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THE PRINCESS AND AP

A Calculus Fairy Tale

Dan Kennedy

Baylor School, Chattanooga, TN

Once upon a time, in a magical fairyland in another part of the universe, there was a divided kingdom called Pitall. The mountain people of High Pitall fought constantly with the valley people of Low Pitall, although nobody could quite remember why.

The ruler of High Pitall was the good King Maxwell, who worked together with his wife, Queen Minerva, to try to restore peace to a united Pitall. Low Pitall's ruler, on the other hand, was a slightly demented elf named Epsilon the Small, who fought against reunification at every turn. He realized full well that the people would choose Maxwell to rule over a united Pitall, and that he, Epsilon, would most probably be overthrown. When Maxwell's peace-making efforts began slowly to win over the people, Epsilon realized that it was time for desperate measures.

In the nearby land of Agnesi there lived a wicked witch, whose evil powers had been quite potent in her youth, but whose strength had diminished with age. She was still quite effective at close range, however, and to prove it to herself she had gradually turned every citizen of Agnesi into a toad. Epsilon was aware of this, but he was also aware that desperate measures called for calculated risks, so he saddled up a small horse and rode off to enlist the help of the wicked witch of Agnesi.

He found her in Agnesi's town center, sitting alone on a park bench next to a bucket of something so foul that he could catch its stench from a hundred feet away. Drawing cautiously closer, he noticed that the stench had also caught the attention of quite a few flies, who in turn had caught the attention of the witch. As a fly would buzz by, she would stun it with a glance, catch it in her outstretched hand, remove its wings, and toss it casually to the ground. Epsilon at first thought this was arbitrary cruelty, until he noticed that the grass around the bench was crawling with toads. The witch was apparently feeding the former citizens of Agnesi.

Suddenly, she spoke. "You are a brave and foolish little man," she said, not bothering to look up from her afternoon diversion. "Before I condemn you to an amphibious life with my friends here, perhaps you would like to tell me why you risked coming to Agnesi."

Epsilon the Small had never felt smaller. "Oh witch," he pleaded, "a terrible peace is about to break out in Pitall, bringing an end to misery as we know it. Unless I can stop him, King Maxwell will end years of fighting and restore freedom to people who might otherwise suffer -- and I know that suffering is close to your wicked heart. If you would but help me with your Magic, I could defeat Maxwell and strike a blow for evil folks everywhere! Please help me, witch!"

He glanced nervously at the toads. The witch threw another wingless morsel their way, then wiped her hands primly on her long, black skirt. She stared silently at Epsilon for a moment, her black eyes peering into the depths of his soul until he thought he would scream.

"Fish heads," she said at last.

Epsilon's heart seemed to leap to his throat. Fish heads? Was that the toad curse? No, he could see that he still towered over the witch's ex-neighbors by a good four feet.

"I use this bucket of fish heads to attract the flies," she continued. "When they get close enough, I zap 'em. Used to be, I could draw them here myself, but now I'm old and I need the fish heads. It's time, you see. Time is the problem."

Epsilon noticed that she was not responding very directly to his request, but he listened respectfully, nodding with all the sympathy he could muster.

"Magic is strong," continued the witch, "but time is stronger. Oh, I thought I knew it all a few centuries ago, but I never learned to control time, and now it controls me. Now I need helpers -- like these stupid fish heads." As if to emphasize her point, she reached into the bucket and drew forth a ripe one.

"Hey, look," he stammered nervously, "I'm still pretty young. We can do this together, can't we? You and I could make a great --"

He saw the fish head flying toward his face. His hand reached out instinctively to deflect the reeking missile and caught -- a book! The witch cackled with delight. Epsilon saw that the book was entitled Calculus.

"It's my latest passion," explained the witch. "Calculus, it seems, is the magic that controls the changing of time. I have yet to fathom its deepest secrets, but I have managed to add some powerful magic to the sorcery contained in that book. Since you seem like a sincerely evil type, I will loan you this book to use against Maxwell and Minerva. Be warned, though: you must also outsmart your opponents, who will attempt to use this same magic against you."

Epsilon was smiling now. "Oh, witch, you can count on me!" he chirped. He thought he detected some disappointment in the unblinking eyes of the toads below.

"Hmmph," snorted the witch, "we'll see about that! Just in case, though, I am giving you one final bit of magic." She reached into the bucket again and pulled out what might once have been the leading end of a very ugly bullhead. When she threw it to him, it turned into a combination lock. Epsilon noticed that there were no numbers on the dial.

"It's an untimelock," she proclaimed proudly. "It's one of my favorite tricks. It actually exists outside of time, and therefore knows the answers to all questions, past, present, and future. To open it, you must ask it a question that it cannot answer. That is the only combination.

Epsilon was understandably confused. "You mean that I can ask this inert object what I had for dinner last Thursday?"

