Creature Club (.doc, 62kb)

“ARRGGHH,” Arvin cries, as the bee lands at the edge of his soup.

He swats at it with his plastic spoon, but the bee buzzes away. His spoon upturns his flimsy Styrofoam bowl, spilling tomato soup onto the table, where it pools lethargically between his tray and his neighbor’s before dripping over the side in thick splatters.

But all eyes instead follow the bee, as it lands on Trundo’s cheese sandwich for a second, as he flings it onto his tray with disgust, his face contorting to match the sound of the ugly “Eww” emanating from his mouth. With a napkin he tears off the spot of bread and tosses it aside, then flicks it away from his tray, then tears off an adjacent, larger portion of bread, for good measure. He sits there, his sullen eyes demanding that someone repay him for the food that the bee has ruined.

But the eyes stay on the bee. No one notices the empty spot where Jamille was sitting. At this moment (because of her bee allergy), she is being whisked away from the scene by Ms. Firran, the female administrator on duty, whose blue flowered dress billows behind the two of them like a protective shield. At the same moment Mr. Brug, the heavyset male administrator, strides down the aisle.

The bee enters Bill-O’s territory. Amid a barrage of neighboring whacks, he calmly uses his mouth to take out the straw from his orange juice, raises his head with exaggerated slowness, and aims at the whirring little body. His fellow gangsters hold their breath in anticipation, and he fires a column of orange juice, which misses the bee entirely and instead lands on Mona’s hair. No one notices her as she dabs at the stain. Bill-O’s two neighbors pat him heartily on the back and laugh as they raise their straws, determined to carry out what Bill-O had planned to do. The pink straws fall in unison from their mouths as Mr. Brug appears on the
other side of the table.

“Move aside, I’ll take care of this,” he booms, the entirety of the Washington Post rolled up in his hand.

Allene jumps up to obey Mr. Brug’s command, while Louis scrambles away and peeks out from behind Mr. Brug’s protective bulk.

As Mona swallows her last piece of qinggangcai and moves towards the last spoonful of rice, she finds the bee has landed in her tin lunch container and is busy sucking at an oil droplet, and that the boys opposite her are chanting, “KILL IT KILL IT KILL IT.” Mr. Brug stands over her, frozen in place, his modesty preventing him from leaning over Mona and dirtying her tin container with his newspaper. She calmly puts down her spoon on her napkin and clangs the
lid shut.

The bee fuzzes frantically from inside the tin.

“Bee got owned,” Bill-O remarks, his friends joining in his snickering. Mr. Brug holds his hand out for Mona’s lunch container. She gets up from her seat without glancing at his outstretched hand, and walks down the aisle.

“I’ll dispose of that for you,” Mr. Brug says, as he follows her. She heads the opposite direction from the trash can. “SIT DOWN,” Mr. Brug says. All eyes are now on Mona; the lunchroom quiets until all can hear the bee.

Mona holds her ancient tin container with two hands as she strides confidently, like a waiter about to present the restaurant’s special dish. She puts it down on the windowsill and struggles to open the window. Finally she figures out how to turn the latch and pushes with both hands, almost falling out. Mr. Brug stands behind, not helping, watching.

Mona upturns the container and removes the lid. The bee buzzes out into freedom, then does a 180° turn and beelines for Mr. Brug’s nose.

“OOOWWW!” Mr. Brug shouts as he tears at his face, his butt hitting Mona and his windmilling feet hitting the cafeteria table, sending it rolling until it hits the neighboring one. The bee falls from Mr. Brug’s nose, hanging dizzily in the air. Mr. Brug’s newspaper comes around again, and this time there is no escape.

Mona fingers the jade Buddha around her neck and whispers a prayer as she walks back to her seat, but no one hears her as the students turn back to their friends and resume their conversations, all thoughts of the bee forgotten.

In this post we give a number theory proof of the following identity

(Prerequisite: knowing how factoring works in .)

Step 1: Approximate the sum with a finite sum.

We claim that

(Bear with me… this is the most technical part of the proof.) Indeed, we have

(Note we used the trivial bound 2 on each term in the first sum and telescoped the last sum.)

Step 2:

where denotes the number of divisors of that are congruent to modulo 4.

Indeed, the LHS equals

The term counts the number of multiples of less than , so counts the number of pairs where and is a multiple of . Summing the number of such pairs over instead we get . Similarly, .

Step 3: is the number of integer solutions to .

One proof of this uses Jacobi’s triple product identity (see http://www.jstor.org/pss/2323169). We give a proof using Gaussian integers.

Each solution to corresponds to a factoring over the Gaussian integers . Thus the number of solutions is the number of such that , or 4 times the number of nonassociated such that . (Two Gaussian numbers are associated if they differ by a unit , so are considered the same.)

Now factor where and are primes congruent to 1, 3 modulo 4, respectively. From knowledge of factoring in we know that

1. 2 ramifies in , that is, it is the product of two associated primes .
2. The split, that is, where is prime in and not associated to .
3. The remain prime.

Now if  and a Gaussian prime divides , then its conjugate must divide . Thus, since we have unique factorization in , each such , up to multiplication by associates, corresponds to a way of splitting the prime factors of into complex conjugate pairs. We note the following:

1. The factors are their own conjugates, so and must each get . If one of the is odd there is no solution. So we suppose they are all even.
2. It doesn’t matter how the prime factors of are split since they are all associates.
3. There are ways to split the factors of , since we can have either , or ,… or  divide . Thus there are solutions to up to associates.

Now if is odd, then the number of factors that are congruent to 1 or 3 modulo 4 are the same: Indeed, for every factor  having an even power of , we can pair it up with the factor , and these two factors are different modulo 4.

