We had Mr. Boyce for math again. He was a little spacy and had trouble keeping the classroom under control, but he knew his stuff. He could take the boring old math book that all the classes used, and show us some of the theory behind the problems that was pretty cool. When Nora complained the year before about the tedium of having to break multiplication problems into sets of hundreds, tens, and ones, Mr. Boyce told us about a culture where very complicated math is done with the fingers, a sort of base 5 approach, and other cultures that use 12 or 20. He showed us how computer code is written in base 2, all 0s and 1s, because computer circuitry is based on millions of simple on‑or‑off switches.

Once he brought in an abacus, and challenged us to a race between his abacus and our calculators, doing big multiplication and addition problems. He could work the abacus faster than most of us kids could punch our buttons. He showed us pictures of Chinese accountants in a bank, using abacuses to do the bank’s bookkeeping, and told us the abacus had been around unchanged for over a thousand years.

Another time he brought in a really cool article about the concept of zero and how it had revolutionized math. He bought in another article about a culture that doesn’t have the zero but yet can do amazingly accurate and quick math calculations without needing paper by a system that I couldn’t even understand.

Anyway, math was our first class of the day. We were some of the last into the room, because of Bob’s slow shuffling, and the only seats left were at the very front. I didn’t much care for that, but it suited Bob just fine. He was usually in the middle of any class discussion, unless it was something uninteresting to him.

After the usual rigamarole of taking roll, giving us a syllabus, and going over class requirements, Mr. Boyce announced that we would begin the semester by studying probability theory. He brought out dice, game spinners, and a few quarters, divided us into groups, and had us try to figure out the probability of any given result with a toss, spin, or flip of each. He also had us figure out what cumulative results were likely to be.

Class discussion was animated, except that I was in a group with a couple of knuckleheads who didn’t have much to say. They just kept spinning the spinner, trying to see how long they could get it to spin. I finally gave up on trying to have an intelligent conversation and turned to watch the next group. Bob and Tony had gotten into an intense argument about whether, if heads had come up nine times in a row, probability was higher, lower, or fifty‑fifty that the next toss would be tails. They got louder and more insistent. Finally Tony threw his textbook at Bob. It skidded across Bob’s desk and hit him in the chest. Bob jumped up and gave Tony a shove. Tony fell out of his seat and ended up sprawled in the aisle, leaning against Nora’s chair.

Bob and Tony spent the last twenty minutes of class in the principal’s office. When I got to my next class, which was science, they were both there in their seats, glaring at each other.

Mrs. Grissom taught honors science, and Bob pretty much worshiped her. She was tall and had a beautiful smile, and was always calm and polite. None of us would think of being rude to her the way Bob sometimes was with teachers that didn’t seem to know as much as he did. She always knew her stuff, and was very organized and methodical, in contrast with Mr. Boyce, for whom it was an archeological expedition just to find his grade book. She also had many cool stories about NASA experiments and discoveries, and how they had changed our daily life through new technology. She used to work for NASA, and had actually married an astronaut’s nephew, Elmer Grissom. His uncle “Gus” Grissom had died in a tragic accident on the launching pad some years ago.

This year in science class, we were studying the scientific method. After we were sitting down and had taken roll, Mrs. Grissom said, “Well, I understand that there was some serious academic debating going on in your last class. Could someone summarize what it was about? Yes, go ahead, Bob.”

Bob scowled at Tony. “Mr. Boyce was teaching us about probability, and the question came up, if you flipped a coin and it came up heads nine times in a row, what was the probability that it would come up heads again? I figured it was less than fifty‑fifty, because the probability is very low that you would get a coin to come up heads nine times in a row, and ten times would be even less likely. At best the chances would be fifty‑fifty, because each time you toss a coin, it’s fifty‑fifty that you’ll get heads or tails. But this bozo,” Bob pointed an accusing finger at Tony, “was saying that it was more likely to come up heads.”

Tony stood up, menacingly. Mrs. Grissom said calmly, “Please sit down, Tony. There is some sense to what you were each saying, because if you have heads nine times in a row, it establishes a pattern, so our human logical tendency would be to guess that the next throw will be similar. A gambler would probably call heads again if he seemed to be on a streak. On the other hand, nine in a row is a mathematical improbability. Does anyone know what the chances are of getting heads nine times in a row?”

Nora held up her hand. “Mr. Boyce said they would be one out of  512.”

Mrs. Grissom smiled. “And how did he come up with that figure, Nora?”

Nora wrinkled her nose and pushed a brown curl back over her ear, the way she did whenever she was thinking. “He said, if the chances were one out of two for each throw, then it would be one out of two, then one out of four, one out of eight, one out of sixteen, and so on, until you’ve divided by half nine times.”

“Right,” said Mrs. Grissom. She wrote on the board: “1/2⁹  = 1/512.”

She continued, “So, strictly mathematically, what would the chances be that you might get heads ten times in a row?”

“One in 1024,” I said.