"Macaroni and pears," said the lock quite clearly.

Epsilon was stunned. "How did it know that?"

"I see through time," said the lock.

"Stop!" warned the witch. "Don't ask another question. There is a fail-safe mechanism that causes the lock to emit a deadly radiation if anyone misses the combination three times in a row. You have asked two questions it could answer; you would be ill-advised to risk asking a third. Remember that it exists outside of time. It knows everything that has happened or will happen."

"Well then, how the heck -- " Epsilon caught himself just in time. The witch cackled, then answered his unasked question.

"It's simple, really. Just ask it for the combination."

Epsilon asked the lock, "What is the combination?"

It immediately fell open.

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Within months, Epsilon and his magic calculus book had brought the twin kingdoms of Pitall to their knees. King Maxwell and Queen Minerva had been captured and locked in a castle in Low Pitall, guarded by the fearsome untimelock. Their daughter, the beautiful and resourceful Princess Limaçon, went into hiding to plot secretly with the loyalists to plan their escape.

Limaçon had enlisted the help of a good sorcerer known as the Light Holder, a pioneer in the sorcery of teleportation, but a magician whose powers were no match for those at Epsilon's disposal. He and Limaçon agreed that they would need to understand Epsilon's strength in order to discover his weakness, so it was essential that they find someone who knew something of this calculus magic. Unfortunately, nobody in the kingdom of Pitall had ever heard of calculus. The Princess herself, although as bright as she was beautiful, had never been exposed to any mathematics beyond the Pythagorean Theorem -- which the Pitallians called the Great Truth, and which many considered to be a message directly from God.

Fifteen weeks into the captivity of Maxwell and Minerva, the Light Holder announced a bold plan. "It is said, my Princess, that everyone in the universe has an exact double, whose every opportunity is precisely complementary to his or her own. It balances the scales of justice: Every opportunity missed in one part of the universe is compensated for in another. If, for example, someone with great leadership is born in a remote jungle and never has a chance to lead a nation, his double might be born somewhere else into a royal family."

Limaçon did not see where this was going. "An interesting theory, Light Holder, but how does it affect us?"

"If the theory is true," he replied, "then somewhere in the universe there is a seventeen-year-old girl with your features and your wisdom, who has been exposed to this magic calculus. If she is your double, then we can assume that she has learned it well."

Limaçon blushed, but she did not argue the point.

The Light Holder continued. "By concentrating on your essence, I believe I can transport your double into our midst. I have made transfers across time and space with inanimate objects before, so I believe I can do it. It would be risky, of course. For one thing, I have never teleported people from unknown places, and, for another thing, I would have to replace matter with matter. It is necessarily a two-way operation."

Limaçon was wide-eyed. "You mean that I would have to switch places with my double?"

The Light Holder shook his head. "No, although that would certainly be easier. You have to be here to lead your double on the search for your parents. Since your knowledge and strengths will be exactly complementary, the two of you will be a formidable force against Epsilon the Small. Don't worry, though. I'll leave my most powerful magic with you when I go."

"Go?" interjected the Princess. "You mean that you would switch places with my double?"

The Light Holder nodded. "There can be no other way."

"But how will you know when it is time to switch back?"

"That," he admitted, "could be the biggest risk of all. I will simply have to know. All I can say is that I have been in this sorcery game a long time now, and I have come to realize that there are forces behind the magic that are more powerful than the magic itself. Sometimes you just have to trust that power." He then added a tantalizing afterthought. "There is no telling what will happen to the passage of time. What might be a minute here could be a week there, or vice-versa. I could be gone ten minutes and return ten years older."

Limaçon had never really thought much about time, but she found that the subject was beginning to get on her nerves.

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All the talk of risks notwithstanding, the decision was quickly made to follow through with the Light Holder's plan. Limaçon found a wig and one of her favorite outfits for the Light Holder to wear during his masquerade, while he gathered some items for her from his alchemy lab. They met at sunset to attempt the exciting experiment.

"Here is the strongest magic I can leave you," said the Light Holder, dropping five golden acorns into Limaçon's open hand. Each acorn bore a number from 1 to 5. She found them to be strangely beautiful, and she could almost feel their power. "Each acorn can grant you one wish," he explained, "but it must involve the number on the acorn. For example, if two soldiers attack you, the 2 acorn can disable them both, the 1 acorn can disable one of them, but the 3 acorn would be useless. Each can be used once, so use them wisely."

Limaçon placed them carefully into her pocket.

"These magic boots are for your double," he continued. "I inherited them from my father, who told me that the wearer of these boots could open any door. I am hoping that she can combine these with her calculus to open Epsilon's untimelock."

She accepted the boots and waited anxiously for his next words, but they were slow in coming. Limaçon saw that there were tears in his eyes. "I wish I could do more," he said finally. "I fear that I am leaving you in great danger."