If all the are even, then we claim . We induct on the number of 3-mod-4 factors . For the case , all odd factors of are 1 mod 4, and has odd factors. For the induction step, note that each 1-mod-4 factor of is obtained by multiplying an even power of (there are choices since is even) by a 1-mod-4 factor of , or multiplying an odd power of (there are choices) by a 3-mod-4 factor. Hence . We get a similar formula for . Thus

by the induction hypothesis.

Thus in either case, the number of solutions to equals the .

Step 4: Counting lattice points in circles

From step 3, we now know

is the number of integer solutions to , i.e. the number of lattice points in the circle centered at the origin with radius excluding the origin. The area formula for the circle gives that there are around lattice points inside, so , which together with steps 1-2, complete the proof.

Note to make the above geometric argument rigorous, for each lattice point  in the circle, shade the square with opposite corners and . Then every point in the circle with radius is shaded (since the square that contains the point is entirely within the original circle) and no point outside the circle with radius is shaded (since the square that contains the point is entirely outside the original circle). The area of the shaded region is . Hence which gives the desired after dividing by and taking the limit.

(Note: The regular calculus proof can be found at http://en.wikipedia.org/wiki/Leibniz_formula_for_pi.)

Check out Online Math Circle, an awesome website started by Shri Ganeshram to provide free math education beyond the classroom, specifically, to help create a more level playing field in the math olympiad realm and offer a chance for students to discuss challenging and higher math. I met Shri when I was an assistant at AwesomeMath and I can vouch that he is dedicated to making his dream of free math education a reality.

We are currently in our third week of lectures, with many more to come!

On another note, here is my final paper for 18.784 (Additive Number Theory) class: an expository account on the finite field Waring’s Problem.

Let be variables. In this post we develop the Newton formulas relating the symmetric polynomials in the

and the power sums

(For example, when we have 3 variables, and . For convenience of notation we don’t restrict the number of variables; if we are working with variables we could just set . By convention .)

We note that the and are the coefficients of powers of of the following generating functions.

Why do these expansions hold? Each term in the expansion of is a term of the form where the are distinct, since we can only get a factor of from the factor . Since each always comes with a factor of , acts as a “counter” giving the total number of variables. Grouping the terms with we get all combinations of products of of the .

For the argument is simpler: expand in geometric series as shown above to get that the coefficients of are the th powers of the .

We want an identity involving and so we look for an identity involving and . First we turn the into : define

Now take the logs of both sides (thinking of the above as a formal series in ):

Now differentiate both sides.

The right-hand side is just (TA-DA!). Thus we get

Now we match coefficients of on both sides. Note

so the coefficient of is . To get the coefficients of the right-hand side, note that a term containing on the RHS comes from multiplying in and a term in . Thus the coefficients of are

which is the Newton sum formula.

These are the notes for the 18.784 (Number Theory Seminar) presentation I’m giving today. We prove estimates on prime-counting functions using elementary methods; in particular, that , the number of primes less than or equal to , grows “like” up to a constant multiple.

Given any two polyhedra with the same volume, is it always possible to cut one of them with a finite number of straight cuts and assemble it to form the other?

In the corresponding problem for polygons, the answer is affirmative, but for polyhedra, the answer is NO. We will show that a cube cannot be cut with straight cuts and reassembled into a tetrahedron with the same volume.

We will assign each polyhedron with an invariant, that doesn’t change when we cut it with a straight cut. This invariant should have something to do with side lengths, and something to do with angles between faces. Indeed, a cut may split a side into two, an angle into two. So we want to do something with adding up side lengths, or adding up angles.

But since there are two pieces of information, not just one, we encapsulate the information about the angles and the side lengths in a tensor.

Given two vector spaces V and W, we form their tensor product  as follows: First form the free product , which has a basis consisting of where and . Then quotient out by the relations

That is, the elements in are sums of elements of the form , and they are required to follow the rules above (and only those rules). In our case will represent lengths and will represent angles.

To any polyhedron define its Dehn invariant in to be

where is the length of and is the dihedral angle between the two planes meeting at . We’re considering the vector spaces over , that is in the tensor product we’re allowed to “shuffle” elements of in (1), but not arbitrary elements in . The second space is is modded out by ; this is just saying that we want angles differing by rational multiples of  to be considered the “same”.

(A) Invariance under cuts

If the cut divides edge into two edges and , then the term in is replaced by

in since and .

If the plane of the cut goes through , then the term in corresponds to the terms

in . Equality follows since the angle measures in and add up to that in : .

Finally, any new edge created is an edge in both and . The angles at in and add up to a straight angle so these terms contribute

Hence if can be cut into and by a straight cut, then .

(B) not rational

Suppose by way of contradiction that is rational. Write with . We have

Then is a th root of unity with

We show by induction that has denominator equal to when written in lowest terms. This is true for . Suppose it true for and ; write and with . Then

Since , this completes the induction step. However, since is a th root of unity, , contradiction. Hence cannot be rational.

An alternate solution is as follows: Suppose and are both rational. Let . Then is a root of unity and hence an algebraic integer; similarly is as well. Hence is an algebraic integer; since it is rational it is a rational integer. Hence we must have ; or . Therefore is not a rational multiple of .

(C) Dehn invariant of regular tetrahedron and cube

The Dehn invariant of a cube is because all angles between adjacent faces are , which is a rational multiple of .

Let be the length of a side in a regular tetrahedron. The distance from the midpoint of an edge to the centroid of an adjacent face (the foot of the perpendicular from the opposite vertex ) is . The distance is . Hence the angle between two faces is . The Dehn invariant is , nonzero by (B).