“Okay. Now, if we look at this equation, assuming you’ve gotten heads nine times in a row, what is the likelihood that the next toss will also be heads?”

“One out of two,” said Junia. She was Nora’s sidekick, a redheaded girl with freckles. We called her “Junior” most of the time. She and Nora were rivals to Bob and me, when it came to grades and reports and stuff, but she was actually pretty cool, not snooty like Nora.

“That’s right. Even though it’s very unlikely to get ten heads in a row, the difference of likelihood between nine and ten in a row is one in two.” Mrs. Grissom wrote “1/2⁹  x 1/2  = 1/2¹⁰ = 1/1024,” on the board.

Bob snorted. “See, I was right!” he said to Tony. “I told you it was fifty‑fifty or less, not more, you knucklehead.”

Tony glared and picked up his book menacingly. He glanced up at Mrs. Grissom and pretended to look something up in the index.

“Now, what does probability have to do with scientific research?” Mrs. Grissom asked.

Nora held up her hand. “The textbook introduction says that in demographic research, a random sample has to be used so that the results reflect the whole population.”

Mrs. Grissom said, “What is a random sample?”

A kid with a blond crewcut, whose name was Eddie, piped up. “It sounds like it would be some kind of collection that you choose by chance, like just grabbing a chocolate candy out of one of those Valentines boxes and you always end up with something that gums up your teeth.”

Nora said, “A random sample is a group chosen at random from the whole population. It has to be big enough that it will be a similar mixture of different kinds of people to what the population is.”

“Right,” said Mrs. Grissom. “What are some ways to get a random sample?”

I held up my hand. “In the book Cry Wolf, Farley Mowat was doing biological research in the Arctic, and he had a hoop that he would throw over his shoulder, and then he would catalog all the species he could find inside the hoop wherever it landed. Until an Inuit threw it for him and it landed in the lake.”

Hiroshi, an Asian‑American whose folks were from Japan or somewhere, said, “Telemarketers and guys who do polls have computers that choose randomly among all the phone numbers. They assume that most people have telephones and that if they call enough numbers, it’s, what do you call it, representative. Only they usually call at supper time, and my folks hang up on them or put my little brother on to talk to them. He’s three.”

The conversation went on like this for a while, and then Mrs. Grissom gave us an outline of the scientific method and explained it. “Class, I want you to study this tonight, and we’ll go over it again tomorrow. Now, the science fair will be in about two months, and you need to have your proposal submitted in about two weeks. You can work in pairs this year, so be working out who your partners are, and if anyone hasn’t found a partner by Friday, we teachers will match you up.”

Tony raised his hand. “Mrs. G, do we have to have a partner?”

She looked at some papers on her desk. “It doesn’t say, but I don’t think so. I’ll check on it.”

Tony said, “I’d rather work alone. Last time I had to do all the work.” He glared at Doug, who was one of the smartest kids in the school but was also one of the laziest.

Doug grinned back at him. “I did the thinking, remember?”

Doug had come up with a really great idea the year before, and Tony had paired up with him thinking it would be an easy way to be a winner. Doug wrote the proposal and the introduction to the project, but then sat back and let Tony do most of the actual research, build the display, and write the technical part of their report. Doug insisted on writing the conclusion, which bugged Tony no end. They did really well, being one of three school teams that got to go to the regional science fair, but Tony grumped the whole time, especially when Doug got to present their project at the school assembly.

The bell rang, and I turned to go to the gym for PE. Bob said, “I’m going to go see the nurse about an exemption.”

Gym class was not Bob’s favorite thing. He wasn’t exceptionally weak or uncoordinated, and he liked sports all right, but he hated calisthenics and running and any repetitive exercises. His attitude made him have problems with coaches and jocks both. He was openly scornful of the school jocks (“Neanderthals!” he scoffingly called them), and since he had trouble climbing rope, chinning himself, and doing pushups, he resented the PE teacher for making him do it.

I went into the locker room and changed into shorts and t-shirt. The coach took roll quickly, and lined us up for calisthenics. After a few minutes, Bob limped in with a scowl on his face, the big bandages on his ankles exposed over his tennis shoes.

Coach whistled. “What happened to you?” he asked.

“Sunburn,” said Bob briefly, handing Coach his tardy slip and shuffling over to get in line.

Neil, a huge black basketball player, leaned over and looked at Bob’s legs, which were scarred above the bandages. “Looks like third degree burns,” he said. “How’d you get them?”

“It was nighttime,” Bob said. “I couldn’t see the campfire because of the dark, and I walked into it.”

He moaned and groaned his way through calisthenics. Coach was easy on him but kept him moving. After stretching, situps, pushups, and all those nice things, we had to run four laps around the gym. Bob shuffled along and was only halfway through when the bell ran. He started towards the door, but Coach said, “One more, Bob. You’re not that lame.”

Bob muttered to himself and turned back towards the track.

