Kominers’s Conundrums: ‘The Phantom Tollbooth’ in Seven Puzzles
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Just like many puzzles, the story illustrates how everyday objects and ideas may change entirely if looked at from a slightly different angle. The book also introduces many foundations of puzzle-solving — especially wordplay.
With the help of my brother (and fellow “Tollbooth” fan) Paul Kominers, this week’s Conundrums presents a tribute to the world beyond the tollbooth. Below, there are seven puzzles (separated by asterisks in the text below). Their types vary, but all are based on the story in the book. The answer to each is either one or two words, as indicated. Once you’ve solved all of them, you will be able to put their seven answers together to form one more puzzle — and the solution to that is this week’s answer.
As you follow the narrative, you’ll discover the different puzzles to unlock. (And of course, the story itself is rife with clues.) Start with whichever one seems most accessible or exciting!
In some cases, it might be difficult to figure out how to extract an answer word (or words) from the puzzling text. But as you sort through the clues in the story, the answers should start to appear.
So what are you waiting for? Let's visit the puzzling lands beyond the tollbooth!
Almost immediately after passing through the tollbooth, you meet a large dog with a clock on his side — the “watchdog” Tock (1 word, 5 letters). "There's no time to waste," he barks, “listen carefully.” And then he barks: “11 PM! 9 AM! 8 PM! 3 AM! 8 AM! That’s the message.”
As you consider what this might mean, a question occurs to you. “Tock,” you ask, “I’ve always wondered why there are twenty-four hours in a day, not twenty-six.”
Tock responds gruffly, “If you really wanted to, I suppose you could divide up time any which way. And you could label it however you want — with letters or symbols, for example, instead of numbers. But the twenty-third hour is important here, and twenty-five and twenty-six aren’t.”
Soon after that, you arrive at Dictionopolis, the kingdom of words. In the castle’s banquet hall, you find the regent King Azaz (1 word, 5 letters) in a tizzy. “As you know, I recently reconciled with my brother, the Mathemagician of Digitopolis,” he explains, “but he’s still teasing me. Just today he sent me this, with a note saying that ‘It would show me how even a word-lover can’t escape numbers completely.’ What do you think that means?”
- When you win a championship, and then another, and then another
- This answer is hazy; try again later
- A European bowling game with a diamond-shaped target
- What a hobbit eats before luncheon, or when you might have an English Breakfast
- By an order of magnitude
- Paper currency featuring the Lincoln Memorial, casually speaking
As you walk down to get a better view, King Azaz grumbles. “I’m so cross with him.”
You set off from Dictionopolis to Digitopolis, and along the way you encounter a helpful-seeming, but nondescript man (2 words, 3 letters each).
“Nice to meet you,” he says, “I’m the world’s shortest tall person!” He does indeed look quite short for a tall person — he appears to be of more or less average height. A few minutes after he walks away, an identical person approaches from a different direction and says “Why hello! I’m the tallest short person in the world.”
You squint confusedly at this newcomer, and ask, “Aren’t you the same man I met a few minutes ago, who declared himself to be the world’s shortest tall person?”
“Of course!” he responds, “You can be many different things, depending on how you start out. For example,” he continues:
- “Some days when I get up in the morning, I feel like those droplets that form on leaves in the fall or spring — or perhaps like part of a Grateful Dead song.”
- “Other days, I feel more like the plant itself — coniferous, with highly poisonous berries, and featured in a song performed by Battlefield Band.”
“What am I supposed to make of that,” you wonder aloud. “Two different things, of course,” he replies. “The only difference,” he reminds you, “is how you start out.”
Next, you come across a man gesturing at the sky, which is awash in strange colors. “I’m glad you’re here,” says color conductor Chroma the Great (1 word, 4 letters). “I’ve been experimenting with digital instruments, and, oh, my sheet music seems to have gotten terribly mixed up.
“For example, can you help me figure out what this says?” He shows you a page with odd-looking musical notes. They go as follows:
As you think these over, Chroma sighs with despair. “I feel as though I’ve hexed myself. My heart aches. Or maybe that’s my chest.”