A regular tetrahedron and cube of the same volume do not have the same Dehn invariant. By (A), if a polyhedron is cut with finitely many straight cuts, the sum of the Dehn invariant of the pieces stays the same. Thus a tetrahedron cannot be cut with finitely many cuts and be reassembled to form a cube.

 

Putnam and Beyond, 926. We play the coin tossing game in which if both tosses match, I get both coins; if they differ, you get both. You have m coins, I have n. What is the expected length of the game (i.e. number of tosses until one of us is wiped out)?

Putnam 1990/B4 Let be a finite group generated by and .  Prove that there exists a sequence such that

1. Every element occurs twice.
2. or for . ()

Putnam 1990/B6 Let be a nonempty, closed, bounded, convex set in the plane. Let be a line and be a positive real. Let be the support lines for parallel to (i.e. they are the closest lines such that the entire figure is contained between them), and let be the line parallel to and midway between and . Let be the band of points whose distance from is at most , where is the distance between and . What is the smallest such that ?

Miklos Schweitzer, ? Suppose is continuous and

Let . Prove that

Solutions

PaB 926. Let . Let be the expected length of the game when the players have and coins. Then . Note . Solving this recurrence gives .

P1990/B4 Consider the graph with group elements the vertices and with a directed edge going from to if or . Then each vertex has indegree 2 (vertices point to from ) and outdegree 2. The graph is connected because generate . Hence there exists an Euler cycle. Such a cycle goes through every vertex twice, so the ordering of the vertices gives the desired sequence.

P1990/B6 (Sketch) It is easy to verify that by considering the equilateral triangle. Take .

First suppose that every three sets of the form intersect. Since these are convex (and compact) sets, by Helly’s Theorem the intersection of all of these sets is nonempty. This intersection must intersect (otherwise, it can be shown that there exists a support line that doesn’t intersect ).

So suppose by way of contradiction that there exist three such sets with empty intersection. Take two of the bands. The support lines corresponding to them form a parallelogram  with lengths three times the length of the parallelogram formed by the intersection of the two bands. The third band must intersect one of the main diagonals, say  at some point. By considering the length of the segment of inside the band, both support lines must intersect the main diagonal at the same side of the midpoint  of , say . But then this support line doesn’t intersect either or , a contradiction (draw the figure to see why).

MS Notice that , so

The lower bound is 0, since . We have

This is 0 if the numerator does not explode. Else using L’Hopital’s rule,

Hence both bounds in (1) are 0, and the integral equals 0.

Problem 1: (Putnam and Beyond, 296) Show that if a (noncommutative) ring with identity has three elements such that

1. 
2. for any , implies
3. but

then the ring cannot be finite.

Solution: Suppose the ring is finite. Let denote left multiplication by . By condition (2),  is an injective function from to . Since is finite, this must be a bijective function. Thus there exists such that , or , i.e. the right inverse exists; then exists.

Next note by (3), . Since is bijective, and are functions , must also be bijective. Thus by the same argument as before exist. Then conjugating by gives using (1); this is a contradiction. Thus is infinite.

Problem 2: (Miklos Schweitzer, 1987/1) The numbers 1 to N are colored with 3 colors such that each color appears more than times. Prove that the solution has a solution with distinct colors.

Frolina glanced at the teacher’s comments on her paper. A circled “F” and the words “Don’t write in green ink” stood mockingly at the top of the page. Scattered throughout were unhelpful comments like “More details,” “Insufficient description,” “Where does this fit with the rest of the story?” and “This is unrealistic,” in the handwriting of someone who made an effort to be neat but thought that exaggerated curls would help in this regard. What did Mr. Cavid mean anyway, “This is unrealistic?” It was supposed to be a fantasy story. He was ranting again about the class’s horrendous writing skills and their lack of effort. What did he expect anyway, assigning them to write a ten-page story in one day? Mr. Cavid’s handwriting ran inconsiderately over her flowery handwriting, even though the lines she skipped offered plenty of room for his comments.

She caressed her letters, and they popped out of the page, rustling their newfound wings. Mr. Cavid’s words looked like big fat red caterpillars; as she ran her fingers over them she could feel their spiky bodies rising from the page. They think they’re feeding on a bunch of leaves, she thought, well, guess again. The mass of green letters, each a cross between a fly and a maple key, rose from the page, overturning the bewildered caterpillars, who landed on the blank paper with a satisfying plop.

“Revisions of the paper are due tomorrow.”

Suddenly Frolina lost her concentration and the mass of green dropped haphazardly back into the page, their order jumbled beyond repair. The caterpillars melted back into the paper, turning into oval-shaped bloody stains.

Mr. Cavid was passing out their morning assignment. He caught a glance of the last fly-letter just before it dived into the paper.

“You must learn to harness your magic, Frolina. Once you’ve lost it you’ll regret all the time you’ve spent fooling around with it.”

Tosk glanced at the morning assignment. It was another impossible list of activities.

Name: Tosk____________ Grade:___/100

• Make one full-color 12” by 18” drawing of a castle. Include the following: At least 5 different types of building material.
• Create one sculpture of: life-sized operational vehicle. Take a photo of it.
• Film a ten-minute video of: stunts. Include the following: Swinging between trees, jumping to the roof, backflips into the river, swordfighting
• Film a twenty-minute magical display. Include the following: Fire, lightning, tornados
• Film two ten-minute movies, on a story with a well-defined plot. Must include dragons. You may work with up to 5 other students.

