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Kominers’s Conundrums: A ‘2,021 Game’ to Kick Off 2021

Scott Duke Kominers
·7 min read

(Bloomberg Opinion) -- Happy new year! At Conundrums, we’re ringing in 2021 with a mathematical challenge: making the number 2,021 in as many fun ways as possible.

This is a little bit like the “24 game” — featured in our 24th conundrum and in the Bloomberg Opinion Instagram Advent calendar — where you have to construct the number 24 from a given set of integers. Except instead of 24, we’re shooting for 2,021.

To make the problem interesting, we’re restricting the set of numbers you can use. This is the 36th Conundrum, so you have to use positive factors of the number 36 (that is, 1, 2, 3, 4, 6, 9, 12, 18, and 36). You can use as many (or few) of those factors as you want, including with repetition.

We’ll be taking submissions in two categories:

First, there is a Challenge Category, in which the goal is to get to 2,021 using as few factors of 36 as possible.

So for example:

2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 – 1 – 1 – 1 – ... – 1 = 2048 – 27 = 2021,

which uses 11 “2”s and 27 “1”s (for a total of 38 factors), would beat

1 + 1 + … + 1 = 2021,

which just uses 2,021 “1”s (and is also pretty boring).

In the Challenge Category, ties will be broken in favor of constructions for which the largest factor used is the smallest, and then in order of submission time.

Second is the Creative Category, where we’re looking for the most mathematically or visually elegant construction of 2,021 from factors of 36.

The Creative Category will be judged by a panel of experts and puzzle enthusiasts including Steve Strong, mathematician Noam Elkies, and yours truly. We’ll be showcasing a number of solutions.

You can use any mathematical operations you want, but if you’re including something really obscure like the Digamma function, please give a brief explanation with your submission. And while you can use functions that give elements of sequences (such as the N-th prime, or N-th Bernoulli number), we reserve the right to disqualify submissions that just give trivial solutions like “1-st element of the sequence 2021 + (N – 1) for N = 1, 2, ….”

(We also might give extra credit for extremely interesting constructions of 2,021 that don’t use the factors of 36 — but no promises!)

Please submit your clever constructions of 2021 to skpuzzles@bloomberg.net before midnight New York time on Thursday, Jan. 7.

Programming note: The next Conundrums will run on Jan. 10.

Previously in Kominers’s Conundrums …

Our “Home Alone” Conundrum imagined a modernization of the film, with seven thieves and 2020-era traps. The first challenge was to identify the traps from the descriptive clues:

A large bucket balanced on the edge of the playroom door is filled with thousands and thousands of JIGSAW puzzle pieces, perfect for burying a thief who hasn’t already gotten the picture. A MAHJONG set has been spread out on the dining room floor, creating 144 opportunities for an unsuspecting thief to slip and skid into the redwood table. An arrow-shaped sign pointing to the computer declares “The secret plans are here!,” but anyone who tries to log in is in for a letdown: A music video starts playing — it’s just a RICKROLL.(4) It looks like a present under the tree, but when you open it, pounds and pounds of GLITTER spew forth, blinding would-be thieves and getting all over everything. That ROOMBA looks cute, but it’s programmed to get underfoot of even the most nimble crooks — pushing them over the edge of the staircase and cleaning the floor all the way. The basement refrigerator contains a milk CARTON that’s been there for decades; a motion sensor trigger flings open the door and grosses out even the most hardened criminal.(1) The den television is on, chilling crooks out by immersing them in a marathon of NETFLIX series. The garden is planted full of ZUCCHINI; they grow near the ground, but they’ve been rigged to shoot up and squash invaders, leaving them green in the face. The movie room is showing Galactic Senate negotiations on rerun, but that’s just to distract crooks long enough for the bust of Emperor PALPATINE to fall on top of them. The smell of baking bread lures hungry thieves into the kitchen, where the floor has been replaced with a giant vat of sticky SOURDOUGH discard. There is a PELOTON in the den, but it’s been re-rigged to run over scoundrels and spin instructors alike. To celebrate their political party affiliation, the house owners got a pet DONKEY; he’ll headbutt any hijackers.

Then, solvers had to fit the clues into the schematic grid we provided. As they did so, paths indicated with arrows spelled out the names of the seven would-be burglars (“tangled up in the traps”).

a casino con man with a taste for eggs and elevenses ([Danny] OCEAN); a cat burglar trying to rocket to the top of the rogues’ leaderboard (MEOWTH); a crafty woman known to detectives, Scotland Yard and at least one bohemian Duke ([Irene] ADLER); a criminal who is always a step and a third ahead of Interpol, at least according to his calling card (LUPIN [the Third]); a fugitive looking for an angel to guard the money from her last heist (TOKYO); a miserly thief-lord who despises peaches (FAGIN); and a small-hearted holiday-hater ([The] GRINCH).

We didn’t reveal which trap was where, so this step took some trial and error. Or alternatively, it was possible to guess a couple of the thieves from the clues we gave and cross-reference them against the trap names. You could figure out, for example, that the “G” in “GRINCH” had to be the start of “GLITTER,” which implied that “GLITTER” had to be the second trap on the top row.

After naming all the thieves, there was one last step: entering them into a second grid “in succession” according to the rainbow colors given, in order to spell out the identity of their ringleader:

That gave the answer “KIERAN C,” or “KIERAN CULKIN,” McAuley Culkin’s brother. Confirmatory clues abounded: Kieran Culkin plays a power-hungry younger brother in “Succession”; he was also the lead in “Go Fish,” which the column referenced obliquely. (“If you get stuck on any part of the puzzle, ‘go fish!’”) And of course, we had suggested you might “get help from Macaulay Culkin himself.”

The legendary Zoz was the first solver for the fifth week running, followed by Dan Kramarsky, Ellen Dickstein Kominers, Eli Russell, Zarin Pathan, and Tanya Otsetarova. The other 19 solvers were Cristóbal de Losada, Jay DeStories, Mark Diehl & family, Andrea Hawksley & Andy Lutomirski, Lazar Ilic, Roy Kimmey, Orlin Kuchumbov, Amine Mahmassani, Belinda Miceli, Dan Morrison, John Owens (who submitted an emoji solution), Ross Rheingans-Yoo, Maggie Schreiter, Georgia Shelton, Murray & Nancy Stern, Michael Thaler, Nathaniel Ver Steeg, Michaela Wilson, and Michael Yin.

The Bonus Round

A new year’s chess problem, courtesy of Noam Elkies; plus the mathematics of the year we left behind. Conway’s Game of Life; and online games for Zoom (both hat tip: Ellen Dickstein Kominers). RIP FarmVille; the bucatini shortage; Shrek on a floppy disk; and a Pompeiian snack bar. Science News superlatives; curling for marmots. The greatest crossword clue correction ever; an intricate paper octopus (also hat tip: Ellen Dickstein Kominers). “Ratatouille: The TikTok Musical” — catch the stream this weekend; then learn about the epic backstory and fabulous costumes. Plus inquiring minds want to know: How do you turn an actor into a reindeer? (“Frozen” content alert! — hat tip: Zoe DeStories.)

(1) If you Rickrolled yourself by clicking on that link: What, precisely were you thinking was going to happen?

(2) There's one of these at MIT as well.

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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