Kominers’s Conundrums: Resequencing a Puzzling Summer

(Bloomberg Opinion) -- September is on the horizon, but we’re still rocking out to summer hits. Here at the Conundrums Cabana, that means looking for ways to remix our favorite puzzles of the season.

This week, we’ll mashup our integer sequence and playlist puzzles. I’ve pulled together nine well-known sequences for you to identify. But there’s a twist: As in the playlist puzzle, I’ve left something out of each.

Your first challenge is to name all the sequences. But once you’ve done that, you'll have to keep going. Believe it or not, there’s actually a word hiding in all those digits – and that’s the answer you’re looking for.

• 1, 2, 2, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, ...
• 11, 17, 29, 41, 47, 59, 67, ...
• 2, 3, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ...
• 3, 5, 11, 23, 29, 41, 53, 83, 89, ...
• 7, 31, 127, 8191, 131071, 524287, 2147483647, ...
• 3, 17, 257, 65537, 4294967297, 18446744073709551617, ...
• 1, 3, 4, 6, 8, 11, 13, 16, 18, ...
• 1, 3, 4, 7, 11, 18, 29, 47, 76, ...
• 1, 1, 2, 42, 429, 7436, 218348, 10850216, 911835460, …

If some of the sequences look unfamiliar to you, feel free to phone your local number theorist or try to look them up on the Internet. As always, when solving Conundrums, you’re free to use any tools at your disposal.

And one note about the names: People often refer to sequences in slightly different ways. For example, 1, 1, 2, 3, 5, 8, 13, … might be called “Fibonacci numbers” or “the Fibonacci sequence.” For our purposes here, don’t worry about articles or words that just indicate that it’s a set of numbers. So 1, 1, 2, 3, 5, 8, 13, … would just be “FIBONACCI.”

If you manage to sort out the answer from among the infinite possibilities -- or if you even make partial progress -- please let me know at skpuzzles@bloomberg.net before midnight New York time on Wednesday, September 2. (Conundrums is off next week, so you get two weeks to solve! If you get stuck, there’ll be hints announced in Bloomberg Opinion Today. Sign up here.) To be counted in the solver list, please include your full name with your answer.

Last Week’s Conundrum

The stars of “Real Genius” accidentally shot a laser through their course registration notes, which were already written in some sort of code.

The solution unfolded from a trick: In each cluster of five-word bits, there was a single set of letters that made each element a word. For the clusters on the left, we needed to add on a “head” at the start; on the right, we needed to add a “tail” at the end.

Putting together those heads and tails and then reading straight down yielded the course name “HONORS BAGGED POPCORN PHYSICS,” which was the answer.

Several solvers remarked that physics made sense in the context of a movie about laser-optics geniuses. But why bagged popcorn?

That was one last tribute to the movie: The iconic final scene (spoiler alert ) involves using a 5-megawatt laser to pop a gigantic bag of popcorn, filling up the professor’s house. (Note to my students this fall: don’t get any ideas!)

Lazar Ilic solved first, followed by Jason Boomer, Anna Collins, Zoz, Noam Elkies, and Jon Delfin. Others among the 25 solvers included Michael Branicky, Philip S. Davidson, Geneal, Kent Holding, Bryan Jones, Karl Mahlburg, Francisco Poggi, Elizabeth Sibert, Eric Suess, Skylar Sukapornchai, Yi Jun Tan, Mike van Dyk, Franklyn Wang, and Elizabeth Wood. Continuing his emoji solving streak, FiveThirtyEight’s Riddler, Zach Wissner-Gross, wrote this.

The Bonus Round

Random walks, but in real life (hat tip: Mitchell Weiss); spot the popcorn kernel; the Diablo loot lottery. Mysterious Mayan temple acoustics; solving a seven-dimensional tiling problem; “The Case of the Top Secret iPod.” AI-generated, but pseudo-profound. Check your basement for auction-worthy fishing gear (hat tip: Ellen Kominers). And inquiring minds want to know: How do they fix windows over at the MIT Media Lab? 

And forget about first names – who calls them the “Leonardo Fibonacci” numbers?

Admittedly, there were a couple of degrees of freedom: “ORS” and “SICS” could have been either singular or plural. We accepted alternate answers like "HONOR BAGGED POPCORN PHYSICS,” but almost everyone figured out the intended answer from context. Also, apologies for the unusual spelling “VIZORS” instead of “VISORS,” which several solvers commented on.

But hey, the movie is literally 35 years old, so if you haven’t seen it yet….

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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