The Volokh Conspiracy

Find a ten-digit number with the following two properties (in base 10, of course): A. The number contains each digit (from 0 to 9) exactly once. B. For every N from 1 to 10, the first N digits of the number are divisible by N.

Thus, for instance, 1234567890 doesn’t work; while 1 is divisible by 1, 12 is divisible by 2, and 123 is divisible by 3, 1234 isn’t divisible by 4.

No fair Googling, or writing a program that just checks millions of possible numbers. The best solution I could find narrowed down the field to 10 numbers; I then had to check something (I won’t say what) for each one, so I suppose that’s a bit brute force, but not much. If you have a more elegant solution, let me know.

Thanks to Cordell Haynes for passing this along.

My friend Haym Hirsh developed these puzzles for G4G9, the ninth Gathering for [Martin] Gardner, and kindly agreed to let me blog them:

Martin Gardner’s relative Dr. Art Renaming traveled widely across the U.S. and the world. Art maintained a diary in which he described some of the puzzling circumstances he experienced over the course of his travels. What follows are excerpts from his diary. Answer the question following each diary excerpt.

1. “I am located in one of the 48 states in the Continental United States. If I go nine miles in a straight line, regardless of direction, I will leave the state I am in.” In what state was Art?

2. “I am located in one of the 48 states in the Continental U.S. If I go 90 miles in a straight line, regardless of direction, I will have needed to move my watch one hour ahead to keep it set correctly.” In what state was Art?

Continue reading ‘Nine Puzzles of Space and Time’ »

The Green Bag 2d has just published my The Numbers of the Constitution, a puzzle that begins:

To what do the following numbers refer in the United States Constitution? ...

1/5:
1/3:
3/5:
...
5:
...
12:
...
Three numbers greater than 1000:
...

Check out the whole item. Thanks to Larry Arnold, Linus Richard Banghart-Linn, Andrew Braniff, Abe Delnore, Peter Durant, Zachary Elwyn, Bryan Gividen, Matt Glassman, Dan Harper, Shaun Hickson, Bart Jacka, Chris Kaiser, Brian Kalt, Don Kilmer, Arne Langsetmo, Ira B. Matetsky, Derek Muller, Allen Pulsifer, Steve Rappoport, Daniel Tilley, Seth Barrett Tillman, Hanah Metchis Volokh, Sasha Volokh, and Jeff Walden for their advice and beta testing.

What is the smallest positive integer that, when written out in standard English, does not yield any Google searches? Do not use “and”s, and do not use the format in which hundreds are counted using numbers greater than ten (e.g., “forty two hundred eighty two” — that’s acceptable standard English, but I just want to set it aside for purposes of the puzzle, because it’s a somewhat less common variant).

I got zero hits with “fourteen thousand nine hundred eighty two” (just a bit of trial and error there), though obviously that number will have hits very soon, as a result of this post. Can you beat that? Please check your spelling before reporting victory.

Legalese puzzle: What is the meaning of “p/n/g,” which is sometimes used in case captions in Pennsylvania and occasionally New York?

I’m not asking about “PNG,” which sometimes stands for “p/n/g,” but more often for Papua New Guinea, the Professional Numismatists Guild, or other things.

UPDATE: Bleh and D.J., in the comments, were the first to get it right; see this Third Circuit opinion for a bit of context. Naturally, it’s not “persona non grata,” which is listed in the dictionary as “p.n.g.” Compare also ppa, which is mostly used in Connecticut but with a smattering of references in neighboring states.

A puzzle for constitutional law buffs: I just read a recent case which explored the interaction between the Second Amendment and part of the Sixth Amendment. What is that interaction?

I realize, of course, that one can dream up all sorts of theories for how any two constitutional provisions might interact — but I’m looking for the one that the court actually discussed, and it also seems to me that this is indeed the most plausible such interaction, given the current interpretation of the two amendments.

UPDATE: Commenter tomhynes wins.

Leonhard correctly finishes a Sudoku puzzle, and tells you: “That’s cool! When I look only at the upper left-hand 3 x 3 square of the puzzle, view each of the three-digit rows as a three-digit number, and add them together, I get 1000.” Is he telling the truth?

An alternative version, if you prefer: You correctly tell Blaise, “I have three three-digit numbers that add up to 1001, and all the digits of all the numbers are different from each other.” Right away, Blaise says, “You didn’t use a 7 in any of them, right?” How did he know?