You then arrive at the mathematical kingdom of Digitopolis, where Azaz’s brother the Mathemagician (1 “word,” 1 digit) is in great distress. “I’m so sad,” he moans. “My brother, that rascal, sneaked in and did this. He replaced all my beautiful numbers with letters and now I can’t see them any more!!”
Indeed, the royal blackboard is showing an equation that seems to have letters in place of digits:
ABCD * E = DCBA.
“I wish I could see even just one of my beautiful numbers again,” the Mathemagician exclaims. “Could you help me figure out what digit ‘A’ is standing in for?”
From there, you progress to the Mountains of Ignorance, which are filled with demons who are, shall we say, not fully informed. But instead of chasing you, they're currently pursuing trivia. The game’s host is the Terrible Trivium (1 word, 6 letters), a demon obsessed with unnecessarily difficult challenges.
- “Who’s the bumbling spy best known for chasing a moose and squirrel?” Trivium asks.
“Oh, oh — I know,” a demon shouts, “that’s Boris.”
“WRONG!!” Trivium snarls.
- “Who’s the Atlanta rapper who won his first Grammys for ‘Ms. Jackson’ and Stankonia in 2002?”
“You must mean Big Boi.”
- “Now who was the Union commander in the United States’ Civil War?”
- “How about the sailor who got trapped on an island with a professor and movie star after a supposed three-hour tour?”
- “The smuggler who travels a far-away galaxy in the Millennium Falcon?”
“NO! You’re awful at this.”
- “Last question: Who’s the performer who hosted ‘Weekend Update’ and then starred in a successful seven-season-long sitcom?”
“Amy Poehler, obviously.”
Tock shakes his head and growls quietly. “It sounds as if these demons know nothing.”
“Or at least,” you reply, “they don’t seem to know half as much as they think they do.”
After figuring out the right answers, you quietly sneak past them.
You finally reach the castle where princesses Rhyme and Reason (2 words, 4 letters each) await. But the demons have finally noticed and given chase. From the ground, they shake their fists and scream nonsense at the castle. However, the princesses explain, “You may be able to learn something even from demons — if you know how to listen.”
As the cacophony continues, you realize that there are two groups of demons, each repeating a different message:
- The first group chants, “Sash! Pot knot bot lot! Lot bot! Brash not!”
- The second screams, “Rash cot mash! Bot clot! Slash knot! Bash stash pot!”
Once you’ve finished your journey, you pass back through the tollbooth. You notice that this time there’s a gentleman sitting there. “I don’t mean to bother you on your way out,” he says, “but I’ve been trying to remember a word. Can you assist me? If you’ve learned anything on your journey, maybe saying it to me a couple times will help.”
The word that the tollbooth operator is looking for is the answer to this week’s Conundrum.
To figure out what it is, you’ll have to put together the different puzzle answers you collected and read them together as he suggests. From there, it might take another type of rhyme and reason (and perhaps a bit of Phantom Tollbooth lore) but you should be able to identify the single word that wraps up our story.
So jump to it!
If you manage to turn this Conundrum into an EZ pass — or if you even make partial progress — please let us know at firstname.lastname@example.org before midnight New York time on Thursday, April 8.
Programming note: This Conundrum is longer than most, so we’re giving you two weeks to solve it. The next Conundrums will run on April 11.
Previously in Kominers’s Conundrums…
In our Pi Day Conundrum, eight of Donald Duck’s friends had requested unusually-shaped pies. But Donald needed some geometry assistance — for each pie, he had messed up one of the dimensions.
For example, “Pumbaa was hoping to chow down on a rectangular saskatoonberry pie with perimeter 44 and area 85” and Donald had written down, “I should be able to do this with side lengths 5 and 16.” But a rectangle with side lengths 5 and 16 has perimeter 42 and area 80 — close, but not quite the correct size.
Solvers had to identify which dimension was off in each instance; in Pumbaa’s case, the solution was to use side lengths 5 and 17. The geometry problems got complicated at times, but the constraint that you could only change one of the two dimensions, along with the rule that the dimensions all had to be integers, made it possible to solve each one:
- 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 [should be 18] 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 be 5] 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 [should be 19] 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 [should be 17].
