Kominers’s Conundrums: Procuring Perfect Pies for Pi Day
(Bloomberg Opinion) -- Happy Pi Day! Math enthusiasts like me commemorate today’s date, 3/14, because it lines up with the mathematical constant Pi, which represents the ratio between a circle’s circumference and its diameter. And how do we celebrate? By eating pies, of course — both because they sound like “Pi” and because they are typically round.
But we have a problem at Conundrums Kitchen. Donald Duck was once again put in charge of pie-making — even after his lackluster performance during our Thanksgiving feast, when he mixed up the pie ingredients to the point of culinary disaster.
A top-flight consulting firm is currently reviewing our management practices, but they can’t solve our most pressing problem: procuring pies on Pi Day. Donald got the flavors right this time, but his friends are very particular about their pie shapes. Unfortunately, he messed up some of the dimensions.
Here’s what Donald was asked to bake, along with his prep notes. Can you figure out what he got wrong, and correct the mistakes? After you do that, you should somehow be able to put wrong and right together to spell out an even more fundamental problem with the pies he produced. That is this week’s answer.
One note as you’re solving: While Donald's geometry is a bit rusty, he still knows an integer when he sees one. Even once corrected, all of the numbers in Donald’s notes should be integers, rather than fractions, irrationals — or even worse, imaginary numbers.
- Daisy asked for a circular walnut pie with area 4 Pi, sliced straight down the middle.
Donald’s notes: Got it — make a round pie with radius 2; then make a radial cut anywhere; count out 20 degrees ten times from there; and slice. - Gaston wanted a hearty pie in the shape of an isosceles triangle with perimeter 16 and total area 12.
Donald’s notes: Easy! Side lengths 6 and 8 should do the trick. - Doc requested a thin, right triangle-shaped treacle pie with area 9.5.
Donald’s notes: This one is a bit harder, but I think I can make it work if I use lengths 1 and 5 for the sides that intersect at the right angle. - Pumbaa was hoping to chow down on a rectangular saskatoonberry pie with perimeter 44 and area 85.
Donald’s notes: I should be able to do this with side lengths 5 and 16. - Colette wanted a trapezoidal prune pie with area 39 and one base of length 5.
Donald’s notes: That’s a bit tricky, but how about 3 for the height and 9 for the longer base? - Esmeralda asked for a parallelogram-shaped cherry pie with total area equal to the square root of 3, and at least one 60 degree angle.
Donald’s notes: Side lengths 2 and 5? - Baloo wanted a buko pie in the shape of a pentagon with three right angles and reflective symmetry, for total area of 153 and perimeter 26 plus 18 times the square root of 2.
Donald’s notes: Honestly I’ve never heard of that type of pie before, and I have no idea how to hit such a bizarre-looking perimeter. But maybe I can just start with a rectangle of side lengths 4 and 19, and tack a triangle onto the long side? - WALL-E of course asked for a trash pie — in the shape of a regular hexagon with area equal to 1.5 times the square root of 3.
Donald’s notes: I’ve never made a hexagonal pie before, but I think I can do this if I set it up so that the length of one side is 1 and the sum of the lengths of the other sides is also 1.
Got all that? Then dust off your high school geometry notes and get cooking!
And this is another week with, shall we say, double trouble: Once you’ve solved the main Conundrum, there’s a second problem with Donald’s pies that you can identify. But to determine what that is, you’ll need to think about the ways in which Donald was right all along — and also figure out a bit about why these characters have such specific taste.
If you manage to get these pies into shape — or if you even make partial progress — please let us know at skpuzzles@bloomberg.net before midnight New York time on Thursday, March 18.
If you get stuck, there’ll be hints announced on Twitter and in Bloomberg Opinion Today. To be counted in the solver list, please include your name with your answer.
Programming note: The next Conundrums will run on March 21.
Previously in Kominers’s Conundrums…
Our cryptic art Conundrum riffed off the NFT craze by presenting the encrypted addresses of seven works by a mysterious “shining star” of the art world. Solvers had to figure out where the pieces were, and how to combine them into a greater whole.
The addresses looked completely abstruse at first glance:
l 05 o 55 c 10.29 A, +07 t 24 25.3 e;
05 T 14 32.27 h, E −08 a 12 R 05.9 t;
05 25 i 07.87, +06 20 N 59.0 s;
T 05 a 32 R 00.40 r, −00 y 17 N 56.7 i;
G 05 h 36 T 12.81 U, s −01 I 12 N 06.9 g;
05 R 40 45.52, A −01 56 D 33.3 E;
C 05 C 47 o 45.39, o −09 r 40 d 10.6 s.
