. Don’t lump Malawi’s cons in with Nigeria’s. Give Lamont an apple to help him stay calm. Don’t panic Ali for niacin. There’s no stigma in eating lots of bread. That’s where you get sinewy or knobby legs. Aramis’s is sip pineapple juice. I can’t remember whether the “architect of German reunification” was Elon Musk or Egon Bahr. Tintin’s dog Milou is IANA’s new mascot! I stayed at as many ryokans as I could afford. You wouldn’t believe what fine bras Kabuki actresses wear. I would love to eat a huge orgiastic dinner, said a hobo.

A Lighter Liter

Another entry for the “everything you know is wrong” file. I was taught that one liter is exactly one thousand cubic centimeters, and a liter of water weighs exactly one kilogram. When I was working with standards, I learned that there was more to it than that. This is the full story as I looked it up about 1997.

When the metric system was devised in the late 18th century, a liter was defined as exactly one thousand cubic centimeters, in other words, the volume of a cube with an edge ten centimeters long. At that time, a gram was defined as the weight of one cubic centimeter (cc.) of water. A meter was defined as 1/10,000,000 of the length of the meridian of Paris between the equator and the North Pole. This length was not precisely known then. In the 19th century, a physical standard meter bar and kilogram weight were made and placed in a safe in Paris. Copies were made, were checked against the standards in the safe, and were distributed to other countries. The technology of that time didn’t allow extremely high accuracy. Later, it was found that 1,000 cc. of water weighed 999.973 grams when measured against the standard. This required a redefinition. The solution that was chosen was to separate the two definitions of volume. A cc. will always remain the volume of a cubical box with an edge one centimeter long. The liter was redefined as the volume of exactly one kilogram of water under standard conditions.

The density of water varies slightly with temperature. Water is densest when it is at 3.98° Centigrade (39.164° Fahrenheit). That peak density is .999973 grams/cc., to the nearest one-millionth. That means that one kilogram of water at that temperature has a volume of 1000.028 cc. In short, depending on how you define a liter, a liter of water occupies more than 1000 cc. or else it weighs less than one kilogram. (At any other temperature, the disparity would be even more extreme.)

The Système International (metric system) is administered by the Bureau International des Poids et Mesures (International Office of Weights and Measures) in Paris. In 1964, the Douzième Conférence Générale des Poids et Mesures (12th CGPM) abrogated the definition of the liter which had been adopted at the 3rd CGPM in 1901. The new definition of a liter is precisely 1,000 cc. At present, the use of liters as a unit of measurement for precise scientific work is deprecated.


Find the States

This very artificial dialogue is a hidden word puzzle. The names of twenty-three states of the U.S. are hidden, the way window is hidden in “Put that twin down.” Puzzles such as this are not uncommon, but they’re fun to construct.

Lamont and Ali are having a chat in the student union when Oscar walks in with a newspaper.

Lamont: Hello! hi, Oscar! There’s an animation festival, and they’re showing the first X-rated animated cartoon, Ralph Bakshi’s “Fritz the Cat”. Want to see it?

Oscar: You wouldn’t ask if you really knew me. X I couldn’t watch.

Lamont: But you go gaga over Monty Python’s Flying Circus. Some of their material is off-color.

Ali: I’m thinking about donating an electron microscope to the bio lab in memory of Princess Di. And yet, I would hate to pain Diana’s fans. What do you think?

Lamont and Oscar (together): Never give a lab a machine. It might emit harmful radio waves.

Ali: Is that your final judgment?

Oscar: As final as kangaroo courts ever get. It’s our personal fiat.

Ali: How about the internal revenue code? Fiat law and code law are at odds here. I need to shelter some of my income from taxes. It’s all about a haven.

Lamont: I just got an e-mail from someone in Malawi. It looks like another 419 scam. It ends up, “you can win a new ham. P.S., Hire the handicapped. It’s good business!”

Ali: Don’t get exasperated. Don’t lump Malawi’s cons in with Nigeria’s. Oscar, give Lamont an apple to help him stay calm.

Oscar: These apples are fortified with niacin. This paper says that if you get too much niacin in your daily diet, it can cause the megrims.

