Kominers’s Conundrums: A Virtual Lecture Hall With Hidden Ties

This semester is only a week or two in, yet the anxieties of Zoom classes are well-known at this point: the awkwardness about knowing whose turn it is to talk; the risk of being Rickrolled by a llama.

But is there a better way to stay engaged, or even become engrossed? This week’s Conundrum is one way to do that. Where some see endless faces stacked atop one another, we see an opportunity: a puzzle that will teach us a bit about the math of online classes.

Suppose that 129 students have “Zoomed in” for a lecture on the mathematics of puzzles. Show that there must be at least two students in attendance who know the same number of other students in the Zoom room (although not necessarily the same other people in the room).

Now suppose that high school friends Pepper and Doug are both there, and figure out that neither of them knows anybody else in attendance. Must there still be two other students who know the same number of students as each other?

If you are able to circle these pixelated squares -- or if you even make partial progress -- please let us know at skpuzzles@bloomberg.net before midnight New York time on Wednesday, September 16.

If you get stuck, there’ll be a hint announced in Bloomberg Opinion Today on Tuesday, September 15. Sign up here.) To be counted in the solver list, please include your full name with your answer.

Programming note: Next week, Conundrums will run on Sunday, September 20. If you have opinions about the optimal release day/time for the column, please let us know at skpuzzles@bloomberg.net.

Previously in Kominers’s Conundrums …

Part II of our end-of-summer remix combined our sequence Conundrums with famous face identification. But there was a twist. As in our pyramid puzzle, each of the four sequences we presented had one red herring added in.

Kominers’s Conundrums: A Virtual Lecture Hall With Hidden Ties

After identifying as many of the images as possible, the goal was to figure out what should come next in each sequence. But they seemed pretty random at first glance. The first, for example started out with Harry Truman, Robert Oppenheimer, and Doris Day.

If you are like millions of others who involuntarily memorized two of those names in succession a few decades ago, you could quickly figure out the trick. This remix, like the prior one, was secretly musical. Each sequence was drawn from a “list song,” whose lyrics present a sequence of people or objects.

And indeed, “Harry Truman” and “Doris Day” are featured in the opening of Billy Joel’s hit song “We Didn’t Start the Fire” – and the subsequent lyrics “Red China, Johnny Ray, South Pacific…” all matched up with the images in the sequence. (“Robert Oppenheimer” was the odd one out.) The next element, then, was “Walter Winchell.”

The full set of answers were as follows, with the red herrings in brackets:

  • Harry Truman, [Robert Oppenheimer], Doris Day, Red China, Johnnie Ray, South Pacific, … Walter Winchell

“Vogue” (Madonna)

  • Greta Garbo, Marilyn Monroe, [Cher], Marlene Dietrich, Joe DiMaggio, Marlon Brando, … Jimmy Dean
  • Leonard Bernstein, [Kelli O’Hara], Leonid Brezhnev, Lenny Bruce, Lester Bangs, Birthday party, … Cheesecake

“The Elements” (Tom Lehrer)

  • Antimony, Arsenic, Aluminum, Selenium, [Nitrogen], Hydrogen, … Oxygen

The next elements in each sequence (italicized above) were all that was required as a solution. But the Conundrum didn’t quite end there. I noted at the end of the column that “once you discover the pattern, it should be pretty easy to identify the red herrings – even if it’s hard to name them initially” (emphasis added). That was secretly cluing that there was something to be done with the red herrings’ initials.

And indeed, the initials of “Robert Oppenheimer,” “Cher,” “Kelli O’Hara,” and “Nitrogen” together spelled out “ROCK ON,” a hidden end-of-summer message from Conundrums … to you.

Lazar Ilic and Ellen Kominers solved first,  followed by Filbert Cua, Suproteem Sarkar & Jennifer Walsh (who submitted images of each element in their answer), Becky Russell, and Daniel, Jeffrey, & Patricia Miron. FiveThirtyEight’s Riddler, Zach Wissner-Gross, submitted a music note emoji and a Madonna/Vogue gif as his solution.

The Bonus Round

Can you find the odd one out (hat tip: Robin Houston)? A puzzle show at Geffen Playhouse (hat tip: Eric Berlin and Ellen Kominers). Puzzlesnacks Plus; a mathematically impossible Super Mario level; a Wonka-like treasure hunt to win a candy factory. Explore the Galactic Puzzle Hunt archives (hat tip: Colin Lu) and the power of the pentatonic scale. Boba Fett dancing. And inquiring minds want to know: How does math help us play a scratched CD?

Ilic is on a three-week first-solver streak!

Thanks also to Zoe DeStories for consulting on the Conundrum draft, and Jessica Karl for making the image.

This column does not necessarily reflect the opinion of the editorial board or Bloomberg LP and its owners.

Scott Duke Kominers is the MBA Class of 1960 Associate Professor of Business Administration at Harvard Business School, and a faculty affiliate of the Harvard Department of Economics. Previously, he was a junior fellow at the Harvard Society of Fellows and the inaugural research scholar at the Becker Friedman Institute for Research in Economics at the University of Chicago.

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