I waited for him in the locker room. We were several minutes late to lunch, which was actually kind of nice because the line was mostly through. Bob was still in a grumpy mood, and didn’t say much as we ate.

After lunch was English class. It looked like mostly Shakespeare this year. Bob slept through a lot of it. I kind of liked poetry and the challenge of figuring out the archaic English. Nora kept talking about having seen three Shakespeare plays during the summer until the teacher finally shut her up.

Then came speech class. Right away, the teacher, Mrs. Brinks, had us get up, introduce ourselves, and give an impromptu speech about how we spent our summer.

Nora got up and talked again about going to three Shakespeare plays. “And then two weeks ago, I earned three more badges at our regional Girl Scout campout,” she said.

Bob, who was dozing at his desk, jerked awake and stared suspiciously at her.

“What were they in?” asked the teacher.

“Well, I got an advanced orienteering one, and one for campfire cooking,” Nora said.

“What was the third one?” asked someone in the front row.

“Fire safety,” said Nora, smiling sweetly at Bob.

Bob snorted loudly. Everyone turned to look at him, and he scowled and turned red. Mrs. Brinks said, “Well, Bob, it sounds like you’re ready to go next. Please come up and tell us about your summer.”

Bob went to the front of the class, looked at the ceiling, and said, “This summer, I spent approximately 21 hours mowing our lawn and the lawn at my dad’s junkyard. I spent about 390 hours in my shop, working on various projects. I spent about 76 total hours camping, and approximately 31 hours in other recreational activities. I read 41 books and 115 journal articles. I slept about 611 hours, and spent about 105 hours at meals. I learned some advanced welding techniques, designed and built seven new mechanical devices, and carried out several chemistry experiments.”

Nora spoke up. “Did you perchance engage in an emulation of the Anglo-Saxon foundling who was raised by Sub-Saharan primates and became the regent of the same?”

The teacher stared at Nora, then at Bob, her eyebrows raised in puzzlement.

Bob glared at Nora warily. “Why do you ask?”

“How many hours did you spend in search and recovery of surreptitiously concealed items?” Nora asked.

Mrs. Brinks looked even more baffled at this.

“I don’t know what you’re talking about, but my suspicions have been verified,” Bob said, a bit huffily.

“Did you engage in any subaquatic reconnaissance utilizing innovative technology of your own creation, and were you intercepted during said reconnaissance and beat about the head and shoulders with assorted damp garments?” asked Junia, completely deadpan.

I snickered. Mrs. Brinks was totally lost.

Bob glowered at Junia. “Shut up, Junior,” he snapped. “You two had better watch your backs.”

Mrs. Brinks said, “Robert! I don’t know what you’re talking about, but you cannot make threats in school.”

Eddie raised his hand. “What happened to your ankles?” he asked. “I saw your bandages in gym class.”

“A wool allergy,” said Bob, and sat down.

On our way out of the school, I said, “So, do you have any ideas for a project this year?”

Bob seemed absorbed in thought. “I have an idea, but I need to work on it a while. Come by my shop when you’re done at home, if you want.”

He sighed and hobbled down the long flight of steps, his ankles looking like elephant’s legs from the bandages. He dropped his backpack into the basket on the back of his motorized bike, threw a leg over the seat, and pushed forward to get the bike off its stand. The stand swung up and whacked him in the left ankle, just as he hit the switch to start the bike. He yelped and kicked his foot out reflexively, making the bike tilt sharply to one side as it jerked forward.

Bob hopped desperately on his left foot for a few yards, trying to regain control, then gave up and dived off the right side of the bike and grabbed for his pained ankle. The bike fell a few feet beyond him, the motor still running, and immediately began to spin wildly in a big circle in the middle of the street, balanced on one foot peg. The rear wheel whooshed and buzzed as it spun against the pavement, and the bike bucked and bounced as it rotated, rolling right up Bob’s back a couple of times before he could scramble out of the way.

About a hundred kids came running to see what all the ruckus was. Bob lay curled in a fetal position on the curb, holding his ankles and cussing, and I stood crouched near the spinning wheels of death, looking for an opening. The cars of parents coming to pick up their kids had to stop. Bob and I found ourselves in the middle of a big circle of spectators, all watching the spinning bicycle.

“Whoa! That is like totally wicked!” shouted one kid.

After getting whacked a couple of times, I managed to snag the front wheel as it was coming around. The back wheel spun against the pavement, making an angry noise and beginning to emit a nasty gray smoke. The bike bucked and started to move in a circle around me. I pushed down the front wheel so the rear one was lifted up in the air. Then I felt my way to the switch on the right handlebar and turned it off.

With some help from the fascinated bystanders, I righted the bike so the batteries wouldn’t drain. Then I went to check on Bob, who was surrounded by a committee of concerned parents who had left their cars and were trying to find out how seriously he was hurt. He managed to get on his feet just as one parent was running inside to call an ambulance.