Ali: Oh, no! I think I get kilograms of niacin from my brand of bread.

Lamont: Don’t panic Ali for niacin. There’s no stigma in eating lots of bread. Anyway, what are megrims?

Oscar: That’s where you get sinewy or knobby legs.

Lamont: You know, after a brisk swordfight, each of the three musketeers has his own way to cool off. Aramis’s is sip pineapple juice.

Oscar: Here’s an article about persistent infrastructure problems in East Germany. Their names are so similar, I can’t remember whether the “architect of German reunification” was Elon Musk or Egon Bahr. Hey, here’s an interesting item. the Internet Assigned Numbers Authority felt it needed a spokesdog. Tintin’s dog Milou is IANA’s new mascot!

Ali: I just got back from a trip to Japan. I stayed at as many ryokans as I could afford. And I went to a Kabuki museum. You wouldn’t believe what fine bras Kabuki actresses wear.

Oscar: And here, in an interview with some street people, “I would love to eat a huge orgiastic dinner,” said a hobo. What nerve!


Nicaraguan Adventure

In the summer of 1974, my brother Steven and I took a road trip around North America.

The Pan-American Highway was generally in good condition. But soon after we entered Nicaragua, we came to a little landslide caused by the seasonal rains. There was a rock right in our lane, and I would have swerved to the other lane, except that a bus was approaching, so I couldn’t avoid the rock. Crunch! It made a hole in the transmission case, and when I stopped to inspect, transmission fluid and oil were pouring out into the dirt.

I walked about three miles back to the nearest town, Somotillo, where I hired someone to tow the car back to a taller mecanico in town. We were going to have to find replacement parts, not so easy in Nicaragua, where automatic transmissions are scarce. We took a bus to a larger town, Chinandega. We heard the driver telling his wife, laughingly, about the accident he had seen earlier: ours.

No one in Chinandega seemed to be able to help us, so we took two taxis interlocales to León and then Managua. In León we shopped around some more for help, but none was available.

It had been about a year and a half since an earthquake had devastated downtown Managua. The baseball player Roberto Clemente had been involved in bringing relief to Nicaragua when his plane crashed. Most of downtown was still off limits because of structural damage.

We found potential solutions for our car trouble in Managua, but we had to stay there a few days without a car. We had dinner at a Chinese restaurant called the Siete Mares. As we left the restaurant, we were accosted by a soldier with a rifle slung over his shoulder. He said something about a curfew, and asked for five cordovas, as a kind of fine. Our smallest bill was a 20-cordova. Steven went back to the Siete Mares for change, but they told him that the “fine” was extortion, and we should just walk away. The soldier followed us. We told him we’d go to the office of the Intercontinental Hotel for change. We asked at the hotel’s Información desk. They’d never heard of such a thing. They told us we could tell the soldier to come in and explain. We went out and hesitated. When I got back to where we left him, he was gone.

Probably he knew he could get in trouble for what he was doing. But if I had to do it over again, I would have given him the twenty (about one U.S. dollar at the time). I bet he needed it more than we did.

We finally got the car fixed and continued our trip.


A Foolish Consistency

In the 1980s, I ate lunch fairly regularly with a liberal man. One of his talking points was that the same people who are against abortion (i.e., conservatives) are in favor of the death penalty. If they’re pro-life on the former issue, they should be pro-life on the second.

My response was the obvious one. The death penalty is carried out against people who have been judged guilty of a heinous crime. Those killed in an abortion are uniformly innocent. They haven’t had the opportunity to harm anyone yet.

But his other point is very questionable. I have known many people who were opposed to both abortion and the death penalty. I myself have qualms about the death penalty. I have read about too many cases where an ambitious prosecutor railroaded a poor person to death row, who was later proved innocent by DNA evidence. I am angry about that kind of injustice. What’s more, the state, in executing a prisoner, is acting on behalf of me and my fellow citizens. I’m uncomfortable with killing even in self-defense, although I think my religion allows that. (The sixth commandment should be translated from Hebrew as “Thou shalt not murder.”) A sentence of life imprisonment is a better solution. It may cost me more as a taxpayer to feed and house a felon. You don’t count the cost in making moral choices.

