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.
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