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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?”
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:
“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:
“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.
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:
As you puzzle over what that could possibly mean, the princesses speak again. Rhyme says, “I’m pretty sure I can help you.” Reason adds, “I’m pretty sure I can help you after that.”
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.
If you manage to turn this Conundrum into an EZ pass — or if you even make partial progress — please let us know at [email protected] 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:
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”:
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!
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.