I believe that abortion is almost always a sin. If it’s clearly a choice between the mother’s life and the baby’s, I would want to save the mother, because she already has a niche carved out that no one else can fill. But I honor the rare women who have chosen to decline treatment for cancer in order to save their fetuses.

They say, “You can’t legislate morality,” but laws aren’t totally arbitrary; most of them are based on someone’s concept of morality. In a democracy, they must be acceptable to the majority, or they will be repealed. I yearn to see the day when almost all Americans are as horrified at the idea of murdering a defenseless fetus as they are at cannibalism and kidnapping, which are both immoral and illegal. When our children first heard the word abortion and we had to explain what it meant, they were horrified, which I think is a natural reaction.

My main point is that it is a fallacy to generalize about clusters of beliefs. If a noted spokesperson says A and B, and if I strongly believe A, that doesn’t mean that I have to believe B as well. I aim for nuanced beliefs that account for significant differences between situations. Call me a casuist.


Lend Me Your Words

Here are some more notes for my manuscript about pronouncing foreign words.

The English vocabulary is a melting pot, always stirring in words from other languages. Recently immigrated words are called borrowings or loanwords. They go through a process of naturalization, and in the long run their foreign origins are forgotten by all but philologists, the genealogists of language.

The Corpus of Historical American English (COHA) is a searchable online collection of texts from twenty decades. That makes it a good resource for tracking the naturalization process.

In English, the ordinary word for an elevated plain is plateau. The -eau ending reveals its origin as a French word. In French it can mean an elevated plain, the pan of a balance scale, or a serving tray, among other things. COHA shows that it was used in English in its current meaning as early as 1833. An 1834 citation in COHA reads, “Where there is no plateau, a salad occupies the middle of the table.” An 1846 citation mentions “the centre of the board, where we should now place the plateau and epergne”; clearly those quotes are about serving trays. With only those two exceptions, the meaning in the COHA quotes is always tableland. The French spelling of the plural is plateaux, which appears in five texts from before 1870, but eleven times before 1870 it’s written plateaus. Over the centuries, the spelling plateaus dominates plateaux by a ratio of 10:1.

Schadenfreude is a very recent borrowing from German. COHA shows examples from 1920, 1958, 1967, 1982, 1992, and after that with a rapidly increasing frequency. The oldest four are all capitalized (despite occurring in the middle of a sentence). Since all nouns in German text are capitalized, this can be taken as an indication of foreignness. The more recent examples are rarely capitalized. The 1920 example was enclosed in quotes and immediately followed by the explanation, “pleasure over another’s troubles.” As recently as 2012, out of 24 citations using schadenfreude, four have it capitalized in the middle of a sentence, and one is accompanied with a definition.

Some of the stages of assimilation are illustrated by those examples. As with plateau, alternate meanings in the donor language drop off. If the word is italicized or contained in quotation marks, those are signs that it’s still considered a foreign word, and the capital S in Schadenfreude is another such sign. It shouldn’t be necessary to define a word with each use, once it’s been assimilated.

The morphology of derived words becomes more like English morphology as assimilation proceeds, as in plateaus. Naïve comes from French. Derived words in French are naïf and naïveté. Since diacritical marks are rare in English words, the spellings naive, naif, and naivete (without accents) would represent at least partial assimilation. The dictionary condones any of the spellings “naiveté”, “naïveté”, “naivete”, “naivety”, and “naïvety”. When I type “naive” in Word, it auto-corrects by changing the i to ï. In English, naive is always an adjective and naif is always a noun; the French words are feminine and masculine, respectively, and in French can be either adjectives or nouns. Sometimes, people who speak French might prefer the spelling that would be appropriate for French in that context. There’s no harm in that.