The Princess smiled and kissed him gently on the forehead. "You have done well, old friend. I only hope that we will prove worthy of your wonderful plan."

The Light Holder smiled and wiped a tear on his sleeve.

"So what are we waiting for?" said the Princess confidently. "Step into this cute little outfit and let's get on with it!"

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In another time in another corner of the universe, Limaçon's double sat anxiously in a school desk waiting for the proctor's signal to begin the free-response section of her AP calculus exam. She had neatly printed "LEE MASON, TOPEKA CENTRAL HIGH SCHOOL" on the scoring sheet and had sealed the flap shut; now she was itching to get under way. The multiple-choice had gone well -- certainly easier than that 1969 test with which the teacher had terrorized the class last week. Arnie Miller had spent most of the ninety multiple-choice minutes with his head on the desk, and had gotten nothing more than a sarcastic comment ("Now here's a sure five!") from Sarge Hampton when she had collected the answer sheets. Lee could hardly believe that Sarge would let him get away with that, but then again, it was beyond her comprehension that anyone would so blatantly waste the exam fee. As Mrs. Hampton completed her instructions to the group, Lee noted with disgust that Arnie's eyes were glazing over again.

"Are there any questions?" intoned Mrs. Hampton in a voice that assured that there would be none. "Good. You may begin."

Lee opened the booklet swiftly and began scanning the first question. Somewhere between the first and second sentences, she sensed that her own eyes were glazing over. When she could focus them again, she found herself sitting on a stone floor in a dimly-lit room, staring into the wide eyes of a girl who could have been her twin.

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Sarge Hampton looked up from her Enquirer, sensing a disturbance in the room. Arnie Miller was breathing rhythmically, his face turned toward the wall, his left cheek pressed firmly against the desk. The Mason girl was deep in thought, her head buried in her hands. Everyone else was writing industriously. Satisfied that all was well, she returned to the story at hand, a well-documented sighting of Elvis yachting near the Bermuda Triangle.

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The Light Holder stared groggily at the strange symbols on the paper in front of him. He had apparently interrupted Limaçon's double in the midst of some group writing task. Peering through his fingers, he noted that most of the young people in the room were writing, but one boy was clearly napping, drawing no negative response from the others. Knowing that he could contribute nothing to the writing task, the Light Holder concluded that the safest thing he could do would be to imitate the sleeping boy. Keeping his face carefully concealed, he slowly lowered his head to the desk.

"Hmmph. Pretty shameful waste of somebody's money," muttered Mrs. Hampton to herself. Turning the page, she began reading an article about a dog that had given birth to a sheep.

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Lee Mason and Limaçon stared at each other for what seemed like several minutes before either could manage to speak. When the words came, however, they spilled out thick and fast, as the two synchronous souls poured forth their complementary experiences into each other until there was virtually nothing left to share. Friendship was too simple a word for the bond that instantly formed between the two girls

When Limaçon had explained the Light Holder's plan, Lee pledged without hesitation to do everything she could to help defeat Epsilon the Small. It was genuinely as if she were being asked to help save her own parents. She thought it was very strange that her calculus knowledge might be the key to their dangerous mission, but then again, everything about this situation was already too strange to comprehend. She kicked off her loafers and tried on the magic boots. To nobody's surprise, they fit her perfectly.

"They're a very nice fit, ma'am," said Lee playfully after a few mincing steps, "but I believe I'll take the patent leather pumps instead."

Limaçon played along, fluttering her eyelashes in mock disbelief. "Why, don't tell me that the Calculus Commando would go into battle wearing patent leather pumps!"

Lee switched to her best John Wayne voice. "Well, I reckon you're right, ma'am. These boots'll do just fine for calculus. Now what do you say we get out there and kick some asymptotes?"

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At nightfall, Limaçon led Lee to the outskirts of the capital city of Low Pitall, where their first challenge would be to get over the wall that surrounded the city. Circling the wall was a moat filled with carnivorous parabaloids, dangerous fish who went after human flesh with a single focus. With the help of Thomas, a trusted page from her father's court, Limaçon and her new friend hoisted a large ladder so that it leaned against the top of the wall, spanned the moat, and rested with its base firmly planted on the bank. Then, with darkness as their cover and Thomas steadying the ladder, the girls began the precarious climb.

As they neared the top of the ladder, Limaçon began to have the uneasy sensation that the wall was growing. Not only was the ladder farther away from the top of the wall than when they had started to climb, but she could even hear the sound of the stones scraping against the wood.

"Lee," she whispered, "this may sound crazy, but that wall is moving!"

"Either that," replied Lee, "or the ladder is slipping!"

