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.