Hiroshi brought his backpack, which had ended up in the middle of the street. Bob shrugged off the adults’ questions, slung the pack into the basket again, climbed gingerly onto the bike, and hit the switch. His head jerked back as usual, his feet came off the pegs, and he and the bike bounced off the curb and into the street, bounced a couple of times from foot to foot, regained control, and headed for home.

I did my chores and homework as soon as I got home, but it was after five when I was able to bike over to Bob’s workshop. I half‑expected him not to be there, because of his injuries, but his bike was parked by the door to his shed.

When I went in, he was sitting on the floor looking at something. He was wearing shorts now, and had some huge, lumpy bandages taped around his ankles.

“Hey, Bob. I didn’t really think you’d be here. I figured your mom wouldn’t let you out, after dealing with your ankle,” I said.

“I didn’t tell her. I just took care of it myself,” he said. “Come look at this.”

He had a squat, square, green machine setting in front of him on the floor. It was plugged into the wall. Something that looked like a funnel rose from the top of it. The machine said “Oster” on the front and had a knob and some buttons. It looked pretty old. Bob turned the knob and the machine whirred to life. The funnel spun rapidly and a bit erratically.

“Okay. So you have a blender with a funnel on it,” I said. “Why?”

“Watch what it does. This is really cool,” Bob said. He picked up a pebble from a pile in front of him and dropped it into the funnel. It spun out and whacked him in the tooth. “Ow!” he yelled.

“Right. What’s it for? Are you going to sell it to dentists?” I asked.

Bob rubbed his lip and felt inside to see if it was cut. “No, you idiot. Here, let me show you.” He went and got a plexiglas face shield and put it on. Then he stood at a respectful distance and began tossing pebbles into the funnel. They went scattering all over the shop floor. I noticed then that there were quite a few pebbles already scattered here and there.

“You’re going to use it to plant grass seed in that bare spot in your yard,” I guessed.

Bob was surprisingly patient. “No, Bumstead. Look at the pattern. See the pattern in the gravel?” He gestured at the floor of the shop.

“Well, I’d say you got a mess of gravel everywhere,” I said.

“Right. Everywhere. There’s no telling when you drop a pebble in where it will come out. It’s completely...”

“Random,” I finished.

“Random,” he said.

“Speaking of random,” I said, “why didn’t you have a twist grip on your bike? I thought you’d gotten rid of that flip switch a long time ago. It’s dangerous.”

Bob grinned ruefully. “I put the twist grip on the submarine, remember? So I left the flip switch on the bike temporarily. Don’t worry, that was the first thing I did this afternoon. I had to patch one of the batteries with epoxy, too. It had a cracked corner.”

I looked at the blender again. “What good is a funnel that scatters gravel?”

Bob said, “The scientific method depends on random sampling. This thing will generate random samples, if you harness it.”

“There are computer programs that will do that.”

Bob snorted. “Pfaah! Computers. Everyone talks about computers. They aren’t really that interesting. I’d rather work with real stuff than with virtual reality. With this, we can put randomness out where people can see it and feel it.”

I grinned. “And maybe lose an eye or a tooth in the process.”

Bob ignored me. “Have you ever heard of minimalism?”

“No,” I said. “Is it some kind of diet?”

“I’ve been reading about it. In art, minimalism is trying to present your message as simply as possible. No huge production, just the minimum it takes to get your point across. This thing is a minimalist way to show randomness visually, using two pieces of junk and some rocks.” Bob rambled on for several more minutes, but I sort of tuned him out because an idea had begun percolating.

“Bob,” I said. “This thing isn’t really random. Where the pebble comes out depends on the wobble of the funnel and the direction you throw the pebble in, and probably the speed of the spinning. If you could isolate those factors, you could predict where a pebble will come out when you throw it in.”

Bob gestured around the floor. “This distribution looks pretty random to me.”

“It does,” I said, “but for each one, it depended on when you threw it in, which part of the funnel it hit, and how fast the funnel was spinning. And maybe the shape and weight of the rock.”

“I know that,” Bob said, “but it’s still a random distribution because you can’t predict it.”

We argued about that until suppertime, when we parted ways and went home.

The next day, Bob raised our argument in Mr. Boyce’s class. “Mr. B, I was thinking about it all night, and even with dice or with flipping a quarter, the individual results aren’t really random. If we knew all the details about which way the dice were held, how high they were thrown, the spin they were thrown with, the surface they landed on, we could predict how they would end up. So isn’t the randomness really in our lack of awareness of those factors? It’s not really that how the dice are going to land or whether the quarter will be heads or tails can’t be predicted. It could be if we had enough information.”

Mr. Boyce thought a minute. “Right. Randomness really has to do with control. If we know enough that we can predict how something is going to come out with some degree of certainty, then whatever we’re making predictions about is not giving random results. If we can’t guess any better than the results you might get pulling numbers out of a hat or numbered balls out of a lottery machine, then we call that random. But if we gain enough information or control that we make a significant difference to the outcome, then it’s no longer random. Randomness is in our minds, not necessarily in nature.”