Slipping? Limaçon looked back and was horrified at what she saw. Thomas, the trusted page, was slowly pulling the base of the ladder away from the wall at a steady rate of two feet per minute.

"Thomas!" she whispered in a moderated shriek. "Stop it! What are you doing?"

But Thomas did not answer. He just kept pulling the base of the ladder away from the wall.

"Thomas, we are going to die!" she whispered, this time more emphatically.

No response from Thomas. Wood scraped harshly against stone as they drew ever closer to the circling parabaloids. They were too far up the ladder to descend back to the bank.

Suddenly, Lee spoke. "Limaçon, forget Thomas! This is Epsilon's work. I recognize this problem from our calculus course; he probably has the kid under some kind of spell. Quick! Get the number 1 acorn!"

Limaçon dug into her pocket and pulled out the 1 on the first try. They could hear the splashing and the snapping of tiny jaws below. The ladder was picking up speed now. Lee dug her fingernails into the wall.

"I'll wish him away," said the Princess softly.

"NO!" screamed Lee, whirling around and grabbing the hand of the startled Princess. She felt the acorn still there. "Someone ELSE!"

Suddenly the acorn was gone. The ladder had stopped moving. Lee could see the stark realization in Limaçon's eyes, and she prayed that she had caught her in time. If indeed she had simply wished Thomas away, then gravity would soon finish the deadly job that he had begun.

But they were not moving.

"Princess?" asked a voice from the darkness at the bottom of the ladder.

Limaçon took a breath for the first time since making her wish. "Obtus?"

"Yes, it is Obtus. I think I am having a strange dream"

Obtus, the strongest page in the court of High Pitall, held the ladder steady with ease. His strength was the main reason that Limaçon had whispered his name, after Lee had prompted her to adjust her wish at the last second.

"No, Obtus, it is not a dream, but thank goodness you are here. We need you to hoist this ladder back to the top of the wall. Can you do that without spilling us into the moat?"

Obtus responded by embracing the ladder in a mighty bear hug, then lifting it gently to the top of the wall as easily as most people would remove a leaf from a dining room table. Lee and Limaçon clambered gratefully onto the top of the wall.

"Will there be anything else this evening, Princess?" asked Obtus, barely breathing hard. Lee squinted through the darkness, trying to get a peek at this Hercules.

"No, thank you, Obtus. Please return this ladder to my father's castle and go back to bed. I'll explain it all to you someday." She shuddered to think how close she had come to wishing Thomas away and leaving the ladder unsupported. Chalk up one big save for the girl in the boots.

After lowering themselves down the wall, Lee and Limaçon crept stealthily through the realm of Epsilon the Small, following forest trails until they came to the edge of a clearing within sight of the castle. As the next day would require clear thinking and good decisions, they decided to sleep at the edge of the forest until daylight.

Later that evening, Limaçon lay on her back looking up at the stars. "Lee?" she asked, wondering if her friend was still awake.

"Hmmmm...." She was, barely.

"Thanks for saving us back there. I guess you knew I was about to wish Thomas away without wishing for someone else."

Lee rolled over to murmur a reply. "Yeah, that's an old calculus trick I learned from Arnie Miller. Probably the only thing he ever taught anyone. If you can't solve the problem, turn to another page."

The two friends giggled themselves to sleep.

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"Lee, wake up!"

It was morning, and there was concern in Limaçon's voice. Lee sat up quickly, and with a glance determined the cause of her friend's concern. They had been moved to the center of the clearing, and were now sitting on an enormous rectangular sheet of some kind of stiff plastic. In the middle of the plastic was a heavy slab of cement, to which they were both securely chained. Towering above them was a huge conical reservoir, from which water was pouring out onto the far end of the plastic rectangle. This vision was already a calculus nightmare, but Lee only grasped the true extent of their peril when she noticed the four men sawing squares off the corners of the rectangle.

Nobody had to tell Lee what would happen next.

"Limaçon, we are in big trouble," she whispered, tugging futilely at the heavy chains. Limaçon, who knew no calculus, was slower to catch on, but it all became distressingly clear to her a few minutes later when the four men folded up the sides of the rectangle to form a watertight tank.

"Can you swim?" Lee asked the Princess as the water splashed up around their knees.

"Not when I'm chained to cement, I can't," said Limaçon. "The men are gone, though, so what do you say we get rid of these two chains?"

She reached into her pocket and found the number 2 acorn.

"You're gonna love this one," she grinned. She closed her eyes and made her wish. The acorn disappeared.

"Limaçon," said Lee, an edge of panic in her voice, "my chains are still on. What did you do?" The water was at her waist and rising fast.

"Relax," soothed Limaçon. "I turned the links into sugar. Watch!"

She tugged at her chain and it came free from the slab. Lee did likewise, then picked up her chain to investigate. Sure enough, the link at the far end had dissolved, and the next link was soft enough to crumble in her hands. Unfortunately, there were still plenty of links to go, and the process seemed to be proceeding a link at a time.

