The Phantom Tollbooth
Shortly after their detour to Conclusions, Milo, Tock, and the Humbug come upon a fork in the road where the meet the Dodecahedron, a man with twelve faces, each of which displays a different emotion. When Milo introduces himself, the Dodecahedron wonders if everyone with one face is called "a Milo" and explains that in Digitopolis everything is named for what it is. He finds Milo's system of naming very difficult and wonders how anything get done if the numbers all had their own names, and one had to do the sum of "Robert plus John."
Milo asks which fork he should take to get to Digitopolis, and the Dodecahedron responds with a series of increasingly ridiculous story problems. When Milo points out how silly they are, the Dodecahedron remarks, "as long as the answer is right, who cares if the question is wrong? Then he explains that none of the three roads is right and points out that it was foolish of Milo to assume that just because there were three choices one of them was right. The Dodecahedron offers to personally escort the travelers to Digitopolis and hops in the car with them.
On the outskirts of the city, they stop to see the numbers mine where digits are chiseled out of the stone much like precious gems. Inside they meet the Mathemagician, an impressive man who carries a giant pencil that he uses as a wand. The Mathemagician explains the functions of the mine to the companions and then invites them to lunch.
Milo's lunch with the Mathemagician proves to be as baffling as his banquet with King Azaz. They dine on subtraction stew, which makes them hungrier rather than fuller and leaves the poor Humbug, who gorges himself on twenty-three bowls, practically starving. After the meal, the Mathemagician magically transports the group to his workshop, where he uses his magic staff to dazzle the group with mathematical tricks. Milo asks to see the biggest number, and the Mathemagician leads him to an enormous number 3. Milo corrects himself and asks to see the longest number, and the Mathemagician opens a closet to display a very flattened number 8.
Milo finally clarifies that he means the number of the greatest value, prompting the Mathemagician to give a lengthy speech about infinity. He leads Milo to a staircase that he claims will lead to infinity, and Milo happily starts climbing, telling Tock and the Humbug that he should be back in just a few minutes.
Milo gives up on his trip to Infinity and returns to the Mathemagician's workshop, where he finally broaches the subject of Rhyme and Reason. When Milo tells him that Azaz has agreed to release the princesses, the Mathemagician refuses to allow it since he and Azaz always disagree. The Mathemagician tells Milo that if he can prove that he and Azaz have ever agreed, he will consent to the release of the princesses. Milo points out that if the Mathemagician and Azaz always disagree then they have agreed to disagree. The Mathemagician gracefully accepts his defeat and even gives Milo a miniature version of his magic pencil as a gift.
Milo and his companions leave Digitopolis and head towards the Mountains of Ignorance. As the clamber up a dark, craggy path, a mysterious voice in the distance keeps making puns with their words. The Everpresent Wordsnatcher, a filth-encrusted bird, eventually presents himself and explains that his purpose is to take the words of other and twist them to make little jokes.
After more climbing, Milo, Tock, and the Humbug finally reach a flat part on the mountain, where the find a well-dressed gentleman whose face has no features on it—no eyes, no nose, no mouth. Despite his frightening appearance, the man seems very friendly and politely asks for help on a few tasks. He instructs Milo to move a pile of sand a grain at a time, Tock to drain a well using an eye-dropper, and the Humbug to dig a hole through the mountain with a needle. Since the gentleman seems so nice, the three travelers accept their tasks and begin working. Soon hours have gone by with no end in sight.
The introduction to Digitopolis, the city of numbers, resembles the introduction to Dictionopolis, the city of words. Milo first meets one of its unusual inhabitants, the Dodecahedron, who has the bizarre habit of cycling between his twelve different faces. His multitude of expressions resembles that of Azaz's five advisors, who also tend to overdo things and confuse Milo. The meal scene with the Mathemagician is also very much like the meal scene with Azaz in that the food seems is quite nonsensical. Subtraction Stew, much like the "half- baked" ideas and "light" meal, does little to satisfy the guests' hunger—rather the opposite, in fact. The hapless Humbug only learns how stew works after wolfing down twenty-three bowls of it, illustrating an interesting role reversal with Milo. Remember that at Azaz's banquet, it was Milo who went hungry because of his misunderstanding of the food, while the Humbug stuffed himself silly.
Just as he learned about letters, perspective, colors, and sound in some of his other stops, Milo takes in a lesson about numbers in Digitopolis. The crux of the education in this section relates to infinity, though only after the requisite puns about the "largest" and "longest" numbers in the Mathemagician's possession. After Milo tricks the Mathemagician into agreeing to release the princesses, the ruler gives him a miniature magic staff as a gift. Notice that once he has learned a lesson from an inhabitant of the Lands Beyond, Milo nearly always gets some kind of gift from them.
The lessons Milo learns from the Mathemagician end this segment of his "education" in the Lands Beyond. As he heads into the Mountains of Ignorance, we will begin to see how Milo puts his learning to use. When he meets the Everpresent Wordsnatcher, we begin a series of demon confrontations, each of which Milo will solve by relying upon his newfound wisdom and the gifts he has been given during his journey.
The Everpresent Wordsnatcher presents the easiest of these challenges, since he is admittedly more of a nuisance than a demon. Notice that the Humbug seems to know the bird rather well, since the Humbug enjoys the sort of annoying word trickery. It is also important to observe that there the kind of games the Everpresent Wordsnatcher plays are very different than those that Juster himself plays with his pun motif. The primary difference is that the Everpresent Wordsnatcher makes puns with the intent of making himself seem intelligent, while Juster uses puns to help teach Milo lessons about the Lands Beyond.
The faceless gentleman does not seem to be a demon at first, but there definitely seems to be something wrong with him. The tasks he gives Milo, Tock, and the Humbug seem unimportant and time consuming. Juster seems to be setting us up for a revelation about the gentleman, a technique called foreshadowing. Though we aren't certain just what he is up to yet, Juster has given us a lot of clues that suggest the gentleman has something up his sleeves.
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