He went on to talk about statistical analysis, and how they calculate whether the results of an experiment are statistically significant. A lot of it was over my head, maybe because I wasn’t that interested, but Bob was absorbed, taking notes and looking in the index of his math book to see what it had about the topic.

Doug raised his hand. “What about chaos theory, Mr. B? How does that fit in?”

Mr. B grinned. “I don’t know enough about chaos theory to discuss it intelligently today, Doug. How about if you and I each read about it tonight, and tomorrow you can give a presentation to the class about it? I think it will fit in well with the things we’ve been talking about.”

In science class, Mrs. Grissom asked us how we would choose among several hypotheses that all seem to explain a phenomenon.

“I would choose the one that seems most sensible or logical to me,” said Junia.

“But what if other people don’t think they’re the most sensible or logical?” asked Nora. “That happens all the time.”

Hiroshi said, “It doesn’t really matter, does it? I mean, if one hypothesis explains it as well as another, you could just choose the one you like the best and go with it.”

Bob said, “I would probably choose the most testable theory. The one you can measure the easiest.”

Doug raised his hand. “Why not choose the simplest? The most efficient explanation would probably be the best, I would think.”

Mrs. Grissom said, “There is good sense in what each of you has said, but in the scientific community, the simplest explanation is considered the best.” She wrote the word “parsimony” on the blackboard, and asked if anyone knew what it meant.

“Isn’t that a preacher who marries people?” wisecracked Eddie.

Nora whipped out a dictionary. “Extreme frugality, reluctance to use resources,” she read aloud.

“That’s right. Scientists sometimes talk about the law of parsimony. They want a simple explanation that matches the observable facts as efficiently as possible and doesn’t require many assumptions.”

Mrs. Grissom went on to talk about what she called Occam’s Razor. Occam was a guy who wrote an article about a hundred years ago proposing that if you had several hypotheses that could explain something, the best explanation was the simplest one requiring the fewest assumptions. I don’t remember what his razor had to do with it; something about the sharp edge. I think peas balanced on it. No, that was another story.

Anyway, Bob got all excited about parsimony and statistical significance. After school, he dragged me over to the high school library, and he buried himself in a pile of books about statistics and the scientific method. I found a couple of books on chaos theory. They were pretty fascinating, what I could understand anyway. The librarian was impressed with our great intelligence and let us check some of the books out.

It was a no-homework day, so we went to the shop. I still wasn’t convinced that the wobbly funnel would spin out the rocks as frequently in all directions, and I wanted to test it. So I found a huge cardboard box and put the machine inside.

Bob got a handful of pebbles and turned on the machine. Then he started dropping the pebbles in, one at a time, and we watched them spin out and whack into the cardboard. Bob stood on one side and I stood on the other, and we kept track of where the rocks went as he dropped them in, each of us watching two walls of the box. After he had dropped in about twenty pebbles, the timer ran out on the machine and it stopped.

“I got six on the right wall and three on the left. How did you do?” Bob asked.

“Five and six,” I said.

“Five, six, six, three,” Bob repeated. He jotted his results on the flaps of the box, and handed me the pencil so I could do the same. “That three is low. Let’s try it again.”

We repeated our experiment. This time it came out seven, five, five, four. “Twelve, eleven, eleven, seven,” Bob said, adding the two sets of results.

“It’s still lower on that section. I bet it has to do with the wobble of the funnel,” I said.

“No, I don’t think it’s statistically significant,” Bob said.

“But it’s only about two-thirds as often as the other sides,” I argued.

We bickered a while, then decided to try it again. This time I dropped the pebbles in, varying how often I dropped them and where I dropped them into the funnel. It did seem to matter where I dropped them, because if they hit the side of the funnel, they came out sooner than if they dropped into the very middle of the funnel, and if I dropped them on one side of the funnel, they usually ejected on the opposite side, depending on the speed at which the blender was running.

By the time I had to leave for supper, we had figured out that pebbles dropped into the very middle of the funnel were the least predictable as to where they would come out and hit the box. Bob pulled out a tape measure and started scribbling something on the floor with chalk, and rummaging in the junk piled in the corner. He didn’t even notice when I said “Goodbye” and left.

When I arrived at school, Bob was just wheeling up with his bike towing the trailer we had built for the submersible. He had the centrifuge in one of his baskets, and some kind of round cage on the trailer. There were some other parts, too, and a bag of marbles. I helped him carry it in, while he ignored the questions of the other kids who wanted to know what was going on.

We told the office that we needed to take some things to Mr. Boyce’s classroom for a presentation, and they gave us a hall pass. Mr. Boyce walked in as we set the equipment in a front corner of the room.

“Hi, Mr. B. Excuse me,” said Bob. He dashed out to go to the bathroom, leaving me to answer Mr. Boyce’s question: “What is that thing?”

“Uh, it’s a machine to illustrate random distribution, chaos theory, and statistical distribution,” I said.