The water was now shoulder high.

"Lee, what's happening?" Limaçon was struggling to break the links of chain closest to her legs, but to no avail. "Some of these links aren't changing! We can't swim with these on our legs!"

But Lee had already figured it out.

"Oh, Limaçon! I don't believe this is happening, but it's more sick calculus. You couldn't have known, and I probably wouldn't have thought of it either, but there's a rule about chains, and it says that any differentiation has to proceed from the outside in. We have to hold on until it finishes happening. Tread water!"

The water was now over their heads. Limaçon, the better swimmer, was able to struggle up to the surface for quick breaths, but the chains would quickly pull her back under. As the water rose, she could see that she was getting closer to the top of the tank. If she could just hold out a little longer, she might be able to grab the edge.

Lee, meanwhile, was weakening fast. She wanted more than anything to rest. She wanted to swim to the surface to tell Limaçon that she would just rest for a minute and be up soon, but her legs would not respond. The last thing she saw before she blacked out was Limaçon's last, desperate lunge to grab the top of the tank.

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With one aching arm draped over the edge of the tank, Limaçon coughed and sputtered, trying to catch her breath. She reached down and felt four links of chain left, one of which crumbled when she squeezed it. She knew that she would have to act quickly now, as Lee had been down for over a minute.

She took a few deep gulps of air and headed for the bottom of the tank.

Lee was sprawled at the bottom, her chain now fully dissolved. Good, thought Limaçon, wrapping her arms around her limp friend, now let's get you up. Lee would not budge.

But her chain is gone, thought Limaçon.

She struggled harder, managing to get her friend upright, but she could lift her no further. Her feet. Her feet were too heavy.

Of course! thought Limaçon, now desperate for air. The boots! Now fully waterlogged, the magic boots were weighing her down like a pair of cement overshoes. It took every ounce of Limaçon's remaining strength to pull the boots off, but she somehow freed Lee and had enough adrenaline left to drag her to the surface. Water was now spilling over the top of the tank. There was no time to worry about the fall; she lowered the limp form of her beloved companion over the edge as gently as she could, then dropped down beside her.

The Princess wasted no time before initiating the lifesaving procedures she had learned through years of training with her father's army -- standard education for royal children. She suppressed her emotions as she had been trained to do, even as she blew the breath of life through lips identical to her own, and felt for the pulse that would signal hope for the heart that might as well have been hers.

She felt a weak pulse on the second check. Stay calm. She listened for breathing; none yet. Six more breaths. Listen. This time there was sound. She barely had time to react before the coughing started.

It was many minutes later, after she had smoothly and efficiently placed Lee into the recovery position, that the realization that her friend was alive washed over Limaçon like a tidal wave. She sat with her knees clutched tightly to her chest and cried like a baby, until Lee woke up and asked her to please stop.

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On the way to the castle later, Limaçon took the 3, 4, and 5 acorns from her pocket and handed them to Lee.

"Here," she said resolutely. "You make the next three wishes. I almost got you killed by not knowing the chain rule, and I don't want to blow it again."

Lee almost declined the offer, but she realized that Limaçon was right. All their challenges so far had involved twists of calculus, and Limaçon had understandably not seen them coming. Lee would clearly be their best hope now. She accepted the acorns without comment and tucked them in her pocket.

They came to a river, which Limaçon identified as part of the Than River system that flowed through Pitall. "This is the Lesser Than, which flows into the Greater Than, which meets the sea at a delta near the Castle of Epsilon the Strong. He's the father of Epsilon the Small, who has built his castle on the Lesser Than."

Something clicked in Lee's mind. "Wait. Which one has the castle on the delta, Epsilon the Small?"

Limaçon shrugged. "Well, they both do, but Epsilon the Strong has a much bigger castle. Of course, his delta is bigger to begin with, so --"

"Ssshhh!" Lee cut her off again. "There's something moving down by the river."

They crept cautiously through the reeds until they were close enough to identify the source of the sound. Both girls breathed a sigh of relief. A family of four beavers were munching contentedly on a fallen aspen tree. "Oh, Lee, they're just beavers! They won't bother us; let's go." She turned to leave, but Lee sat staring at the beavers, temporarily lost in thought.

"Remember what you told me the Light Holder said about magic? That stuff about trusting your instincts and knowing the right time to act?"

"Yes," replied the Princess tentatively. "What? Are you thinking of acting now?"

Lee fingered the golden acorns in her pocket. Three left, and they were not even inside the castle yet. They could certainly not afford to waste a wish. "Never mind," she sighed. Just a silly calculus idea. I'll tell you later; let's go."

