Our puzzle this week was suggested by Marco Moriconi, a regular Numberplay contributor and professor of physics at the Universidade Federal Fluminense, in Niteroi, Rio de Janeiro. (He also writes a monthly math column for the Brazilian national science magazine, Ciência Hoje (Science Today), called Qual o Problema, or “What’s the Problem?”). Dr. Moriconi found this week’s puzzle in a blog by its creator, Roy Cook, a professor of philosophy at the University of Minnesota who tells how he momentarily stumped the famous paradox master Raymond Smullyan with the challenge. (Check out Dr. Cook’s recently-published Paradoxes and Dr. Smullyan’s Beginner’s Guide to Mathematical Logic, selected by Amazon as a favorite book of 2014.)
We’ll take on Dr. Moriconi’s slightly-altered form of the puzzle, which presents the challenge in three different scenarios. “Since the answers are so different,” he says, “it shows how precise statements in mathematics can make a huge difference.”
Let’s try —
Truth-Tellers and Liars
In a party there are truth-tellers and liars. At some point, they all form a huge conga line and start singing. The first one goes, “There’s at least one liar here!”, the second one goes, “There are at least two liars here!” and so on, until the end of the line. Who is lying? Who is telling the truth?
In a second version of the party, the first in line sings, “There’s exactly one liar here!”, the second goes, “There are exactly two liars here!” and so on. In this case, who is lying, and who is telling the truth?
In the third version, the first sings, “There’s at most one liar here!”, the second, “There are at most two liars here!” and so on. What then?
We also have a bonus puzzle suggested the mathematician and computer scientist, Bill Gosper, who proposes the following:
An Interesting Angle
Where would you go to find a physical object having an angle of 2π/37? (Hint: If you don’t look far enough, you might only find π/19.)
Also, in case you missed it: a beautifully-written piece (click here) by Adam Higginbotham about the skeptic James Randi (with its stunning photograph of Mr. Randi by Jeff Minton), the popular investigator of claims of paranormal abilities. If you enjoy math puzzles you may be familiar with Mr. Randi from his connection to recreational mathematician Martin Gardner, who helped create the modern skeptical movement through his landmark 1957 book, Fads and Fallacies in the Name of Science.
That concludes this week’s puzzle. As always, once you’re able to read comments for this post, use Gary Hewitt’s Enhancer to correctly view formulas and graphics. And send your favorite puzzles to gary.antonick@NYTimes.com.
Solution
Check reader comments on Friday for solutions and commentary by Marco Moriconi and Bill Gosper.