But if you squinted at them enough, you could notice that the letters seemed to spell something:
l o c A, t e;
T h, E a R t;
i , N s;
T a R r, y N i;
G h T U, s I N g;
R , A D E;
C C o , o r d s.
Or rather, adjusting the spacing:
locAte ThE aRt iN sTaRry NiGhT UsINg RA DEC Coords.
The reference to “Starry Night” seemed promising given the art theme, but the next step wasn’t to go look at the painting. The instruction to use “RA DEC Coords” pointed solvers to the equatorial coordinate system for identifying stars based on right ascension and declination. They happen to be expressed using pairs of three-number sequences that look just like the numbers in the addresses.
For example, the numbers in the first address — 05 55 10.29, +07 24 25.3 — indicated the star with right ascension 05h 55m 10.29s and declination +07° 24′ 25.3″, which is Alpha Orionis, more commonly known as Betelgeuse.
Looking up the stars indicated in this way revealed them to be the seven principal stars in the constellation Orion — and that “combined work” was the answer.
Then there was an extra puzzle: Trying to figure out our cryptographic artist’s name. We hinted that this would have something to do with “Norse roads,” “horses and toads” or something else that sounded similar. But that still left the task of figuring out where the artist’s signature might be.
Astute solvers realized that there was information in the addresses that hadn’t been used in locating the art: The bizarre-looking capitalization of certain letters in the instructions, as well as the way those letters were spread out across the different parts of the coordinates.
Looking at the original message again, but with the letters broken up according to the parts of the coordinates they came from gave us:
locA te Th EaRt i Ns TaRr yNi GhTU sINg R ADE CCo ords.
Reading that in Morse code with lower case interpreted as dots and uppercase interpreted as dashes (i.e., as ...– .. –. –.–. . –. –.–. .–. –.–– .––. – ––– ––. ....) spelled out the name “VINCENCRYPTOGH” — a pun with a nod back to “Starry Night.” (And yes, “Morse code” more or less rhymes with “Norse roads.” Don’t @ me.)
Lazar Ilic* solved first, followed by Zoz*, John Owens*, Nathaniel Ver Steeg, Zarin Pathan* and Michaela Wilson. The other 18 solvers were Tommy Bell, Jeff Fossett & Ari Rodriguez*; Scott Beveridge*; Ayush Bhargava, Fraser Simpson, & Mimi Sheng*; Bryce DeFigueiredo*; Darren Fink & Dina Teodoro*; Rafael Frongillo*; Craig Harman; Maya Kaczorowski*; Ellen Dickstein Kominers; Vera Mucaj & Boris Zinshteyn*; Ross Rheingans-Yoo*; Christiane Schwind*; Nancy Stern; Elton Tavenner; Michael Thaler*; Eric Wepsic*; Ryan Yu; and Dylan Zabell. (Asterisks indicate solvers who also identified the artist’s name.) And thanks especially to Adam Rosenfield* for test solving!
The Bonus Round
The bowling drone sensation that’s sweeping the nation; a women’s history month crossword; mathematicians discover a new way to add to 3. Bring art to your backyard with augmented reality; 25 years of Pokémon; more virtual escape rooms; and what we know about black holes. Obscure Disney movie facts; Optimus Prime not welcome here (hat tip: Ellen Dickstein Kominers). Hacking the hackers; a crossword-based graphic novel; the classical music in old cartoons (hat tip: Elizabeth Sibert). Plus inquiring minds want to know: How does paper fold under pressure (hat tip: Giorgio Gaglia)?
Okay fine, it only lines up to two digits past the decimal point. But that's more digits than some state legislators would have had given us.
Plus heck, I'll take pretty much any excuse to eat pie!
All measurements are in inches or square inches, as appropriate.
And yes, you read that right: on your way to the answer, you'll somehow have to turn numbers into letters.
This of course was also clued by the reference to our artist as a "shining star."
Fun fact: Orion is my favorite constellation, and apparently it is also the favorite of solver Zarin Pathan.
The stars were sorted in Greek letter order (rather than, say, order of brightness) to prevent solvers from mistakenly thinking that the order of the addresses was meaningful for the puzzle.
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.
©2021 Bloomberg L.P.