But as they walked on toward the castle, Lee kept her hand in her pocket, rolling the three acorns around in her fingers. By the time they had arrived at the castle gate, she had secretly made another wish. She did not tell Limaçon that there were only two acorns left in her pocket now.

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The two girls stood staring at the untimelock. Lee looked absently at her bare feet and wondered what difference the boots might have made, but there was no use fretting about that now.

"Do you think it can read minds?" whispered Lee, wisely keeping her words inaudible to the untimelock.

"No, it just sees all times and places at once."

"I'll bet it doesn't know about Topeka," said Lee smugly. "I could ask it a question about my family."

Limaçon agreed that it was worth a try. "Go for it."

"What is my brother's birthday?" demanded Lee of the untimelock.

It did not even hesitate. "March 3rd, 1959."

Lee's eyes widened. "It's right! Limaçon, this is serious."

Then she suddenly had another hunch.

"Play your instincts," she muttered to herself. Then she said, "A particle moves along the x-axis with position function x(t) = sin t. What is its velocity at time t = 2?"

Again, the untimelock replied instantly. "The cosine of 2."

Limaçon looked hopefully at Lee, recognizing that some calculus had just passed her by. "Is that it? Is it right?"

Lee nodded.

"Lee, that is two questions. I know I don't need to remind you that if it answers one more, we are both dead!"

But Lee looked strangely in control. "Don't worry," she whispered, "I wasted one."

Limaçon wanted to scream. Her best friend had gone insane and was about to get them both killed.

"You were right about the cosine of 2," said Lee almost casually to the untimelock. Then her eyes narrowed and her hands went defiantly to her hips. Adopting her John Wayne voice for the second time in 24 hours, she demanded, "Justify your answer!"

The untimelock fell open.

"Just like Arnie Miller," sneered Lee. "Long on facts, short on proof."

"Lee, you did it!" squealed Limaçon.

"Yup." She removed the lock and tossed it aside. "I figured that a lock that existed out of time might have a problem explaining the concept of changing time. But we can't waste our time congratulating ourselves; we have to find your parents before Epsilon finds out that we've gotten in."

Limaçon, who had lived in castles all her life, had no problem maneuvering unseen through back corridors, until she had led Lee to a window overlooking the main courtyard. When Lee saw the fearsome creatures guarding the tower on the opposite side, it was immediately obvious to her where Maxwell and Minerva were being held.

"Those ugly creatures are graffs," said Limaçon, sizing up the situation. "They're very mean, and nearly invincible. A graff has a third eye in the back of his head, so they make excellent guards. They also have five arms, and they pack a sword for each one."

"You said," noted Lee, "that they're nearly invincible."

"Right. See those red patches on their necks? They breathe through those, right through the skin, and they're very sensitive there. we call them the critical points. If you can hit a graff there, he's dead."

"Fine," groaned Lee. "And what, pray tell do we hit them with? A pop quiz?"

"No, these. I picked them up by the river while you were communing with the beavers." She handed Lee a hollow reed and a handful of hard, plump seeds.

"Pea shooters!" exclaimed Lee. She could not help laughing.

"You have these in the land of Topeka?"

"Yeah. I guess I was a pretty good shot once, but it's been a while. You sure you can kill one of those huge beasts with a little old seed?"

"I've done it," said Limaçon proudly. "I'm as well-trained as any soldier in my father's army. I may not know calculus, but I can hit the critical points on a graff!"

To illustrate her point, she put the reed to her mouth and launched a practice shot at a portrait of Epsilon the Small hanging at the far end of the corridor. It ricocheted cleanly off his chin.

She turned to Lee. "Let's do it!"

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As they crept cautiously into the courtyard, Lee noticed that there were three graffs patrolling, each with five swords at the ready. She felt the acorns remaining in her pocket: 3 and 5. "Only in an emergency," she promised herself.

They were actually halfway across to the tower before the first graff spotted them. Five arms shot forth and began slashing in Lee's direction.

"Wun, Wee!" screamed Limaçon, trying not to lose her mouthful of seeds. The graff wheeled to face the sound and was met by the first salvo from Limaçon's humble weapon. A flash of crimson from the neck, and it was over. The huge creature slumped lifeless to the ground.

Before Lee could compliment the shot, she saw the other two graffs rushing at them from opposite directions, swords slashing every which way. She would have to take one of them on herself. Popping a mouthful of seeds, she spat them out in rapid fire, hoping that at least one of them would find a critical point.

None did.

Just as the first of five swords was about to run her through, she heard Limaçon behind her. "DUCK!" She ducked. Like a miniature helicopter, a sword came whirling over her head and caught the charging graff squarely in the neck. It struck handle first, but it found a critical point. The graff collapsed like a house of cards.

Slowly, Lee turned to face her battle partner. "Did you learn that from your father, or have you been dating Rambo?"

