Kominers’s Conundrums: A Game That You Can’t Stop
(Bloomberg Opinion) -- This week, “stonks” broke the market (and the internet), as retail traders egged on by Reddit memes drove GameStop, AMC Theaters and other public companies to stratospheric levels. And Conundrums has received an anonymous tip about the newest force in the revolution: A secretive forum where everyone’s so excited about trading that they only talk in ticker symbols.
A senior Conundrums analyst and I have been able to unearth some of these cryptic communications. Can you decode them and figure out what they’re suggesting as the next big investment opportunities?
Once you do, you should be able to put those together to figure out what someone who’s thinking about starting up a stonk revolution might want to do with their portfolio instead — and that is this week's answer.
First, we’ve been seeing this message repeatedly:
- HOG EA DAL GME ETSY HBIO EXPE DKS GPRO EB H ENR DBX GS EQBK.
This one, meanwhile, seems especially hard to translate even once received — it appears somewhat foreign:
- ORA NTIOF RY INVN STMEF GMVHF.
We’ve also had trouble making sense of this readout:
- ALLT BOOT DENN GGG RRGB UUUU INN DSSI CTTH GOOG CCNE KKR SSET.
This message seems to have been started in mixed company:
- TACO PODD SPCE XOM O LUV NVTA RACE STAR KODK PLAY.
And some of these symbols have cropped up in numerous places:
- ONEM INTC FOUR AMZN QTWO ADI MMM UBER F NFLX IIIV ANF FFIN UA DDD TWNK ONEM DELL TWOH TSLA.
New posters seem to issue this statement repeatedly for about a year before maturing and losing interest:
- TT TBI TIPT TLYS TLRY TSN.
Also, here is a disclaimer I never thought would be necessary in a puzzle column: Neither the final answer, nor anything else herein, is intended as investment advice!
If you manage to figure out what makes these symbols tick — or if you even make partial progress — please let us know at firstname.lastname@example.org before midnight New York time on Thursday, February 4.
Programming note: The next Conundrums will run on February 7.
Previously in Kominers’s Conundrums …
Conundrums fashion week had us looking to identify fourteen stylish but unusual articles de mode:
From upper left to bottom right, solvers recognized
After that, solvers had to figure out how to fit these words into the grid provided. CUFFLINKS and BOUTONNIERE helped when getting started because they each had a unique numbers of letters. A little trial and error from there quickly led to the name of the “hot accessory” that “everyone’s been talking about:” BERNIE’S MITTENS, which was the answer.
Zoz solved first, followed by Zarin Pathan, Lazar Ilic, Dan Rubin & Jennifer Walsh and Ellen & Bill Kominers. The other solvers were Darren Fink, Dina Teodoro & Amanda Abado; Alexander Haberman; Paul Kominers & Amelia Aboff; John Owens; Fernando Raffan-Montoya; Murray & Nancy Stern; Skylar Sukapornchai; Sanandan Swaminathan; Michael Thaler; Barbara & Nathaniel Ver Steeg; Kevin Waterman; and Alan Zhu. Owens submitted an all-emoji solution. Thaler submitted an answer written out in copies of the Bernie’s Mittens meme.
The 2,021 Creative Construction Competition
Our New Year’s Conundrum asked for mathematically clever constructions of 2,021 from factors of 36 (1, 2, 3, 4, 6, 9, 12, 18, and 36 – possibly using some factors multiple times). We announced the results of the Challenge Category a couple weeks ago; here are some standouts from the Creative Category, as judged by Noam D. Elkies, Steve Strong, and yours truly.
Some solvers figured out ways of encoding 2,021 using number bases: Zoz, for example, observed that 2,021 is the base-3 representation of the 18-th prime.# (Here and hereafter, a “#” sign denotes solutions that were also recognized in the Challenge Category.) We also appreciated Skona Brittain’s expression of 2,021 as “(1 + 9 + 3/4) in base 3/2,” although one panelist joked that might be “too clever by half” for its use of a half-integer number base.