Submit all assignments by 12PM (including original tape for all film assignments). Must include all elements to get full credit.

He tried hard to finish all the assignments every day, but it was impossible to get everything done to Mr. Cavid’s satisfaction. The only time when he made something that Mr. Cavid liked was when he spent the whole 4 hours working on a picture of a space battle (complete with 5 different types of aliens). That day, however, he got a 19% because he had failed to get anything else done. Now he always made sure to time himself: 48 minutes for each activity, and he could go back to them if he had time. The twenty-minute magical display would invariably take longer than twenty minutes to set up.

He tried to smile as he picked up the video camera and headed outside. Maybe today he would break his previous record of 76%. His major obstacle was the second item…

As he watched the students of Fountain Secondary School start to work from his office window on the second floor, Mr. Cavid couldn’t help but feel jealous. He thought back to when he had lost his magic… His students didn’t realize this, but they would be thankful for all that they had created with their magic, that they would not have done unless he had given them so many challenging assignments. They never seemed to appreciate what he taught them.

Nobody had ever appreciated his efforts…

“That was the best picture I’d ever drawn!”

“Yeah, right, you put it away because you didn’t want anyone to see how bad it was. It couldn’t have been close to as good as mine.”

“I can draw much better than that!”

“Prove it then. If you’re so good at drawing then you can recreate it.”

He closed his eyes and concentrated hard on the picture but there was only emptiness, an emptiness that stretched forever and yet was cut off too soon by the scathing laughter…

Frolina was sprouting flowers from her fingertips again. Mr. Cavid shook his head. Beautiful, but not very creative; she’d already submitted the same magical display several times already. He’d have to take off five points for non-originality. So what if the flowers had helicopter rotors? That was only a minor variation.

If he were young again… he’d spent all his time using his magic again. How he yearned to be able to mind-draw again, to half-close his eyes and let the pictures in his imagination flow onto the canvas unmediated… He had a few pictures left from his childhood, but they were boring—pictures of his house or of sunsets or something, mostly blurry because he never really spent much time on mind-drawing. Why had he never drawn things like dragons or castles or, or something more creative? He’d taken it all for granted, just like these kids, until that fateful day… Why hadn’t he created something beautiful while he had the chance? Now if he wanted to draw he’d have to learn to use brushes and paint, and if he made a mistake he couldn’t just erase it with his mind…

He remembered his art teacher’s wrinkled hand, holding on to his left hand, leading him to an empty classroom. The teacher stuck a paintbrush in his left hand and bent down to look into his face.

“You’re in for a treat, I’ll teach you how to really draw…”

He held the paintbrush limply; he could only concentrate on holding in the watery heat welling up in his eyes.

He glanced away from his window, to the pile of framed 12” by 18” posters stretching from floor to ceiling, ordered neatly by date and author. Well, he had these. He pushed back his chair and walked over, climbed carefully on his stepladder and slid off the top three posters. Yesterday’s topic was landscapes, and he had thoughtfully required at least five types of terrain. There were mountains, oceans, forests, grasslands, canyons, glaciers,… He held one at arm’s length and scanned it… it was breathtakingly beautiful… He looked closer, but already knew what he would find; the trees looked somewhat blurry, and their colors were too uniform. But it was still more beautiful than any mind-drawing he had made… if only he could have that magic back again, he’d draw a picture better than any of his pupils! But he did make help make these pictures in a way; if he hadn’t required five types of terrain they probably would have just drawn a bunch of mountains.

Initially he’d planned to scan all of the pictures so he could keep a copy, but that couldn’t capture the texture of the dried mind-paint, or the way they captured the light… And he had gotten too behind on digitizing them. Oh well, no one had asked for them back yet. He promised himself that he would give a picture back if a student asked. He really needed to buy a higher-resolution scanner.

He slid the posters back and sat back down in his chair to watch Illusen conjure up a fireworks display and a lot fantastical flying creatures.

Illusen stayed at the back as the students lined up to submit their tapes and posters. She stayed behind Wylon; she vaguely noticed that he was gripping his posterboard coiled up tightly.  When everyone else had left, she cautiously glided over to the desk, poster in one hand, and video camera in the other.

“Mr. Cavid…”

“Yes?”

“I was taping my assignments and… and… the video camera ran out of batteries.”

“You what?”

“So I only got half a minute and the rest didn’t get recorded.”

Mr. Cavid stared at her, hand grasping the table until his knuckles were white, his face slowly reddening.

“I noticed you looking through the window, sir, so I was wondering whether you could just grade me based on what you saw…”

Illusen blinked her eyes, and Mr. Cavid’s face was getting greener. She stifled a giggle.

“NO!” Mr. Cavid shouted, jolting Illusen out of her self-made hallucination.

Noticing that he was at the edge of his seat, he sat back before continuing, “I made it clear that there’s no credit without the tape,” he said. “You must do it again this afternoon—I mean, if you want credit, you can tape the assignment again this afternoon, exactly like you did it this morning, and I’ll give you credit for it.”

She turned Mr. Cavid’s moustache purple before she got the courage to continue.

“I think I’ll just take the zero, it was too tiring to make all those fireworks.”

“No, no, I’m sure you can do it again. I know, you can integrate it into your movie assignment this afternoon.”

“All right,” Illusen said wearily, relaxing her magic, turning Mr. Cavid’s moustache back to black.

Mr. Cavid hit the microphone forcefully with his left hand and pulled it off its stand. The lunchroom quieted down.

“One student forgot to change the battery in her video camera and did not capture her assignments on film,” he started.

Illusen would have made his nose appear to grow longer, except that she felt daunted with having to recreate her morning displays again in the afternoon. She lowered her head and concentrated on slopping mashed potatoes into her mouth.