Tell me, please, which version you like better. Thanks to my father Vladimir for the Sudoku frame for the first problem. I thought up both problems yesterday, but I’m sure someone else has beaten me to them, maybe by centuries.

People who follow federal criminal law know that it occupies Title 18 of the U.S. Code. Copyright law buffs know that the Copyright Act occupies Title 17. Why are they next to each other?

A puzzle from Michael Lorton:

Most words that end in -tion or -sion are nouns, but some are also verbs: consider mention, proposition, question, section, and station. What words that end in -tion or -sion are verbs but not nouns?

UPDATE: “Stunning, overwhelming thread victory.”

(but not of most other things) — a fun puzzle from Glen Whitman (Agoraphilia).

Fill in the blank in the following list:

1. Vermont, Connecticut, New Hampshire

2. Pennsylvania, Delaware, New Jersey

3. Indiana, Ohio, Kentucky

4. Arkansas, Mississippi, Tennessee

5. Iowa, Missouri, Nebraska

6. California, Colorado, Arizona

7. Washington, ______________, Oregon

A puzzle from Yefim Somin (yes relation to the coconspirator) — name three countries whose names are based on the points of the compass, but whose English names do not have the point of the compass as a word in the country name. (East Timor and North Korea, for instance, wouldn’t work.)

Here’s a new Oklahoma statute, 21 Okla. Stats. sec. 839.1A:

Any person, firm, or corporation that uses for the purpose of advertising for the sale of any goods, wares, or merchandise, or for the solicitation of patronage by any business enterprise, the name, portrait, or picture of any service member of the United States Armed Forces, without having obtained, prior or subsequent to such use, the consent of the person, or, if the person is deceased, without the consent of the surviving spouse, personal representatives, or that of a majority of the adult heirs of the deceased, is guilty of a misdemeanor. This section applies to the name, portrait, or picture of both active duty members as well as former members of the Armed Forces of the United States. Every person convicted of a violation of this section shall be punished by a fine of not to exceed One Thousand Dollars ($1,000.00), or by imprisonment in the county jail for not to exceed one (1) year, or by both said fine and imprisonment.

Consider three possible applications of the statute: (1) Advertising of nonspeech products that isn’t misleading — i.e., doesn’t suggest an endorsement that isn’t there — for instance if someone sells “Jarhead Beer” with a picture of some generally unknown marine on the label.

(2) Advertising of books, movies, or newspapers, e.g., an unauthorized biography of Colin Powell that has his name and likeness on the cover.

(3) T-shirts, bumper stickers, pins, prints, and the like that contain a servicemember’s name or likeness (either an anonymous servicemember’s or a more famous one’s, such as Powell’s or McCain’s), and that are used to advertise themselves (for instance, when the T-shirt is hanging in a store window or sitting on the shelf).

And in considering them, ask two questions:

(A) Could a general right of publicity law, which purports to impose civil liability on the use of people’s names and likenesses for commercial purposes, be constitutionally applied in these cases? Many states do indeed have right of publicity laws that differ from the Oklahama statute chiefly in that (i) they impose civil liability, not criminal, and (ii) they don’t limit themselves to soldiers.

(B) Even if such a general law would be constitutional, would this narrower law still be impermissible, either because its narrowness makes it impermissibly underinclusive under the relevant standard of scrutiny (Central Hudson scrutiny for commercial advertising or strict scrutiny for otherwise fully protected speech), or because of R.A.V. v. City of St. Paul?

“Add” is an example of a word that’s 2/3 composed of one letter. “Lull” is 3/4 composed of one letter. But this is easy enough with short words (“I” is 100% one letter!). What words can you think of that are 6 letters long but are more than 50% one letter? 7 letters? 8 letters? Words with internal special characters (hyphens or apostrophes) are fine, and the special characters don’t count towards the word length.

I include some of my own answers below; post yours in the comments, if they yield results that are better than mine, and than the others you see in the comments. If you think someone’s answers (including mine) are incorrect, please check the usual sources before so asserting.

In April I posted this puzzle about rearranging letters in the name of someone in the news.

Today’s challenge is very similar. A person who was in the news recently has first name that is 6 letters and last name that is 4 letters. Take the letters in the last name, add two letters and rearrange them all, and you get that person’s first name. Who is it?