How are borrowings pronounced? For example, rendezvous is pronounced in English with z and s silent, as it is in French. But a French speaker would trill the r and nasalize the first e, which are hard for an English speaker, so in English conversation you don’t even try. Trilling the r and nasalizing the e would probably be regarded as showing off, and justly so. Similar comments apply to genre, but now the English speaker can freely choose between /ȝ/ (zh sound) and /dȝ/ (j sound) for the initial g. Borrowings from Italian are often food- or music-related, and are pronounced more or less Italian style. In pizza, the z’s are unvoiced, and don’t buzz the way they would in an English word. Lasagna is pronounced lah-zahn-ya, with no audible g. Scherzo is pronounced like scare-tso. Will the process of assimilation eventually harmonize the spelling with the pronunciation? Will there come a day when quesadilla is spelled kayzadeeya, or when it’s pronounced as it’s spelled, kweh-sad-illuh? Probably not, because English speakers learn not to expect phonetic spellings.

In other languages, the spelling of borrowings is often modified to make the pronunciation come out right. In German, “lustig” means “merry.” (“Till Eulenspiegels Lustige Streiche”, a German folk tale and a tone poem by Richard Strauss, is translated “Till Eulenspiegel’s Merry Pranks”.) The French adopted the word, but they respell it using French phonetics, “loustic”. Similarly, Portuguese for “fan” (in the sense of a fervent admirer) is “fã.” Swedish for “tape” is “tejp.” Norwegian for environment is “miljø”, from the French “milieu”. Japanese for “club” is “kurabu,” when Romanized.

So, how can you tell whether a borrowing has been assimilated? The surest way is to consult a dictionary. Dictionary makers apply some of the tests I’ve suggested above. The Merriam-Webster dictionary lists “esprit de corps” in the main vocabulary, and “esprit de l’escalier” under foreign words and phrases. That accords with my intuition. It’s hard to say why, except that “esprit de corps” is a phrase I’ve known longer and seen or heard more often. In COHA, out of 400 million words of text, there are 184 occurrences of “esprit de corps” (going back to 1830) and only one of “esprit de l’escalier” (1968).

In “Buckley: The Right Word,” an anthology of essays by William F. Buckley, he writes,

… the lawyers’ nolle prosequi, which has become so thoroughly transliterated as to have acquired English conjugational life: thus, “The case against Dr. Arbuthnot was nol-prossed.”

I am abashed at finding fault with the peerless Buckley vocabulary, but “transliterated” ordinarily means converted phonetically from one writing system (alphabet) to another. The original alphabet for nolle prosequi is the same as ours.


Mathematics Spoilers

These are the answers to yesterday’s math questions. If you want to look at the questions before the answers, scroll down to the previous post quickly, or click here.

To space the answers further down the page, I will insert some of my favorite spoonerisms. Reverend William Archibald Spooner (1844-1930) was a dignified clergyman who had an unfortunate mental quirk. He would unintentionally switch the initial sounds of two words in his sentences, often with ludicrous results. It is said that he once proposed a toast to “our queer old dean” (dear old queen). Many of the quotes from him are probably apocryphal.

I had an uncle who loved to spoonerize. His wife, my aunt, would sometimes react to a sentimental story with “Doesn’t that harm your wart” (warm your heart). An annual fund-raiser at our church was called “One great hour of sharing,” which I liked to speak of as “One great shower of herring.”


1. Walter must have said that the trip began on a Tuesday and ended on a Monday, and the Tuesday must have been February 21, 1888. Given that the trip took place after the start of the American Civil War (April 12, 1861) and before Lindbergh’s solo flight (May 20, 1927), if Walter had said any other combination of days, the answer would have been either ambiguous or impossible.

The reasoning used to solve this problem hinges on recognizing that Walter’s information would have allowed Everett to deduce the length of the month (28, 29, 30, or 31 days). You see, if Walter had said it took from Monday the 21st to Tuesday the 19th, from a Monday to a Tuesday is some number of weeks and one day, and from the 21st of one month to the 19th of the next month is two days less than the length of the first month. For example, August has 31 days, so from Aug. 21 to Sept. 19 is 29 days. That would work, because 29 days is four weeks and one day. But it could be August of any year from 1861 to 1926, as long as Aug. 21 was a Monday. There were 66 Augusts in that period, and the 21st fell on a Monday in 10 of them. Everett would not have been able to get the year right with certainty.