Limaçon shrugged. "Sorry. I ran out of seeds on the second graff."

Again, there was no time to savor temporary victories. Someone would soon notice three dead graffs in the courtyard, and their mission was still far from over. They each picked up a sword and ran across to the tower.

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At the top of a long flight of steps, they found themselves in a corridor with five locked rooms. In the middle of the corridor was another graff, but this one had some sort of mechanical breathing device on its neck.

"Uh-oh," warned Limaçon. "This graff has no critical points!" It was also charging at them, with five drawn swords.

Lee had an idea. "Yo, Three-eyes! Come and get me! Yeah, you! Let's go, you big, ugly land squid!"

The enraged graff rushed at the insolent little intruder, sweeping right past the stunned Princess. Lee, with a good head start, bounded down the steps of the tower with the graff in hot pursuit. When she had led it out into the courtyard, she circled widely around, stood behind it, and waited. The graff stared at her with its third eye, seemingly unable to move.

"Can't turn around, can you, big fella?" taunted Lee. "You should study more calculus if you want to live in this crazy place. This is what happens to a graff with no critical points!"

Leaving the bewildered beast to ponder his monotone fate, Lee ran back up the steps to rejoin her friend. Limaçon had already figured out the purpose of the five doors.

"We have the same security gimmick in our castle," she explained. "Three of these doors are decoys. If you try to open a wrong door, it sets off an explosion that kills the prisoners in the real cells. The real doors in our castle are 2 and 4, but these might be rigged differently."

"They are," assured Lee. "The real doors are 1 and 5, and I'll tell you why. I think that Epsilon's magic calculus spells have changed Low Pitall in ways that even he doesn't understand. Everything seems to be vulnerable to some sort of twisted theorem from that magic book! Like those chains. That was your wish, not his magic, and yet the chain rule applied and we almost drowned. And that graff in the courtyard unable to turn around. Surely Epsilon didn't plan that, but the book made it happen! Now here we are, right after dealing with a graff with no critical points, and the next problem is to find a Max, your father, and a Min, your mother. In the book it will say to look at the endpoints."

Limaçon was catching on. "And the end doors are the endpoints."

"Exactly. Epsilon had to put them at the endpoints whether he knew it or not, because the book would have made him do it! It's calculus!"

Lee was so confident that Limaçon had to believe her. "Well, the Light Holder said that we would find Epsilon's weakness by understanding his strength. But can you open the two doors? We've already used the number 2 acorn."

"I think so," said Lee. She reached into her pocket and took out the 3. "I wish that the only doors left on these cells would be the middle three!" The acorn vanished, and so did the end doors. There was no explosion.

Limaçon saw King Maxwell emerge from the far cell. "Father!" she cried as she ran to his arms.

"Limaçon, darling," cried Queen Minerva, rushing toward the embarrassed Lee.

"Uh, no your majesty, I'm just a friend. Your daughter's over there."

Limaçon and her parents were ecstatic, but Lee reminded them that it was not party time yet. She herded them down the steps, pausing on the way down to peer out the window. She could see that the waters of the Lesser Than River were over the banks and still rising, thanks to the dam built downstream by the beavers. Perfect, she thought to herself. Let's just hope it works.

When they emerged into the courtyard, the first thing Lee noticed was that the graff with no critical points had still not changed directions. The second thing she noticed was that the courtyard was ringed with soldiers, each with a drawn crossbow aimed directly at her and her companions. She also heard Limaçon scream as the door to the tower slammed shut behind them.

But she might never have noticed the small elf standing directly in front of them if he had not addressed her directly.

"Well done, Princess," said the little man with bitter sincerity. Lee had not been formally introduced, but she knew from the monogram on his breastplate -- ETS -- that they were about to face their most challenging test.

"You have surprised me by coming this far in your doomed mission to save your parents," sneered the elf. "You must have some pretty strong magic, but you will soon learn that mine is stronger!"

The curious thing, thought Lee, is that he thinks I am the Princess.

"You don't scare me, little man!" responded Lee. "I know your calculus tricks, and I know how they can be turned against you. See how I have doubled myself -- and I was already twice as big as you to begin with!"

Lee winked at Limaçon, who was speechless. Epsilon the Small was furious.

"Yes, a good trick indeed, but to no avail if both your bodies should bleed from the wounds of a hundred arrows! Perhaps if I let you die slowly enough, you will explain this trick to me before your breath seeps away!"

Lee thought she detected a twinge of fear in Epsilon's eyes, but it was quickly gone. The hand in her pocket clutched the remaining acorn tightly as she searched the situation for a 5 upon which to base her final wish.

She was also keeping a very close eye on the length of his sleeves.

"My soldiers are less easily distracted than these dim-witted graffs," continued Epsilon. "I have decided that none of you are worth keeping alive, so you will all be dead in five minutes. Maybe now you would like to explain how you defeated my untimelock."