Others constructed 2,021 from famous integer sequences. We really liked Tanya Otsetarova’s constructions:
2,021 = 43*47 = LuckyPrime(6)*LucasPrime(6),# and 2,021 = 43*47 = (Catalan(6) + 1)*(Catalan(4) + Catalan(6)),
which used the Lucky and Lucas prime sequences and the Catalan numbers, respectively. Bob Day found 2,021 = 2,584 – 563 = Fibonacci(18) – NthPrime(NthPrime(3^3)),# with an “unexpectedly economical route to the large difference Fibonacci(18) – 2021.” Meanwhile, Ross Rheingans-Yoo wrote 2,021 = (Triangular(9))^2 – 4, where Triangular(9)) is the ninth triangular number,# and Skylar Sukapornchai wrote 2,021 = 6^4 + 3^6 – 4, making use of the Pythagorean triple (27,36,45).
We also enjoyed Zarin Pathan's visually elegant rendition:
2,021 = (3*3*2*2+2+2+3)*(3*3*2*2+3+3+3+2).
That happened to be the submission we received with the largest number of factors — even more, as panelist Elkies notes, than 2*2*2*2*2*2*2*2*2*2*2 – 3*3*3.
Michael Thaler won the “chutzpah and humor prize” with a solution involving concatenating numbers of shaded pixels (that is, putting them side-by-side):
Michaela Wilson gave a more kosher solution using integer concatenation (denoted by ||): (18+2) || 2 || 1 = 20 || 2 || 1 = 2021.#
Lazar Ilic suggested it might be possible to construct 2021 using repeated iteration of the floor (i.e., the “greatest integer less than or equal to”), factorial, and square root functions in some order. He wasn’t able to give a construction, but panelist Elkies found a way to make this strategy work:
2021 = 6!!v!v!v!v!v!v!v!v!v!v!v!v,
Here, v[N] is the floor of the N-th iterate of the square root function. This requires 13 factorials in all: 6!, 720!, 2576!, 74!, 2290!, 1871!, 153826!, 615!, 460806!, 39451!, 102478!, 3784!, and 644859!.
But we were especially impressed by the submissions that managed to use each of the nine factors of 36 exactly once. John Mullahy found such a construction using the binomial coefficients, which indicate the number of ways of choosing M items from a set of N items (here denoted by parentheses, with N on the top and M on the bottom):
Thaler, meanwhile, found a construction using all nine factors in order:
2,021 = 1*2 + (3^4 – 6) – (9 – 12)*18*36.
And sometimes with math, the most elegant solution is the one that’s simplest. Our pick for overall Creative Category winner was Scott Wu, who found a construction using all nine factors with only the basic arithmetic operations of addition, subtraction, multiplication, and division — and on top of that, he used each operation precisely twice:
2,021 = 6*(12 + 2/(1 – 9))*(36 – (18+4)/3).
Particularly miraculous in this construction is that the two fractions, once expanded and simplified, have different denominators — but the multiplication cancels that out.
The Bonus Round
A documentary about KenKen puzzles, streaming February 1 via MoMath; a mesmerizing tiling clock. Real-life transcription puzzles; Bernie the Wellerman; a piano drop; and “the scientific case for two spaces after the period.” All of Zelda in seven minutes; “Kinetic polarized light art”; and supermassive black holes, getting more massive? (hat tip: Ellen Dickstein Kominers) A “Hidden Cows” 1,000-piece. And inquiring minds want to know: How many holes does a straw have?
This last bit is not part of the main Conundrum; just a little "Easter Egg" courtesy of our sponsors.
The items in the scarf family (PASHMINA and SNOOD) seemed to give people the most trouble. One solver pointed out that LEGGINGS worked as well as JEGGINGS. That said, we're not sure whether anyone would have a problem with jeggings.
He used the glove emoji since there isn't one for mittens, but we decided to let that slide.
Carney Hawks and Paul Chamberlain came close to this, with 2,021 = 18*6*3 + 36*12*4 – 9*2 – 1 – 12 and 2,021 = 36*18*3 + 6*12 + 4 + 1.
Other solvers included Don Brown, Mario Galindez, Orlin Kuchumbov, Gerardo Cesar Medina, Holden Mui, Alex Ognev, Mary J. Riegel, and Suproteem Sarkar.
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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