The city was just as clean and regular and beautiful on the inside as it looked from the outside. There were street signs, indicating that the gate was at the intersection of the curved DeMorgan Boulevard and, straight ahead leading to the center of town, Modus Toll Road. The streets weren't occupied by cars. Instead they were thronged by pedestrians on the edges, and down the middle lanes various items more like the curiosipede - conventional bicycles, but also scooters and trikes and skateboards - trundled along. The inhabitants were mostly human, or close enough, but there were irregularities - Nyssa saw animals, or at least things that looked like animals the way the Barbet did. And, quivering as it nearly overflowed a single roller skate, what looked like a quantity of caramel-drizzled pudding. She was so bewildered by the pudding that the helpful curiosipede sent her after it till she'd nearly run it over. "Excuse me!" Nyssa called. "You, in the roller skate, would you tell me please what you are? I'm sorry if that's a rude question."

"Oh, not at all," said the pudding. Nyssa had quite given up expecting things not to speak to her. "I'm a proving pudding."

"I've never heard of a proving pudding."

"Well, we go where there's demand, of course. If no one wants anything proved, they won't care to have puddings except for alternative uses I'd really rather not think about," replied the pudding. Its roller skate fell into the curiosipede's pace as they went up the boulevard. "But we're very good at generating guarantees that things must or must not be the case."

"Is there a lot of call for that in Ference?" asked Nyssa.

"Plenty!" replied the pudding. "In fact, I'm in a bit of a hurry now. If you'll excuse me." And it turned right to zoom down into a subway entrance marked Commuter's Rail.

"At home," Nyssa confided in Pomodoro, "a pudding is a dessert."

"Well, you can see why it wouldn't care to think about that," Pomodoro said reasonably.

Nyssa directed the curiosipede around the boulevard, which appeared to circuit the whole city - fence on the left, buildings and more streets radiating between them on the right. She didn't know what she was looking for, exactly. But at least speculating about it kept them going at the local speed limit. This was posted at regular intervals as a nine-term algebraic formula, but Nyssa expected she could approximate it by matching the bicyclists.

When they had gone nearly halfway around the city, the fence jutted out away from its previous purely convex curve. Nyssa spotted in the space thus made a tall white palace, formed out of a cluster of hexagonal stone towers, connected up until the points between each pair when one proved higher than the other. On top of each tower was a flag in a different color, bearing a different symbol in bold white print. It had no windows, and instead of a door per se one face of the frontmost, shortest hexagon was missing, displaying an entryway inside.

The curiosipede swiveled to roll to the very front of the palace. Thinking it impolite to ride it indoors, Nyssa disembarked, scooped Pomodoro onto her shoulder, and tentatively stepped inside.

"Welcome!" chorused several voices, and from three stairwells leading up and ahead, left, and right into the palace, there appeared three identical men; Nyssa supposed that in a palace they might be footmen. Each was dressed in a smart black suit and a tall top hat, and each carried a different object: a pen, a book, and a calculator.

"Thank you," Nyssa told them. "I've been enjoying the city very much, so far, it's beautiful. It was all right to come in the castle, then?"

"Of course!" they all said simultaneously. "The main chambers of the Castle of Queen Qed are open to the public. You may wander as you like. Allow us to introduce ourselves. Holding the pen is," (they were still all speaking in unison), "the Minister of the Department of Working Memory; with the book is the Minister of the Department of High Fidelity Storage; and with the calculator, the Minister of the Department of Accurate Reckoning. We are all delighted to see a visitor in Ference, as we get so few."

"Thank you," Nyssa said again. "Um, if I may ask, why do you all talk at the same time?"

"Given sufficient common knowledge of the premises at work, any agent may come to an agreement with any other agent whose reasoning they trust, about the correct expectations to have of any topic," announced the Ministers together. "Because we are all here listening to you, and can all see one another, we can agree on what to say. So as a symbol of the unity of the Queen's subordinates in enacting her will within Ference, and the superior logic of its citizens, we speak as a group."

