The Volokh Conspiracy

Consider the product 1! x 2! x 3! x ... x 99! x 100! — a very big number, but that doesn’t faze us mathematicians (since you won’t need to multiply out in any event).

The puzzle: Can you, by omitting exactly one of the factorials from the product, produce a perfect square? (For instance, omitting 3! would make the product be 1! x 2! x 4! x 5! x ... x 99! x 100!.)

The heads of executive departments have many powers and many duties. But what are the powers and duties that the Constitution expressly assigns to them?

China is a country. China is a kind of dish. According to dictionary.com, “china” is also “a playing marble of china, or sometimes of porcelain or glass.” But it turns out that, in some states, a term pronounced — but not spelled — “china” is a bit of legalese. What does it mean?

You and I have 12 pennies, arranged in a row, heads up.

We take turns. On each turn, the player can flip either (A) one coin or (B) two coins that are adjacent to each other. The player can choose on his turn which of these do, and which coin or coins to flip.

Once a coin has been flipped, it can’t be flipped again.

(Example: I start by flipping coin 5. Now it’s your turn; you can then, for instance, flip coin 12, or coins 9 and 10, or coins 3 and 4. But you can’t reflip coin 5, and you can’t flip coins 4 and 6, because they aren’t adjacent.)

The goal is to be the one to flip the last coin or coins, leaving the coins all tails.

What’s the winning strategy, and who’s the winner — the person who goes first, or the person who goes second?

Lawyers know that the Federal Reporter, which started out just as “F.,” is now on “F.3d.” The Northeastern Reporter is “N.E.3d.” Which reporters have a series number that’s higher than 3d?

This number can be seen, if you’re imaginative, as representing a particular online work (or set of works, if you prefer to view it that way). Which one? [UPDATE: Please explain your answer.]

Pretty easy, but I found it amusing. Treating “^” as meaning exponentiation, and treating the exponentiation chain as going on infinitely, solve for x:

x^(x^(x^(x^(x^(x^...))))) = 2

What vegetable’s name is etymologically connected — distantly, to be sure — to an astronomical concept (not just the name of a particular object, such as the name of a planet or a star)? There might well be many answers, but I have one in mind.

Cool math puzzles, which I remember from my childhood (though the ones I did were in Russian). The principle is that you must solve a puzzle such as OLD+OLD+OLD = GOOD or COUPLE+COUPLE = QUARTET, where each letter stands for a digit and no two letters standing for the same digit. A fun project for kids who like math, and maybe even for adults. Thanks to my colleague Doug Lichtman for the pointer.

Here’s an easy one: White Russia, Yellow China, Black Turkey, Red ___.

My friend Haym Hirsh passes along this problem: Prove that every positive integer has a multiple (other than 0) that consists only of zeros and ones. (Thus, for instance, one such multiple for 6 is 1110.)

Say that the only point-scoring events in a football game are field goals (3 points) and touchdowns with one-point conversions (7 points). Some point totals cannot be scored in such a game — for instance, 1, 2, and 4. What is the highest integer point total that cannot be scored using just 3-pointers and 7-pointers?

Now say that we exclude field goals, but allow touchdowns with missed-conversions, so the only point-scoring events are 6 points and 7 points. What is the highest point total that cannot be scored using just 6-pointers and 7-pointers?

And now let’s generalize. Say that there are two point-scoring events, one which yields a points and one which yields b points. If a and b have a common divisor, then of course there are an infinite number of positive integer point totals that can’t be scored; for instance, if all you have is 4-pointers and 6-pointers, then all the scores will be even, and any odd score will be unachievable. So let’s assume a and b are relatively prime, which is to say that they don’t have any common divisors. What is the highest point total that cannot be scored using just a-pointers and b-pointers?

UPDATE: Thanks to commenter Nick, I now know this is the Frobenius coin problem.

Here’s a puzzle I came up with this morning (though I’m sure others had thought of it before).

We know that the sum of integers from 1 to n — let’s call it S₁(n) — is n × (n+1) / 2.

We know that the sum of the squares of integers from 1 to n — let’s call it S₂(n) — is n × (n+1) × (2n+1) / 6.

We know that the sum of the cubes of integers from 1 to n — let’s call it S₃(n) — is the square of S₁(n).

Of course the sum of the cubes (S₃) is always divisible by the sum of the integers themselves (S₁).

The sum of the squares (S₂) is sometimes divisible by the sum of the integers (S₁). For instance, consider n=4: 1+4+9+16 = 30, and 1+2+3+4 = 10.

When is the sum of the cubes (S₃) ever divisible by the sum of the squares (S₂), for n > 1?

What are the earliest five constitutions of independent countries? [UPDATE: Why limit ourselves to five? Let's just go no further than 1799. Also, I forgot the most important part: they must be written. Every country has a constitution, but most old ones are unwritten!]

A bit of definitions and clarifications to start us off:

1. The countries may or may not still exist.
2. The constitutions may or may not still be in force.
3. The constitution must self-consciously be a constitution; i.e., the Magna Carta doesn’t count. [UPDATE: Perhaps this point might be said to imply the "written" point that I've also clarified above. Note that most early codes are just law codes, not "constitutions" in the modern sense. To qualify here, a constitution should, at a minimum, purport to establish the state, define its officers, etc.]
4. The country involved must consider itself independent; if there’s debate over whether the country really exists (like if many countries don’t recognize it), I resolve the doubt in favor of independence.
5. There might be some debate over the status of the earliest U.S. state constitutions, e.g. the South Carolina constitution of early 1776. Therefore, exclude the original 13 states from the answers.

I thought I’d repeat a puzzle that I first posted five years ago.

Put these in order by moving the fewest names:

Washington, Wilson, Johnson, Johnson, McKinley, Davis, Jackson, Taft, Kennedy

Moving a name to between two other names, before the first name, or after the last name all count as one move. Thus, exchanging #2 and #7 requires two moves (since you’d have to move #2 and then move #7).

Note that “order” need not necessarily mean chronological order.