This and many other wonders Princess Nell saw during her long ride up the spiral road. The clouds cleared away, and Nell found that she could see great distances in every direction. King Coyote's domain was in the very heart of the Land Beyond, and his castle was built on a high plateau in the center of his domain, so that from his windows he could see all the way to the shining ocean in every direction. Nell kept a sharp eye on the horizon as she climbed toward the King's inner keep, hoping she might get a glimpse of the faraway island where Harv languished in the Dark Castle; but there were many islands in the distant sea, and it was hard to tell the Dark Castle's towers from mountain crags.

Finally the road became level and turned inward to pierce another unguarded gate in another high wall, and Princess Nell found herself in a green, flowery court before the King's keep-a high palace that appeared to have been hewn from a single diamond the size of an iceberg. By now the sun was sinking low in the west, and its orange rays ignited the walls of the keep and cast tiny rainbows everywhere like shards from a shattered crystal bowl. A dozen or so messengers stood in a queue before the doors of the keep. They had left their horses in a corner of the yard where a watering-trough and manger were available. Princess Nell did likewise and joined the queue.

"I have never had the honor of carrying a message to King Coyote," Princess Nell said to the messenger preceding her in the queue.

"It is an experience you will never forget," said the messenger, a cocky young man with black hair and a goatee.

"Why must we wait in this queue? In the stalls at the Cipherers' Market, we leave the books on the table and continue on our way."

Several of the messengers turned and looked back at Princess Nell disdainfully. The messenger with the goatee made a visible effort to control his amusement and said, "King Coyote is no small-timer sitting in a stall at the Cipherers' Market! This you will soon see for yourself."

"But doesn't he make his decisions the same way as all the others-by consulting rules in a book?"

At this the other messengers made no effort to control their amusement. The one with the goatee took on a distinctly sneering tone. "What would be the point of having a King in that case?" he said. "He does not take his decisions from any book. King Coyote has built a mighty thinking machine, Wizard 0.2, containing all the wisdom in the world. When we bring a book to this place, his acolytes decipher it and consult with Wizard 0.2. Sometimes it takes hours for Wizard to reach its decision. I would advise you to wait respectfully and quietly in the presence of the great machine!"

"That I will certainly do," said Princess Nell, amused rather than angered by this lowly messenger's impertinence.

The queue moved along steadily, and as darkness fell and the orange rays of the sun died away, Princess Nell became aware of colored lights streaming out from within the keep. The lights seemed to be quite brilliant whenever Wizard 0.2 was cogitating and dropped to a low flicker the rest of the time. Princess Nell tried to make out other details of what was going on inside the keep, but the countless facets broke up the light and bent it into all directions so that she could get only hints and fragments; trying to see into King Coyote's inner sanctum was like trying to remember the details of a forgotten dream.

Finally the messenger with the goatee emerged, gave Princess Nell a final smirk, and reminded her to display proper respect.

"Next," intoned the acolyte in a chanting voice, and Princess Nell entered the keep.

Five acolytes sat in the anteroom, each one at a desk piled high with dusty old books and long reels of paper tape. Nell had brought thirteen books from the Cipherers' Market, and at their direction, she distributed these books among the acolytes for decipherment. The acolytes were neither young nor old but in the middle of their lives, all dressed in white coats decorated, in golden thread, with the crest of King Coyote. Each also had a key around his neck. As Princess Nell waited, they deciphered the contents of the books she had brought and punched the results onto strips of paper tape using little machines built into their tables.

Then, with great ceremony, the thirteen paper tapes were coiled up and placed on a tremendous silver platter carried by a young altar boy. A pair of large doors was swung open, and the acolytes, the altar boy, and Princess Nell formed into a procession of sorts, which marched into the Chamber of the Wizard, a vast vaulted room, and down its long central aisle.

At the far end of the chamber was-nothing. A sort of large empty space surrounded by elaborate machinery and clockwork, with a small altar at the front. It reminded Princess Nell of a stage, empty of curtains and scenery. Standing next to the stage was a high priest, older and wearing a more impressive white robe.

When they reached the head of the aisle, the priest went through a perfunctory ceremony, praising the Wizard's excellent features and asking for its cooperation. As he said these words, lights began to come on and the machinery began to whir. Princess Nell saw that this vault was, in fact, nothing more than an anteroom for a much vaster space within, and that this space was filled with machinery: countless narrow shining rods, scarcely larger than pencil leads, laid in a fine gridwork, sliding back and forth under the impetus of geared power shafts running throughout the place. All of the machinery threw off heat as it ran, and the room was quite warm despite a vigorous draught of cold mountain air being pumped through it by windmill-size fans.