“How does it work?” Mr. Boyce asked, looking down at it. Besides the machine, which now had the funnel pretty much upright, Bob had made a sort of cage out of two hoops connected with eight legs. There was clear plastic wrapped around the cage.

“Well, it’s a centrifuge,” I said, “This cage is to catch the items that spin out, and if you keep track of where they hit, you can determine whether the distribution is random or predictable. I better not say much more or Bob is going to get upset.”

Bob came in a minute later and gave a very technical explanation to Mr. Boyce. “I was wondering if we could experiment with it in class,” he finished.

Mr. Boyce said, “Well, I think so. I promised to have Doug discuss chaos theory, so we’ll have to leave time for that, as well as for the math lesson from the book.”

“This will tie in real well with chaos theory,” I said.

Mr. Boyce wrinkled his forehead. “Maybe so,” he said dubiously.

The bell rang then, and the classroom began filling up with noisy kids.

Once things settled down, Mr. Boyce took us through the math lesson, which was pretty simple, and then he had Doug give a presentation on chaos theory. Doug was pretty good, being about as smart as Bob and Nora, and since he’s lazy, he didn’t talk too much but just got straight to the main points.

He mainly quoted from the encyclopedia.

In mathematics and physics, chaos theory deals with the behavior of certain nonlinear dynamical systems that (under certain conditions) exhibit the phenomenon known as chaos, most famously characterized by sensitivity to initial conditions (see butterfly effect). As a result of this sensitivity, the observed behavior of physical systems that exhibit chaos appears to be random, even though the model of the system is 'deterministic' in the sense that it is well defined and contains no random parameters. Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economies, and population growth.

Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder. When we say that chaos theory studies deterministic systems, it is necessary to mention a related field of physics called quantum chaos theory that studies non‑deterministic systems following the laws of quantum mechanics

A non‑linear dynamical system can, in general, exhibit one or more of the following types of behavior:

·           forever at rest

·           forever expanding (only for unbounded systems)

·           periodic motion

·           quasi‑periodic motion

·           chaotic motion

The type of behavior a system may exhibit depends on the initial state of the system and the values of its parameters, if any. The most difficult type of behavior to characterize and predict is chaotic motion, a non‑periodic complex motion which has given name to the theory.

In order to classify the behavior of a system as chaotic, the system must exhibit the following properties:

·           it must be sensitive to initial conditions

·           it must be transitive

·           its periodic orbits must be dense

Sensitivity to initial conditions means that two points in such a system may move in vastly different trajectories in their phase space even if the difference in their initial configurations is very small. The systems behave identically only if their initial configurations were exactly the same. An example of such sensitivity is the so‑called "butterfly effect", whereby the flapping of a butterfly's wings is imagined to create tiny changes in the atmosphere which over the course of time cause it to diverge from what it would have been and potentially cause something as dramatic as a tornado to occur. The butterfly flapping its wings represents a small change in the initial condition of the system which causes a chain of events leading to large‑scale phenomena like tornadoes. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different. Other commonly‑known examples of chaotic motion are the mixing of colored dyes and airflow turbulence.

When Doug was done, Bob looked at him and said, “You don’t really know what any of this means, do you?”

Doug grinned and shook his head. “Not really,” he said cheerfully. “I read all about it, but a lot of it was Greek. What made some sense to me was the part about a butterfly flapping its wings possibly causing a hurricane on the other side of the world because it affects something that affects something else that affects something else, a whole series of small changes that eventually make a huge change. They talked about turbulence, like when you mix dye and water, and I know what that looks like from science class last year. It’s sort of predictable, sort of not, and there are a lot of factors that can affect it.”

I butted in. “That article also talks about how, when you have, say, three planets whose gravities affect each other, the pattern of their motion doesn’t necessarily settle into a pattern that repeats like our solar system does, but it doesn’t keep increasing or getting smaller. Like if you had two solar systems merging together, you could end up with a really complicated set of orbits that keep changing all the time without collapsing into the sun or spinning off into space. It’s too bad they don’t have pictures of that. I read another article that was much more interesting.” I stood up and talked about the details of the weather simulation program that started everything, how Lorenz found that dropping a couple of decimal places in his figures and restarting his simulator in the middle of a cycle changed his outcome completely.

“One of the cool things about chaotic systems is that, if you look at a small segment in them, the same thing is going on in that segment as in the whole thing. Like, one guy, Mandelbrot, started figuring out that if you tried to figure out how long a coastline is, measuring all the bays and everything, the closer you looked at it, the more you would have to measure, smaller and smaller bays and points and hollows and rocks, until you’re measuring around grains of sand, and even the grains of sand have irregularities that you’d have to measure.”

I showed them a picture of a Koch curve, that looked like a magnified snowflake, made of triangles with smaller triangles attached, and those smaller triangles with even smaller triangles, on and on. I held a magnifying glass to the picture to show them how the tiny details of the figure looked just like the big picture of it. “They’re starting to use these formulas in computer graphics, to have an economical way to generate detail in pictures,” I said.

