# Archive | Puzzles

## Football Math Puzzle

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. [...]

## Math Puzzle

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? [...]

## Constitutional puzzle

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.

## Puzzle

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. [...]

## Math Puzzle

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. [...]

## Nine Puzzles of Space and Time

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? [...]

## “When Numbers Get Serious, You See Their Shape Everywhere”

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. [...]

## Time-Wasting Puzzle of the Day

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. [...]

## “P/n/g”

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. [...]

## The Second and Sixth Amendments

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. [...]

## Math Puzzle:

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. [...]

## Puzzle:

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? [...]

## -Tion/-Sion Verbs That Aren’t Also Nouns:

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.” [...]

## A Puzzle, from My Friend Warren Usui:

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