The priest took the first of the thirteen rolls of paper tape from the platter and fed it into a slot on the top of the altar. At this point, Wizard 0.2 really went into action, and Princess Nell saw that all the whirring and humming she'd seen to this point had been nothing more than a low idle. Each of its million pushrods was tiny, but the force needed to move all of them at once was seismic, and she could sense the tremendous strains on the power shafts and gear boxes thundering through the sturdy floor of the keep.

Lights came on around the stage, some of them built into the surface of the stage itself and some hidden in the machinery around it. To Princess Nell's surprise, a seemingly three-dimensional shape of light began to coalesce in the center of the empty stage. It gradually formed itself into a head, which took on additional details as the machinery thundered and hissed away: it was an old bald man with a long white beard, his face deeply furrowed in thought. After a few moments, the beard exploded into a flock of white birds and the head turned into a craggy mountain, the white birds swarming about it, and then the mountain erupted in orange lava that gradually filled up the entire volume of the stage until it was a solid glowing cube of orange light. In this fashion did one image merge into another, most astonishingly, for several minutes, and all the time the machinery was screaming away and making Princess Nell most anxious, and she suspected that if she had not seen less sophisticated machines at work at Castle Turing, she might have turned around and fled.

Finally, though, the images died away, the stage became empty again, and the altar spat out a length of paper tape, which the priest carefully folded up and handed to one of the acolytes. After a brief prayer of thanks, the priest fed the second tape into the altar, and the whole process started up again, this time with different but equally remarkable images.

So it went with one tape after another. When Princess Nell became accustomed to the noise and vibration of the Wizard, she began to enjoy the images, which seemed quite artistic to her— like something a human would come up with, and not machinelike at all.

But the Wizard was undoubtedly a machine. She had not yet had the opportunity to study it in detail, but after her experiences in all of King Coyote's other castles, she suspected that it, too, was just another Turing machine. Her study of the Cipherers' Market, and particularly of the rulebooks used by the cipherers to respond to messages, had taught her that for all its complexity, it too was nothing more than another Turing machine. She had come here to the Castle of King Coyote to see whether the King answered his messages according to Turing-like rules. For if he did, then the entire system— the entire kingdom— the entire Land Beyond-was nothing more than a vast Turing machine. And as she had established when she'd been locked up in the dungeon at Castle Turing, communicating with the mysterious Duke by sending messages on a chain, a Turing machine, no matter how complex, was not human. It had no soul. It could not do what a human did.

The thirteenth tape was fed into the altar, and the machinery began to whine, then to whir, and then to rumble. The images appearing above the stage flourished into wilder and more exotic forms than any they had seen yet, and watching the faces of the priest and the acolytes, Princess Nell could see that even they were surprised; they had never seen anything of the like before. As the minutes wore on, the images became fragmented and bizarre, mere incarnations of mathematical ideas, and finally the stage went entirely dark except for occasional random flashes of color. The Wizard had worked itself up to such a pitch that all of them felt trapped within the bowels of a mighty machine that could tear them to shreds in a moment. The little altar boy finally broke away and fled down the aisle. Within a minute or so, the acolytes, one by one, did the same, backing slowly away from the Wizard until they were about halfway down the aisle and then turning away and running. Finally even the high priest turned and fled. The rumbling of the machinery had now reached such a pitch that it felt as though an epochal earthquake were in progress, and Nell had to steady herself with a hand on the altar. The heat coming from back in the machine was like that from a forge, and Nell could see a dim red light from deep inside as some of the pushrods became hot enough to glow.

Finally it all stopped. The silence was astonishing. Nell realized she had been cringing and stood up straight. The red glow from inside the Wizard began to die away. White light poured in from all around. Princess Nell could tell that it was coming in from outside the diamond walls of the keep. A few minutes ago it had been nighttime. Now there was light, but not daylight; it came from all directions and was cool and colorless.

She ran down the aisle and opened the door to the anteroom, but it wasn't there. Nothing was there. The anteroom was gone. The flowery garden beyond it was gone, and the horses, the wall, the spiral road, the City of King Coyote, and the Land Beyond. Instead there was nothing but gentle white light.