“She did not get any credit for her assignment. I want to make clear that you must make sure to capture all your assignments on film! Because otherwise you make your magic displays and what will you have to show for it? Who will you know what beautiful things you made? No one! And when you lose your magic… you’ll wish that you had captured that moment.”

His eyes lingered on the students for several more seconds before an abrupt whine from the microphone jerked him back. He awkwardly fitted the microphone on its stand and left the room.

Sekrid reached over Illusen’s tray for the bottle of ketchup, accidentally knocking over her cup of orange juice.

“Sorry,” he mumbled.

 “What, aren’t you going to make the ketchup grow wings and fly over like you usually do?” Frolina teased.

Suddenly all eyes were on him.

Sekrid let the question hang in the air before he sputtered, “I’m just trying to conserve energy, you know!”

Several heads from the next table looked up.

“I mean, don’t you all feel like it’s harder and harder to finish all these assignments?”

“I didn’t know you actually worked on the assignments, Sekrid,” Sarsa said as she stuffed the last bite of her cheese sandwich into her mouth, but no one paid her any attention.

“Yeah, I don’t like to admit it, but it’s true,” Tosk said dejectedly, “I only got done three and a half of them. I totally gave up on the sculpture assignment.”

“Really, I didn’t think mine was that bad. What was yours?” Sarsa asked.

“No, really, what’s the point of all this?” Sekrid interrupted, “We do all these creative assignments and Mr. Cavid keeps saying we’ll all lose our magic, so why are we even bothering? It’s not like we even get our tapes back! Who will care what we did?”

“We all will!” Frolina interjected, “Maybe we won’t have tapes of what we did. But we’ll remember that we did it, and we’ll remember the magic displays that our friends did, and we’ll remember working on these movies together.”

“Exactly!” Sekrid said. “We’re not doing this because to get better at anything, we’re only doing them for ourselves and for each other! And this is obviously not the best way to be using our magic. When was the last time you made a mind-drawing just for fun? When was the last time you had a friendly magical display contest? When was the last time you willingly helped your parents with a bit of magic?”

He looked at Sarsa, who hastened to pick up crumbs from her tray. Everyone knew that Sarsa used her magic to save her ailing grandmother when she was in primary school.

“We’re doing it for a grade,” Tosk contributed.

“Yeah, and that’s why school is so screwed up,” Sekrid said as he plunged his fork into his sandwich.

As Mr. Cavid took lunch in his office he considered what to give as an afternoon assignment. He’d assign them to film a full-length movie. He had originally wanted to turn their ten-page stories into movies, but after reading them he had decided they were mostly meaningless. He glanced at the pictures from the morning, dissatisfied. Tosk had done pretty well, but his castle was just made of stone, with one turret made of brick and one made of slate and one made of metal and one made of clouds. Obviously they had been thrown in there just to satisfy his requirements. The metal turret clashed with the surrounding peach-colored stone. The picture had been beautiful in spite of Mr. Cavid’s instructions, not because of them. Everything else was drawn from Tosk’s own inspirations. Mr. Cavid was hoarding artwork but he couldn’t put a stamp on any of it; none of it was his…

He had an idea. He’d write a plotline for the students’ afternoon movie. They’d put in the special effects with their magic but the basic idea would be his. He’d be able to say that he came up with the idea for the movie.

Put those brain cells to work, turn on those creative juices! He racked his brain. What message would he like to send?

He typed (with his left hand):

Make a movie on the following story: Once upon a time there was a kingdom of magic. A wicked witch took away the magic from all the adults, so that only the kids had magic. But they were inexperienced so they sought the help of a wise king who was knowledgeable about magic (although he had lost it due to the witch). Under his training, the children learned to use their magic. They went on an adventure to defeat the witch and release magic back into the land. The adventure should include the following: poison/ lava rivers, a princess, a dungeon, and fireworks and flying creatures. NO CREDIT UNLESS ALL ELEMENTS ARE INCLUDED.

What a beautiful idea! He needed to make his own movie ideas more often. And he could take credit for the both the general outline and the major plot ingredients…

“This is utterly ridiculous!” Sekrid said, “This time he forces us to do a movie on some stupid story he made up!”

“At least that means we don’t have to think up our own one,” Frolina said.

“Well, I don’t know about you but I don’t intend to just follow his storyline,” Sekrid said.

“We have to if we want a good grade…” Tosk piped up.

“Can’t we just have some fun for once? How about this instead: Once upon a time there was an evil king. He has video camera robots all over his kingdom who spy on the children. Every time a robot sees a kid who can do magic, the king takes the kid away. He forces these kids to perform for him, and when they tire out, he uses this machine which sucks the magic out of them. And he uses his store of magic to make himself live forever. So one day these kids manage to evade the king and they run away to this wise witch who teaches them how to defeat the evil king.”

“And the king is Mr. Cavid!” Sarsa suggested, surprising everyone.

“I like this idea,” Frolina said. “Let’s do it!”

“Okay, let’s go!”

“Wait, aren’t we going to make up the script first?” Wylon asked.

“We’ll just make it up as we go along,” Sekrid said dismissively, “Who wants to be which character?”

First scene: Tosk, Sekrid, Frolina, and Sarsa sat in the clearing. Illusen hid in the forest, orchestrating the illusion of video cameras hanging in the air. They drifted around aimlessly, carried by the wind.

“Action!” Wylon called as he flipped the switch on the real video camera.

“We can’t go anywhere without these video cameras watching us,” Tosk said, “Last night several got in our house and Mom had to throw them out, then she got in trouble with the police for doing that.”