The class talked about it for a while, but most people were completely lost. Bob stayed out of it since he hadn’t read about it yet. I could see him lost in thought. Finally Mr. Boyce said, “Thank you, Doug and Mike. Now Bob has a machine he wants to show us.”

Bob quickly set up the machine in the open space between the teacher’s desk and the front row. I helped him. The cage he had built was eight-sectioned and covered with plastic, and fit around the machine. Bob had labeled each panel A, B, C, and so on up to H. The other piece of equipment was another funnel on a frame that hooked on top of the basket so the funnel hung right over the other funnel.

“What is that thing, part of your bicycle?” Tony wisecracked.

Bob ignored him and set the bag of marbles on the teacher’s desk.

“This is a simple centrifuge. When something is dropped into it, it spins out.”

“Kind of like you spinning off your bicycle yesterday?” Tony said. Mr. Boyce shushed him while Bob gave him a glare.

With an effort, Bob calmed himself down. “The basket-like thing is to catch whatever spins out. There’s no reliable way to predict where the things will go when they’re dropped in, so theoretically they should end up distributed randomly around the machine.”

Bob called up four kids to sit around the machine. Each kid put his hands on two panels, so all eight were being touched. Bob instructed them to keep track of how many marbles hit each panel. Then I plugged in the machine, and Bob started dropping marbles in. The first one spun out and whacked against the plastic.

“Ow!” yelped Ernie, shaking his hand. “Right on the knuckle!”

“Just touch the plastic lightly,” suggested Mr. Boyce.

Ernie rubbed his knuckle and put just his fingertips back on the plastic.

Bob dropped a dozen marbles in, and then had the kids tell how many they had felt with each hand. After he had the results, he read: “Two, one, one, three, two, none, one, two.”

Mr. Boyce made a chart on the board with eight columns, and put the results under each one. “How does that distribution look?” he asked, looking at the numbers.

“It’s too few to tell,” said Doug. “The higher the number, the more the pattern will show.”

“Right,” said Mr. Boyce. “Do it again, with more marbles this time, Bob.”

Bob dropped all the marbles in this time, about thirty of them. The results were marked on the board. Now every panel had been hit at least once, and most of them three times or more.

The bell rang then, and we gathered up the equipment and took it to Mrs. Grissom’s classroom. Junia saw that we were overloaded, so she offered to carry the other funnel. We put it in the front of the classroom, set up and ready for science class. I saw Junia put a yogurt on her desk. Mrs. Grissom let us eat healthy snacks in class sometimes, and Junia usually had yogurt, jello, or fruit to eat.

We went out to our lockers to stash our math books. Bob said, “I should have read about chaos theory. It sounds interesting.”

“It is,” I said. “What I could understand of it. I haven’t figured out yet how it relates to randomness, because it’s not random. It’s predictable, but really complicated.”

“Hmm,” grunted Bob. He didn’t say anything else until the second bell rang, and we filed down the hall.

When we walked into science class, Junia asked us if we had taken her yogurt.

“I hate yogurt,” Bob said.

“It was on your desk when we left ten minutes ago. We haven’t been back in here since then,” I said.

Mrs. Grissom talked about how accidental discoveries had provided many breakthroughs. She mentioned Newton’s apple and the law of gravity, Euripedes and his overflowing bathtub, how Masonite was the result of leaving wood fibers in a machine with a steam leak while the inventor went to lunch, safety devices resulting from tragic accidents, and other products that resulted from things not going as expected. She talked at length about how discoveries took patience and perseverance. She quoted Thomas Edison, who said his inventions were “ten percent inspiration and 90 percent perspiration.” She said sometimes knee-jerk reactions lead to abandoning fruitful directions, and she gave examples of several people quitting on the verge of a breakthrough, only to have someone else complete their experiments and receive the credit and benefits of the discoveries.

It was unusual for Mrs. Grissom to talk so much at one time, and to sound so preachy. A number of the kids got bored, and some even dozed off. I got a little sleepy myself. But Bob hung on every word. I saw him taking notes feverishly.

Finally, with about ten minutes to go in class, Mrs. Grissom let Bob show off his invention. Bob gave a little introduction and talked about how a centrifuge was a simple way to generate random results. He got four kids to gather around the plastic screen, and got his handful of marbles ready to drop in. I took the plug and stood by an outlet, ready for his signal. When everyone was ready, Bob pointed to me, and I plugged it in.

The blender whirred to life, and white gooey liquid came flying out of the top of the funnel. It spattered all over the inside of the plastic, and trickled down to the floor. Bob’s jaw dropped. I quickly unplugged the cord.

“Ooh, yuck!” said one of the girls who was right by the machine. She held up her hand, which had the goo on it from a loose place in the plastic.

Someone handed her a Kleenex, and she wiped it off, with her nose wrinkled in disgust.

Hiroshi, who was one of the people by the machine, touched a little that had trickled out from under the screen, and held it to his nose. “It smells like sour milk,” he said.