She turned around. The Chamber of the Wizard was still there.

At the head of the aisle she could see a man sitting atop the altar, looking at her. He was wearing a crown. Around his neck was a key-the twelfth key to the Dark Castle. Princess Nell walked down the aisle toward King Coyote. He was a middle-aged man, sandy hair losing its color, gray eyes, and a beard, somewhat darker than his hair and not especially well trimmed. As Princess Nell approached, he seemed to become conscious of the crown around his head. He reached up, lifted it from his head, and tossed it carelessly onto the top of the altar.

"Very funny," he said. "You snuck a zero divide past all of my defenses."

Princess Nell refused to be drawn by his studied informality. She stopped several paces away. "As there is no one here to make introductions, I shall take the liberty of doing so myself. I am Princess Nell, Duchess of Turing," she said, and held out her hand.

King Coyote looked slightly embarrassed. He jumped down from the altar, approached Princess Nell, and kissed her hand. "King Coyote at your service."

"Pleased to make your acquaintance."

"The pleasure is mine. Sorry! I should have known that the Primer would have taught you better manners."

"I am not acquainted with the Primer to which you refer," Princess Nell said. "I am simply a Princess on a quest: to obtain the twelve keys to the Dark Castle. I note you have one of them in your possession."

King Coyote held up his hands, palms facing toward her.

"Say no more," he said. "Single combat will not be necessary. You are already the victor." He removed the twelfth key from his neck and held it out to Princess Nell. She took it from him with a little curtsy; but as the chain was sliding through his fingers, he tightened his grip suddenly, so that both of them were joined by the chain. "Now that your quest is over," he said, "can we drop the pretense?"

"I'm sure I don't take your meaning, Your Majesty."

He bore a controlled look of exasperation. "What was your purpose in coming here?"

"To obtain the twelfth key."

"Anything else?"

"To learn about Wizard 0.2."

"Ah."

"To discover whether it was, in fact, a Turing machine."

"Well, you have your answer. Wizard 0.2 is most certainly a Turing machine-the most powerful ever built."

"And the Land Beyond?"

"All grown from seeds. Seeds that I invented."

"And it is also a Turing machine, then? All controlled by Wizard 0.2?"

"No," said King Coyote. "Managed by Wizard. Controlled by me."

"But the messages in the Cipherers' Market control all the events in the Land Beyond, do they not?"

"You are most perceptive, Princess Nell."

"Those messages came to Wizard-just another Turing machine."

"Open the altar," said King Coyote, pointing to a large brass plate with a keyhole in the middle.

Princess Nell used her key to open the lock, and King Coyote flipped back the lid of the altar. Inside were two small machines, one for reading tapes and one for writing them.

"Follow me," said King Coyote, and opened a trap door set into the floor behind the altar.

Princess Nell followed him down a spiral staircase into a small room. The connecting rods from the altar came down into this room and terminated at a small console.

"Wizard is not even connected to the altar! It does nothing," Princess Nell said.

"Oh, Wizard does a great deal. It helps me keep track of things, does calculations, and so on. But all of that business up there on the stage is just for show— just to impress the commoners. When a message comes here from the Cipherers' Market, I read it myself, and answer it myself.

"So as you can see, Princess Nell, the Land Beyond is not really a Turing machine at all. It's actually a person— a few people, to be precise. Now it's all yours."

King Coyote led Princess Nell back into the heart of his keep and gave her a tour of the place. The best part was the library. He showed her the books containing the rules for programming Wizard 0.2, and other books explaining how to make atoms build themselves into machines, buildings, and whole worlds.

"You see, Princess Nell, you have conquered this world today, and now that you have conquered it, you'll find it a rather boring place. Now it's your responsibility to make new worlds for other people to explore and conquer." King Coyote waved his hand out the window into the vast, empty white space where once had stood the Land Beyond. "There's plenty of empty space out there."

"What will you do, King Coyote?"

"Call me John, Your Royal Highness. As of today, I no longer have a kingdom."

"John, what will you do?"

"I have a quest of my own."

"What is your quest?"

"To find the Alchemist, whoever he may be."

"And is there . . ."

Nell stopped reading the Primer for a moment. Her eyes had filled up with tears.

"Is there what?" said John's voice from the book.

"Is there another? Another who has been with me during my quest?"

"Yes, there is," John said quietly, after a short pause. "At least I have always sensed that she is here."

"Is she here now?"