“They’re just like flies…” Frolina said, staring at one intently. Her mind felt for the essence of the illusions, tracing the source back to Illusen. She formed up an image of a fly. Illusen seemed to hear, and the cameras suddenly grew wings and buzzed in circles around them.

“What just happened?” Sekrid asked, afraid.

“I—I don’t know…” Frolina said. Suddenly her eyes grew large and her tongue darted out, thin and skinny, wrapped around a camera, and brought it back, seemingly of its own accord. She felt the metal breaking apart in her mouth—Illusen always made her illusions very realistic—but after she coughed once it dissolved in her mouth. She stuck out her tongue again; it looked normal.

“You—you can do magic!” Tosk said in wonder.

The drone of the video cameras increased as hundreds of them popped out of thin air (slightly beyond the view of the camera) and descended as a wall upon the children.

Tosk sliced his hand through the mass of cameras, leaving behind a trail of fire and sizzling cameras. Sekrid swatted a camera coming towards him, and suddenly blobs of ink were coming from his hand, hitting the camera lens; they knocked into each other in confusion.

A few seconds passed; then they looked at Sarsa expectantly.

“RUN!” she shouted, and they dove into a gap in the cameras, running towards the forest.

The cameras’ lens turned into hooked beaks; their plastic fly wings turned into black feathered ones, and claws erupted from their bottoms as they swarmed towards the children.

Wylon followed the four actors, struggling to keep up. When Sekrid had asked, he’d volunteered to be the cameraman again, as well as edit the footage afterwards. None of the others knew it but Wylon had lost his magic already. One morning he was trying to mind-draw; he closed his eyes and thought… and when he opened them there was nothing on the canvas, nothing at all. He’d been ashamed to reveal it to anyone so when everyone went outside for morning work he just stayed in the empty classroom. No one else but Mr. Cavid knew, and he had been quite nice, giving him full credit for his half-finished drawings—done with pencil and crayon—even though they couldn’t quite compare with the others’ mind-drawings done in the same amount of time.

He’d kept silent during the conversation at lunch. The loss of magic was an uncomfortable topic, and he didn’t want to reveal himself. Soon the others would lose their magic too… the average child lost his or her magic sometime around graduation from secondary school. He felt sad for them, and hoped they would enjoy it while they could. If only they would run slower; then the video wouldn’t be so jerky.

He followed them to the creek, which, under Illusen’s magic, had turned into a river of lava.

A vine shot out from Frolina’s hand, gripped onto the opposite branch, and she swung over along with Sekrid. Tosk followed, by temporarily growing wings, leaving Sarsa alone on the opposite shore.

“Come on, I’m sure you can do magic too, just try!” Frolina shouted.

Why are they doubting me? Sarsa thought, then realized that they were just acting, and that she hadn’t done any magic in the movie yet. She’d hesitated to attack the video cameras because she couldn’t think of a good trick—everything she thought of just seemed to cliché, like she’d done it a thousand times for Mr. Cavid already. Now it was the same thing again. Maybe she could make a giant fish jump out of the lava river and she’d ride it—no…, that was totally unrealistic; after all it was a river of lava, wasn’t it? The video cameras were swarming around her; her friends were making plans to rescue her. Unsettled by the buzz of expectancy in the air, her mind latched onto what she knew would work. She raised her hand and a vine shot out, wrapping itself around a branch on the opposite side; she swung over, feeling bad for copying Frolina.

“Good job, I knew you could do it…”

Suddenly lava erupted from the gorge, melting the video cameras.

Sarsa tagged behind the rest of her friends. The gorge had vanished, turning into the creek once more; Wylon jumped from the last stone in the creek to the bank and followed.

She felt uncomfortable with the lack of planning. The beginning had just been the four of them suddenly discovering they could do magic and then running. There was no introduction, and if they were supposed to be in a land where magic was rare, what was the chance of all four of them discovering magic all at once? She needed time to think up more magic tricks for the rest of the movie. How were they going to defeat the king? They needed a more elaborate plot…

Before them was a log cabin with a field of colorful spotted mushrooms growing in the front yard. An unnecessarily long and snaky grass path led to the front door; the path was marked simply by the absence of mushrooms.

Sekrid touched the tip of a mushroom; suddenly it shot a jet of spores into the air that exploded into silver fireworks. The other mushrooms followed—one shot up eggs instead; they hatched into dragons which disappeared when they left the camera’s view—and they watched until the show subsided. Frolina stepped up to the front door to knock but a voice came from inside before she did.

“Who is it? I’m making FUNGUS SOUP!”

“We need your help,” Frolina replied.

The door creaked open, revealing Illusen clad in a black hat and black robes.

“Come in, come in, then…”

The door shut behind them. Illusen resumed stirring a large pot in the middle that was giving off green steam.

“No one’s come here since I surrounded my cabin with a lava river,” Illusen said, “So, what brings you here?”

“CUT CUT!” Sekrid shouted, “Wylon’s still outside.”

Suddenly the pot, and the mushrooms outside the window, disappeared, Illusen’s clothes faded to brown, and her wrinkles melted into her face. The cabin stayed but it now resembled the empty shed that it was in reality. Wylon came in.

“Sorry about that…”

Sarsa took this opportunity to ask how they were going to finish the rest of the story.

“I’ll turn the school into a castle,” Illusen said, “and I’ll make a copy of Mr. Cavid, with a crown on his head. Just try and find him. We’ll improvise for the rest.”

Tosk took the time to voice his concerns as well: “We’re going to be in trouble if we submit this video,” he said.