"Can you do that with anyone?" asked Nyssa, fascinated.

"Only with people of the very highest quality of wit," the Ministers said, "who wish to participate and will make the corresponding changes to converge on a decision."

"It's very impressive," Nyssa assured them.

"Thank you! Is there any part of the castle you particularly wish to visit today?"

"What is there?" Nyssa asked.

"Behind the minister on your left, there are ballrooms!" they proclaimed. "Behind the minister on your right, the kitchens. Behind the minister directly in front of you, the throne room! The halls of study and research! The treasury, the armory! And the remainder of the palace, much of it unavailable for tour. We, or a subset, will be happy to conduct you to anything you wish to see."

"I'd love to have a look at them all," said Nyssa.

"If you'll follow the minister on your left first, then," the ministers all replied, "he'll conduct you in that wing." And Nyssa did so, following the Minister of the Department of Working Memory.

The ballrooms, of which there were six full sized and one much smaller, proved to contain (in more than enough space to hypothetically dance in) balls. That is to say, spheres: sitting on the floor, dangling from the ceiling, adorning tables and shelves each on its own decorative cushion, sitting on slowly revolving pedestals, rolling eternally on treadmills, heaped in baskets or glasses or coolers lined with snow. The spheres were as small as pinheads up through the size of an entire ballroom, so big that walking all the way around it took about a minute. "Um," Nyssa asked the Minister of the Department of Working Memory, "why do you have all these balls?"

"Oh, Her Majesty collects them," he replied. "They're mined from the Ball Pit, which isn't in Ference itself, it's some miles south, and used in various commercial applications, but the prettiest ones wind up here, or sewn into one of the Queen's ball gowns. Mathematical objects are always in fashion in Ference."

"Ah." Nyssa reached out for a ball; her hand slipped across it almost like it wasn't there, like it resisted her hand less than even air, though she couldn't stick her arm through it. "It's so smooth."

"It's a mathematical object; it's frictionless," said the Minister, "that's part of why they're so valuable! They're difficult to transport."

They returned to the entryway and Nyssa followed the Minister of the Department of Accurate Reckoning into the kitchens, where she found no food at all. Though there was a delicious smell in the air, and many workers in aprons and chef's hats were bustling around with pots and pans and ovens and in one case something very like a rotary cheesegrater, what they handled was not vegetables or cheese or pastries but rather assorted letters and other shapes.

They were in all sizes, from a great big square that someone was broiling that took up the entire baking sheet, to tiny sprinkles coming out of the grater. The grater was emitting hundreds of tiny shavings, each itself shaped like the letter P. When the cook holding the implement noticed Nyssa looking, she said, "This is a double negrater!"

"What's inside it?" Nyssa asked.

"It isn't not P," said the cook.

"- so it's P?"

"It isn't not P," the cook replied patiently. "What's going on the biconditionals, now, those are P." The biconditionals, Nyssa saw, were three parallel lines, like an equals sign with a bit extra. The cook decided the biconditionals had had enough P, reached for a grinder, and ground out little copies of the letter S onto them too until every biconditional had a healthy crust of both S and P. "Here, would you like one now? I'm sure you'll be invited to stay for dinner but what's the harm?" She pressed a warm one into Nyssa's hand and Nyssa took a bite, expecting it to be a cunningly shaped biscuit, but instead it tasted like nothing she'd ever had before, and melted beautifully on the tongue. The seasonings gave it an excellent depth of flavor.

"This is great," she told the cook.

"They're my favorites too," said the cook, now grating letters onto a new batch. "They're a balanced meal, though admittedly a diet of nothing but biconditionals won't help you grow, so be sure to try lots of things at the banquet."

"When is the banquet?" asked Pomodoro.

"Oh, very soon now," the cook said.

"We just about have time to finish your tour," the Minister said, "and end it in the dining hall!" And he led Nyssa back to the entryway, and handed her off to the third Minister, who led her up the center staircase.