In the period 1861-1927, there would be too many 28-day, 30-day, or 31-day months to determine which of them was the right one, but there were only 15 29-day months (the leap-year Februarys). Looking up the day on which the 21st of each of these Februarys fell, we find that only one of the days (Tuesday) was unique, permitting Everett to solve the problem.

Note that if you knew that the Civil War started between 2/22/1860 and 2/18/1864, and that Lindbergh flew between 3/20/1924 and 3/18/1928, you could have answered the question without an almanac. You didn’t need to deduce that the days of the week were Tuesday and Monday. Just the facts that a normal year has one day more than 52 weeks, a leap year has two days more than 52 weeks, and 1900 was not a leap year, would have enabled you to construct a table showing which leap February started on a unique day of the week for that period.

You might have hit on this answer faster if you regularly used my trick for calculating day of the week in the near future. Today is Saturday, January 23. Someone in my family has a birthday coming up on March 6. What day of the week will that be? Well, March 6 is the same as February 29+6=35, because this is a leap year and February has 29 days. And that’s the same as January 31+35=66. January 66 is 66-23=43 days from now. Forty-three days is six weeks and one day, so March 6 will be one day later in the week than Saturday, i.e. Sunday.

2. The first answer is yes. The second is no. Basically, it’s because kilometers are smaller than miles, so a distance given in kilometers conveys more information. If you convert kilometers to miles by dividing by 1.609344 and rounding to the nearest mile, either 6 km. or 7 km. will give 4 miles, and there’s no single conversion method that can transform 4 miles into both 6 km. and 7 km., as needed. Multiplying 4 miles by 1.609344 and rounding gives 6 km.

To prove that converting miles to kilometers and back will always produce the number you started with, let the conversion factor be called f (in this case, f = 1.609344, but all we need to know is that f > 1). If m is the number of miles, multiplying by f and rounding to the nearest whole number produces the value k = [ fm + .5 ], where brackets represent the greatest integer function. This means that k is an integer such that fm – .5 < k <= fm + .5. Doing the reverse conversion, we calculate M = [ k/f + .5 ]. That means that k/f – .5 < M <= k/f + .5.

Is it possible that m < M? If so, m < M <= k/f + .5 <= (fm + .5)/f + .5 = m + .5(1/f + 1) < m + 1, since f > 1. Ignore the intermediate inequalities and look at m < M < m+1. Since m and m+1 are consecutive integers, it’s impossible for M to be an integer that falls in between them. This disproves the original assumption that m < M. In much the same way, we can disprove that M < m, and it follows that M = m, as desired.

3. Just three farthings. A farthing was a quarter of a penny. In those days, they wrote a money amount with a pound sign (£) first, then the number of pounds, then a space, then the number of shillings, then a slash (called a solidus, Latin for shilling), then the number of pence, then d (for denarii). If there were no pence, there would be a hyphen after the slash. The cashier would probably have figured it out about like this: 39 * 2/6¾d = 78/- + 234d + 117*¼d = 78/263¼d. There were 12 pence in a shilling, so 263¼d is the same as 21/11¼d. Adding the 78/- makes the total price 99/11¼d. There were 20 shillings in a pound, so £5 = 100/-. The change would be 100/- minus 99/11¼d. Cancelling out the 99 shillings, the change is 1/- minus 11¼d. Remembering that 1/ is 12d, that’s 12d – 11¼d, which leaves just ¾d, which is three farthings.

British cashiers these days just don’t know how good they have it.

The United Kingdom converted to decimal currency in mid-February, 1971. I was touring in Great Britain for four weeks in January, 1971. My travel journal reports, “About 5 times I’ve been given incorrect change, always in my favor.” I suppose I took some glee in correcting the pros in their work.

4. No. If you learned in school about casting out the nines, you can see why not. Take a number like 2076. If you add its digits (2+0+7+6 = 15), the result has the same remainder on division by 9 as the number you started with (2076/9 = 230 r. 6; 15/9 = 1 r. 6). This fact is used, among other things, to test whether a number is divisible by 9. If a ten-digit number contains every digit from 0 to 9 once, then the sum of its digits is 0+1+2+…+9 = 45. Since 45 is divisible by 9 with no remainder, the ten-digit number must also be divisible by 9, so it can’t be a prime.