“Who said anything about submitting the video?” Sekrid said. “We’ve made too many movies for Mr. Cavid already; it’s time we made one for ourselves. We’re going to keep it. Just tell him the video camera ran out of battery. So what if we get no credit for today?”

“You mean I did all that work for nothing?” Illusen said in a small voice.

“What about our grade?” Tosk demanded, “We’re going to have to do a whole other movie after this one! I don’t want a zero for today!”

Illusen looked at Sekrid pleadingly. She didn’t have the energy to go through all of this again.

“How about this,” Wylon said. “I’ll find a way to edit the movie so that it fits Mr. Cavid’s criteria. We’ll keep the original version, and he gets what he wants.”

Sekrid reluctantly agreed, and the movie continued.

“Lights”—Illusen stood up, and her witch robes appeared around her again; the pot entered with a clang and the mushrooms silently invaded the field outside—“camera, action!”

“To help you on your quest I will give each of you one item that will help you concentrate your magic,” Illusen rasped. “For you, Frolina, the Seed of Healing,” she said, plunging her hands into the pot and picking out a warm, pulsing green seed to hand to Frolina.

“For you, Tosk, the Time Orb,” Illusen continued, producing a gold sphere with one section sliced off to reveal a clock face, “When time becomes your enemy.”

“For you, Sekrid, the Flying Eye,” Illusen continued, plucking an eye out of the cat that suddenly appeared in her hands (who promptly grew another one), “When you need to see beyond yourself.”

“For you, Sarsa, the Grab Bag,” Illusen continued, teasing a small pouch out of her pocket—it looked rather like a strawberry, the part above the string green and leafy, the part below red with black dots—“When you are out of ideas.”

“CUT CUT!” Sarsa interrupted. “You’re not supposed to know our names.”

“Don’t be silly, Sarsa,” Frolina said, annoyed, “Witches always know people’s names without asking them.”

“Okay, start again…”

“Now go! Do not delay…”

Mr. Cavid watched the children return from the forest. The lava river and the fungus fireworks were fabulous—though to his annoyance, the fireworks were silver, not gold like the one in the morning. Well, he could take off a few points for that. This time Wylon was filming; he was always the best cameraman; he could rest assured that all of it was caught on tape. Though, the illusions often seemed somewhat blurry on tape—well, maybe this time would be better; anyway, it was good enough. They seemed to have the plot mixed up too; why were they visiting the witch first? And what was the deal with all the video cameras? They’d probably taken some creative license with his plot—maybe the video cameras were sent by the evil witch and they were under the witch’s spell.

A moat now surrounded the school building and the front door was replaced by a drawbridge. The kids were coming inside. He opened the door to his office and found the hallway paved with red carpet and tapestries lining the walls. Illusen had outdone herself this time; perhaps he could ask her to keep the school like this; that would be a huge improvement.

Standing to one corner, unmoving, was an illusion of him (though older), with a crown and a scuba diving pack on his back. They’d made him into the wise king! He felt so proud, that his students regarded him so highly… maybe they complained about him occasionally, but it was nice to know they appreciated his instruction. Why didn’t he act as the king instead? He took the gold crown from his doppelganger’s head and placed it on his own. He didn’t know what the scuba diving pack was for so he ignored it. Finally, he could star in his own movie—the same one he wrote the plot for!

After they entered the castle, Sekrid sent his magical eye to scout ahead. That was a brilliant idea—Illusen made the object but he channeled his own magic into it to operate it; he must ask her how she managed to do that.

He was vaguely aware of walking along with his friends, and some commotion going on there with some guards, but his mind was with his other eye outside Mr. Cavid’s office. None of them had ever seen the inside. Was it true that he stored all of their work inside? The door opened and Mr. Cavid—the real one—came out. He made sure to hide on the other side of the door, and flew in as Mr. Cavid left.

There was the pile of posters—the ones that they had been drawing every morning for a year and a half!—stacked up from floor to ceiling. A shelf of videotapes covered the wall on the other side of the room; strips of tape stretched across the wood panels, acting as labels. A bulletin board by the window was filled with photos arranged in a rectangular array, pinned to the board with thumbtacks. Some he recognized as ones they had taken during their various assignments; others looked as if they were taken from Mr. Cavid’s window.

Nothing here was Mr. Cavid’s; he had stolen them all of them from the kids; he’d pinned their memories and accomplishments onto the board like butterflies and made everything his. Sekrid’s eyes scanned the shelves. He’d made sure that Wylon had made copies of the past few videos they’d made as he was editing them, but he didn’t have the older tapes or the posters. One of the piles held his masterpiece on a cityscape—Mr. Cavid had required 5 types of buildings but he’d put in 43, he thought with pride. But no, he had only an eye; he couldn’t lift it out of the middle of the pile. No! Now he knew all these were here but he had no way of seeing any of them…

They’d gotten past the guards, avoided a bunch of traps, and saved the prisoners from the dungeon (though they had to stop a few times to plan ahead—for example Frolina had to lay the traps that they would cunningly avoid—since they couldn’t leave all the work to Illusen), one of which was a princess from a faraway land who was explaining to them how to kill the evil king: he relied on the pack of magic on his back to keep him alive, as he was already 250 years old.

“There he is!” Tosk shouted; Mr. Cavid came around the corner, wearing the crown.

“Get him!” Frolina shouted.

“Where’s his pack?” Sarsa asked, “He’s alive but he hasn’t got his pack!”

“Hi, children!” Mr. Cavid said, “You’ve obviously fallen under the spell of the evil witch! Here, let me help! Abracadabra Ala—”

“No, that’s not him!” the princess shouted.