Lee was still staring at Epsilon's sleeve. She thought she had seen it earlier; now there was no doubt about it. His thumb, clearly visible when he had first addressed them, was now fully covered. The pleated trousers were bagging up around his ankles. His helmet was falling over his eyes. Way to go, beavers. We'll see who has five minutes.

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"You have five minutes," said the woman at the front of the room. The Light Holder kept his head pressed to the desk, but he noticed that the boy who had been sleeping had stirred at the sound of her voice and was now looking through the papers on his desk. There was a general increase in activity among the others as well. He had no way of knowing what would happen in five minutes, but he sensed that the time to switch back was approaching.

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Lee knew that it was time to make her move.

"Silence, Epsilon!" she thundered. Limaçon was aghast. "Soldiers! You should know that you are serving a master whose power is fast fading into oblivion. See how the power of High Pitall magic is causing him to shrink! Look! he disappears before your very eyes!"

By now, Epsilon was painfully aware that she was right. He was shrinking -- and fast. The soldiers dropped their crossbows and began muttering nervously among themselves.

"What sorcery is this?" screamed the panicky elf, growing more elfin by the second. "Stop this! I command you! No, I beg you! Stop! I'm disappearing! Help me! Hellllllllp!!"

His voice grew weaker, until he was the size of a cat -- a squirrel -- an insect. Then, with a final peep, he simply ceased to exist.

Lee turned to the horrified soldiers. "Now, who wishes to pledge loyalty to me and my parents, Maxwell and Minerva of United Pitall?"

Every soldier dropped quickly to his knees.

"Very wise, my friends. Now go out and tell the people what you have seen. Tell them that a treaty is at hand that will restore Pitall to one rule, and that there will be peace in this land forever. Then return with fresh food and horses so that my parents and I can return to our castle."

As the soldiers filed out of the courtyard, Lee congratulated herself on a pretty fine speech. Limaçon quickly pressed her for the details of the latest calculus lesson.

"Remember the beavers in the river?" began Lee. "Well, I used the number 4 acorn on them. You had said that Epsilon the Strong lived near a large delta, while Epsilon the Small lived near a small delta. Then I remembered from calculus that epsilons and deltas are usually related. With the two Epsilons coming from the same royal lineage, I figured that the relationship might be linear, with each Epsilon varying directly as his delta. I had the beavers build a dam across the mouth of the Lesser Than, which flooded the river basin and wiped out the delta. Sure enough, as his delta went to zero, so did Epsilon!"

Limaçon smiled and shook her head. "I'll probably never understand that, but it was clever of you to pretend to be me. I guess I'll be getting a lot of respect around here after you leave."

"Speaking of leaving," interjected Lee, "how and when will I get back to Kansas? I seem to have lost the magic slippers."

Limaçon missed the allusion. "You mean the boots? They were never the key to going back. It's all up to the Light Holder now, but since our mission is accomplished here, I guess the time for switching back is near. Oh, Lee, I'll miss you!"

They embraced and tried to hold back the tears, but tears flowed freely anyway. "I'll probably have missed graduation," sobbed Lee, trying to change the subject.

"Not necessarily," choked Limaçon. "The Light Holder said that hours here could be minutes there. You might have only been gone for one minute!"

"One minute?" sniffled Lee.

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"One minute." The woman was getting up from her desk now. The Light Holder sensed that a confrontation was imminent. Whatever the situation in Pitall, the time in this place was certainly right for the girl's return. He closed his eyes and reached out into the void, making the connection as easily as he had made it before.

He found himself kneeling in a courtyard with the Princess crying on his shoulder.

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Lee jerked her head up from the desk with a start. The class was still taking the AP test! Had this all been a dream? How long had she been out? Had she written anything in her test booklet? No, she could see that it was blank. Rats. She glanced over at Arnie Miller, who was now awake and grinning at her like a well-fed cat. He winked and gave her a thumbs-up sign. Lee rolled her eyes in disgust and dug her hand into her pocket for a pencil. Maybe there was time to salvage a few points on the free response.

There was no pencil, but her fingers closed around the acorn.

"Time!" exploded Mrs. Hampton. "Put your pencils down. Do not write anything more in your test booklets. When I have collected the papers, you will be free to go. Please leave quietly and do not disturb the other classes."

Lee closed the empty booklet and stared at the golden acorn in her hand. Staring back at her was the number 5. As Sarge Hampton approached her desk, she closed her eyes and wished.

She handed the booklet to Sarge.

"Hmmmph. Now here's a sure five!" intoned Mrs. Hampton sarcastically.

Lee looked at her hand, from which the acorn had vanished.

"Yes ma'am," she said in her best John Wayne voice, "I believe it is."

1995 Dan Kennedy