- 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 [should be 21] 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 [should be 1]?
- 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 [should be 18], 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 [should be 5].
The answer to the puzzle was “an even more fundamental problem with the pies he produced,” which we had hinted you could obtain by putting “wrong and right together.” What did that mean? (And how were we supposed to obtain a word from all those numbers in the first place?)
The trick was to treat both the incorrect and corrected dimensions as letters of the alphabet, using the standard mapping with 1 = A, 2 = B, and so forth. Reading first all the incorrect dimensions and then all the corrected ones in order gave “20 8 5 16 9 5 19 1 18 5 19 17 21 1 18 5,” or rather, “THE PIES ARE SQUARE,” both a pun on the formula for the area of a circle (Pi R-squared) and a real problem for Donald since none of his friends wanted square-shaped pies.
And then as we hinted there was a bonus puzzle pointing to a second problem with the pies, which would somehow involve “the ways in which Donald was right all along” and “why these characters have such specific taste.”
Astute solvers noticed two types of information that hadn’t been used at all in solving the main puzzle: the characters’ names and their preferred pie flavors. Looking at the letter position in each corresponding to the dimension Donald hadn’t messed up spelled out the bonus answer — for example, in the case of Gaston, the relevant dimension was 6, indicating an “N” from “Gaston” and a “Y” from “hearty.” This revealed that the pies were square “AND ALSO WAY TAU HOT”:
DAISY (2) / WALNUT (2)
GASTON (6) / HEARTY (6)
DOC (1) / TREACLE (1)
PUMBAA (5) / SASKATOONBERRY (5)
COLETTE (3) / PRUNE (3)
ESMERALDA (2) / CHERRY (2)
BALOO (4) / BUKO (4)
WALL-E (1) / TRASH (1)
Lazar Ilic* solved first for the second week in a row, followed by Zoz*, Dan Rubin & Jennifer Walsh, Hirsh Jain, Zarin Pathan, and Scott Wu. The other solvers were Karolyne DeBriyn*, Noam D. Elkies*, Scott Hopkins, Yousef Ibreak, Maya Kaczorowski*, Colin Lu, Vera Mucaj, John Owens, Ross Rheingans-Yoo, Spaceman Spiff, Sanandan Swaminathan*, Michael Thaler*, Nathaniel Ver Steeg, Ryan Yu*, Dylan Zabell*, and Dylan Zhou. (Asterisks indicate solvers who also figured out the second problem with the pies.) Both Owens and Thaler submitted emoji solutions. And thanks especially to Eric Price* and Adam Rosenfield* for test solving!
The Bonus Round
“The puzzling Antikythera mechanism”; a regular expression crossword puzzle; and a virtual Harvard Museum tour. A not-so-cryptic encryption algorithm (hat tip: Skeet Singleton); vaccination fashion (hat tip: Ellen Dickstein Kominers); and a bitcoin clock (hat tip: Mike Nizza). Integer overflow causes a rather negative Pokémon Go experience (hat tip: David Martin Warsinger). Making a self-quoting tweet; using letters we don’t need; “How James Holzhauer Broke ‘Jeopardy’”; “Larry, I’m on Ducktales”. Plus inquiring minds want to know: Does Bordeaux taste different after going to outer space?
If the princesses' clues aren't enough, you might also want to look to Norton Juster's other most famous work – The Dot and the Line.
If you hadn't heard of saskatoonberry pie before, then that's more news you can use, courtesy of Conundrums!
To figure this out, you could set up the pair of equations 44 = P = 2L + 2W and 85 = A = L * W; write L = 85/W, so that 2W + 170/W = 44; and solve to find that W has to be either 5 or 17.
This one looked like it might be particularly hard because it involved a hexagon, but in fact in a regular hexagon, if one side has length L, then the other sides must have total length 5 * L. Since all of Donald's dimensions had to be integers, that immediately implies that we must have L = 1 here.
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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