“She’s trying to trick us!” Frolina said, “He obviously has the crown. And he’s trying to put a spell on us. Take this!”

She put both her hands forward; vines twisted around her arms and shot out at Mr. Cavid. The vines grew a toothed mouth that latched on to his face. Tosk shot a stream of fire at Mr. Cavid; his shirt caught fire; he rolled on the floor and the fire caught on the wooden floorboards.

“NO NO STOP STOP STOP”

 Illusen breathed heavily, the world around her swirling like a whirlpool. One moment the layout of the school was clear in her mind, the individual tapestries and carpets; the detention room-turned dungeon, the back stairs-turned secret passage; she was shouting through the voice of the princess, then everything was fading; she clutched at it, with the helplessness of trying to imagine the continuation of a dream when she was rising to consciousness; there was the sound of fire, fading away, replaced by the sound of birds and she knew she was in the shed, she was afraid to open her eyes, but the world was still spinning…

As Mr. Cavid was engulfed in flames he was reliving his childhood, it was the day that he had lost his magic Recess, the kids were running around on the playground. He was sitting on the swings, pushing his feet hard so that he was almost horizontal. He glanced down at his classmates running around below, playing duel-tag, then to his backpack, where he had placed his rolled-up mind-drawing, positioned so it would not fold. It was almost finished; just one more day and he could show his masterpiece to everyone. But then, just as he was reaching the back of his swing, one of his friends ran over; the one pursuing him shot a stream of fire from his palm; the fire engulfed his backpack; he let go of the swing and jerked forward, falling to the ground. A pain shot up his leg, woodchips pierced his knee, but he was crawling desperately towards his backpack; he reached through the flames, oblivious to pain, the zipper was white hot, he ripped at his backpack desperately, the fabric tore, he yanked out his drawing, already unrecognizable, burning away into nothingness, he clutched at it, even though his hands were on fire, his shirt was burning away, he was rolling on the floor in agony as the world receded…

As the hallway caught fire, dusty, cracked yellow paint swept across the sparkling white paint of the corridors and the portraits of Mr. Cavid, in a smooth stroke like a huge paintbrush was operating in reverse, sucking all the magic away. The only thing missing was a clock striking midnight.

“No! That’s the real Mr. Cavid! See what you’ve done!” Sarsa screamed as smoke spread in the corridor and the shrill fire alarm went off.

“No, no,” Frolina said weakly, her vines melting away.

“Quick, Sarsa, do something!” Tosk shouted, immobile as the flames raged behind him, “Save him! Save him like you saved your grandma!”

She searched her pockets frantically and came out with the Grab Bag. She jerked open the string and plunged her hand in—it was larger on the inside. She felt something like a wooden post, stuck in mud, and pulled, but it wasn’t coming out. Something was pulling at her mind; the more she pulled the more tired she was getting.

“Your amulets! We need to use them!”

Tosk flipped the switch on his watch and suddenly the smoke was spreading towards them in slow motion; Mr. Cavid was flailing around on the ground slowly; Tosk felt as if he was moving through water but the three of them were moving around faster than everything else around them.

Frolina placed her seed on Mr. Cavid’s stomach—he was now moving only imperceptibly—and imagined her strength flowing towards Mr. Cavid through the seed, Mr. Cavid, the teacher she’d always hated, whose right hand was a repulsive, blackened stump, who gave her F’s; the seed was sprouting, its shoots growing along Mr. Cavid’s skin, but no, it had spikes on it…

“We need to combine our powers!” Tosk yelled, “Here, everyone hold hands… You too, Wylon. Where’s Sekrid? Sekrid!”

Suddenly smoke obscured his vision, coiling between his eye and the posters, the photos, the tapes.

“No!” he screamed with a mouth that seemed foreign. He tried to grab the poster but he was just a flying eye butting against the wooden frame. There was a flash of red, and suddenly he was awake again, hammering against the wall, grabbing smoke with his hands, coughing.

“Sekrid!”

He searched his mind for the connection, to that eye, but it was gone.

“Help us!”

But now he had his hands and legs; he tried to run towards the office but he felt oddly detached from his body; he hardly seemed to move. A wall of fire blocked his way down the hallway. He ran towards it, coughing; he’d use his magic to blow the fire away, but no, nothing was happening, he stumbled and fell…

Frolina, Tosk, Sarsa, and Wylon put their hands together over Mr. Cavid’s stomach. One of Sarsa’s hands was still grabbing at the stubborn item in her Grab Bag; Wylon’s video camera, still on, hanged from his neck, pointing downwards, like dead weight.

“Just use whatever magic you have,” Tosk said desperately.

“I’ve already lost my magic,” Wylon said, looking away.

“It doesn’t matter, just, just believe!”

The sprouts from the seed stopped growing; blood was oozing from the spot on Mr. Cavid’s stomach where the spikes had punctured. Sarsa concentrated her energy on their hands; what was supposed to happen? A surge of light, maybe, some tingling sensation? There was nothing but their hands stacked one on top of another, wavering in front of their sweaty, crouched bodies.

No, nothing is happening, Sarsa realized, and then the fire alarm blasted again in their ears, a prolonged pain in her ears. She lessened her grip on the bag and felt the tension in her body released. The smoke lunged towards them, and the bag dissolved into wisps of air; Tosk’s clock blew away from his hand like sparkling sand, only the wilted sprout remained on Mr. Cavid’s dead body. The fire alarm resumed its rhythm.

Sarsa shared a look with Tosk; he lowered his hand from the pile. They all knew what had happened, even before Sarsa spoke.

“Our magic is gone…”