"Only if you build a place for her," John said. "Read the books, and they will show you how."

With that, John, the former King Coyote and Emperor of the Land Beyond, vanished in a flash of light, leaving Princess Nell alone in her great dusty library. Princess Nell put her head down on an old leather-bound book and smelled its rich fragrance. One tear of joy ran from each eye. But she mastered the impulse to cry and reached for the book instead.

They were magic books, and they drew Princess Nell into them so deeply that, for many hours, perhaps even days, she was not aware of her surroundings; which scarcely mattered as nothing remained of the Land Beyond. But at some length, she realized that something was tickling her foot. She reached down absently and scratched it. Moments later the tickling sensation returned. This time she looked down and was astonished to see that the floor of the library was covered with a thick gray-brown carpet, flecked here and there with splotches of white and black.

It was a living, moving carpet. It was, in fact, the Mouse Army. All of the other buildings, places, and creatures Princess Nell had seen in the Land Beyond had been figments produced by Wizard 0.2; but apparently the mice were an exception and existed independently of King Coyote's machinations. When the Land Beyond had disappeared, all of the obstructions and impedimenta that had kept the Mouse Army away from Princess Nell had disappeared with it, and in short order they had been able to fix her whereabouts and to converge upon their long-sought Queen.

"What would you have me do?" Princess Nell said. She had never been a Queen before and did not know the protocol.

A chorus of excited squeaking came from the mice as commands were relayed and issued. The carpet went into violent but highly organized motion as the mice drew themselves up into platoons, companies, battalions, and regiments, each of them commanded by an officer. One mouse clambered up the leg of Princess Nell's table, bowed low to her, and then began to squeak commands from on high. The mice executed a close-order drill, withdrew to the edges of the room, and arrayed themselves in an empty box shape, leaving a large open rectangle in the middle of the floor.

The mouse up on the table, whom Nell had dubbed the Generalissima, issued a lengthy series of orders, running to each of the four edges of the table to address different contingents of the Mouse Army. When the Generalissima was finished, very high piping music could be heard as the mouse pipers played their bagpipes and the drummers beat their drums.

Small groups of mice began to encroach on the empty space, each group moving toward a different spot. Once each group had reached its assigned position, the individual mice arranged themselves in such a way that the group as a whole described a letter. In this way, the following message was written across the floor of the library:

WE ARE ENCHANTED

REQUEST ASSISTANCE

REFER TO BOOKS

"I shall bend all my efforts toward your disenchantment," Princess Nell said, and a tremendous, earsplitting scream of gratitude rose from the tiny throats of the Mouse Army. Finding the required book did not take long. The Mouse Army split itself up into small detachments, each of which wrestled a different book from the shelf, opened it up on the floor, and scampered through it one page at a time, looking for relevant spells. Within the hour, Princess Nell noted that a broad open corridor had developed in the Mouse Army, and that a book was making its way toward her, seeming to float an inch above the floor.

She lifted the book carefully from the backs of the mice who were bearing it and flipped through it until she found a spell for the disenchantment of mice. "Very well then," she said, and began to read the spell; but suddenly, excited squeaking filled the air and all the mice were running away in a panic. The Generalissima climbed up onto the page, jumping up and down in a state of extreme agitation and waving her forelegs back and forth over her head.

"I understand," Princess Nell said. She picked up the book and walked out of the library, taking care not to step on any of her subjects, and followed them out to the vast empty space beyond.

Once again the Mouse Army put on a dazzling display of close order drill, drawing itself up across the empty, colorless plain by platoons, companies, battalions, regiments, and brigades; but this time the parade took up a much larger space, because this time the mice took care to space themselves as far apart as the length of a human arm. Some of the platoons had to march what was, for them, a distance of many leagues in order to reach the edges of the formation.

Princess Nell took advantage of the time to wander about and inspect the ranks, and to rehearse the spell.

Finally the Generalissima approached, bowed deeply, and gave her the thumbs-up, though Princess Nell had to pick the tiny leader up and squint to see this gesture. She went to the place that had been left for her at the head of the formation, opened up the book, and spoke the magic spell.

There was a violent thunderclap, and a rush of wind that knocked Princess Nell flat on her back. She looked up, dazed, to see that she was surrounded by a vast army of some hundreds of thousands of girls, only a few years younger than she was. A wild cheer rose up, and all of the girls fell to their knees as one and, in a scene of riotous jubilation, proclaimed their fealty to Queen Nell.