For the general case, let b be the base of notation (b is an integer ≥ 2). A number in base b has the same remainder as the sum of its digits when they are both divided by b-1; when b=10 and b-1=9, this is the same thing as casting out the nines. Given a b-digit number in base b containing every digit from 0 to b-1, the sum of its digits is (b-1)b/2. For example, I asked about base 5. The sum of the digits in that system is 0+1+2+3+4 = 10, so any number formed with those five digits has the same remainder when divided by 4 as 10 does. That remainder is 2. So any such number is even. In the general case, if b is even, then (b-1)b/2 is b-1 times b/2, and b/2 is an integer; therefore the sum of the digits is divisible by b-1; therefore the number itself is divisible by b-1. If b = 2, that doesn’t help, because a prime can be divisible by 1; but if b > 2 (and b is still even), then b must be at least 4, b-1 ≥ 3, and if the number is divisible by b-1, it can’t be a prime. If b is odd, on the other hand, then (b-1)/2 is an integer. Say d = (b-1)/2. The sum of the digits is divisible by d. But b-1 is also divisible by d (two times). It follows that the original number is divisible by d. If b = 3, then d = 1, and the number could still be prime, but if b > 3, then d ≥ 2, and the number can’t be a prime.

So far, we’ve seen that if b is an even number > 2 or an odd number > 3, the number in base b formed with every digit from 0 to b-1 can’t be a prime. Since 1 is not a possible base, that leaves only 2 and 3 as possibilities. It turns out that they actually work. In base 2, the number 10 uses all the digits and is prime. In base 3, the numbers 012, 021, 102, and 201 use all the digits and are prime (in ordinary base-10 arithmetic, they are equal to 5, 7, 11, and 19, respectively).

5. Surprisingly, no. The two Archimedean solids shown below are not mirror-symmetric. Their shaded faces are twisted to the right; if they had been twisted to the left instead, that would produce their mirror images. (L. to r.: snub cube, snub dodecahedron)


6. Eight. There are deltahedra with 4, 6, 8, 10, 12, 14, 16, and 20 sides. The ones with 4, 8, and 20 sides are the Platonic solids, tetrahedron, octahedron, and icosahedron. The one with 6 sides looks like two tetrahedra glued together; the one with 10 sides looks like two pentagonal pyramids with their pentagonal sides glued together; and the octahedron is equivalent to two square pyramids with their square sides glued together. The one with 16 sides can be described as a square anti-prism with a square pyramid glued to each of its square sides. The one with 14 sides is a triangular prism with square pyramids glued to each of its square sides. The one with 12 sides isn’t so easy to describe.

You might ask, how about two hexagonal pyramids glued together? Well, a hexagonal pyramid is flat. If you erect six equilateral triangles on a hexagonal base, they will only meet if they lie in the same plane as the hexagon. How about three tetrahedra glued together? No, the resulting figure is not convex. It will have one edge common to all three tetrahedra, and at that edge the two triangles will form an angle of more than 180°. How about an octahedron with a tetrahedron glued to one side? If you glue a tetrahedron to an octahedron, their two adjacent faces will lie in the same plane, forming a single rhombus-shaped face.

7. You may have thought it was a cryptarithm, where you have to replace each letter by a digit (the same digit for the same letter throughout). If so, surprise! It’s just a straight long division problem in hexadecimal notation (base-16 arithmetic, where the digits are 0 1 2 3 4 5 6 7 8 9 A B C D E F). I replaced the digit 0 with the letter O, because it was a giveaway that the digit 0 is slashed in this font.

8. Real! Euler’s well-known formula says e = -1, and is often used to define powers of numbers when the exponent is not an integer. Take the square root of both sides: eiπ/2 = √-1 = i; raise both sides to the ith power: ii = (eiπ/2)i = eiiπ/2 = e-π/2. Since e > 1 and e and π are both real, the answer is real. Its value is about 0.20788.