“So that’s where my yogurt went!” Junia said.

Bob looked around the classroom, his face dark with fury. Doug was staring in fascination, as were most of the kids. Everyone was chattering. I noticed Tony snickering and saying something to the guy next to him.

Bob turned to Mrs. Grissom. His face was tight. “I’m sorry, Mrs. Grissom. This was an act of sabotage, and I’m going to figure out who did it.”

“Well, for right now, please go to the restroom and get some paper towels for the floor. Your equipment will need to go outdoors until it can get cleaned. Mike, you and Hiroshi please take it out back. There’s a hose there that you can use to rinse it off during lunch break, and then you can put it back in here for safety, if you like.”

We got things cleaned up just as the bell rang. Bob and I and Hiroshi went back out and hosed off the equipment, and put it back in Mrs. Grissom’s classroom. Bob was still mad, but didn’t say anything. He seemed to be deep in thought. He poked at his hamburger casserole at lunch, making swirls on his plate and looking at them.

By the time we got to English class, Bob was completely zoned out. He was calm now, but only paid enough attention to the teacher to be able to answer questions. Otherwise he was busy jotting things in a notebook. I glanced over and saw he was drawing a diagram that included the centrifuge and some other equipment.

“What’s that?” I asked in a whisper.

He said, “Did you hear what Mrs. Grissom said about breakthroughs coming from mistakes and failures? That yogurt gave me a great idea.”

After school, I helped him load the centrifuge into the bike trailer. It still smelled a little of yogurt, but it was clean.

When we had it all strapped on, Bob said, “Do you have those chaos books with you? I don’t want to stop by the library with all this on my bike.”

I fished through my backpack and handed him the two books. He stuck them into his pack, and said, “I’ll be reading these and working out some issues on the centrifuge, so it’s probably better if you don’t come over today, and maybe tomorrow.”

. . .

School was calm the rest of the week. We went to our classes as usual. Bob’s ankles were getting better, but he got to sit out when we played dodgeball. Tony hassled him about the centrifuge and yogurt, but Bob ignored him. He was still absorbed in making diagrams and reading chaos theory.

I went to his shop on Saturday, after doing my chores. Bob had the centrifuge set up, with the funnel over it, only there was another contraption above that. Bob was adjusting its position when I walked in.

“Good heavens, professor! What have you done?” I exclaimed.

Bob had a couple of levels and a plumb bob taped to the contraption he was adding. “This is a feeder for the centrifuge,” he explained. He tightened some screws and checked his instruments. “I’m positioning it so it will be exactly centered over the middle of the funnel.”

“What does it do?” I asked.

He sat back. “This thing has a timed release for the marbles, and inside it has a centrifuge of its own that shoots the marbles out in a direction I can’t control,” he said. “Then they hit the side, and drop into the funnel below. The other thing it does is coat the marbles with graphite so they leave an imprint on the plastic when they hit.”

“Let’s see it work,” I said.

Bob plugged in the centrifuge, dropped a handful of marbles in the upper machine, set the screen around it all, and flipped a switch on the cord to the upper machine. It began to whir, and a marble dropped out of it and into the lower funnel, which spun it out to whack against the plastic. Where it hit, it left a gray imprint. Then another dropped, and another. Soon there were a dozen gray imprints scattered around the plastic.

Bob shut off the machines. “This way we have a more complex system. There’s the timed release, the upper centrifuge, and the lower centrifuge. Each centrifuge has variable speed. We can also change the weight of the marbles, by using ball bearings or beads or something. The graphite helps us keep track of where things hit without having to touch the plastic and involve several people.” He sat back. “I got the idea for the graphite from the yogurt. The yogurt made a really cool pattern on the plastic, did you notice? I was mad about having to clean it up, but it got my mind going. And the stuff on chaos theory made me think of having two systems interacting, so instead of us feeding in the pebbles, one mechanical system feeds into the other. It’s a very simple set of machines, but it illustrates several important scientific principles and theories at once.”

“How does the upper machine work?” I asked.

Bob unscrewed the top from it and showed me the little machine that pushed the marbles out every so many seconds, the chamber where they got coated with graphite, and the chute that fed them into the spinning saucer. The marbles would spin out, hit an outer metal wall, and drop down into the lower funnel, where they spun out to hit the plastic.

“Pretty cool!” I said. “Have you tried it with smaller or bigger marbles yet?”

“Not yet,” he said, “but I think it will be interesting.”

He went to his shelves and found a can of rusty ball bearings. He put a handful in the canister, and we started up the machines again.

The ball bearings left a line of small gray dots higher up on the plastic than the marbles. Bob located a handful of plastic beads, and ran them through. They didn’t pick up much graphite, but they hit higher on the plastic than the ball bearings.

We fooled around with the machine for several hours, trying different timings and speeds to see what difference they made. The speed made a difference to how high the projectiles hit on the plastic, but not to the distribution around the circle, from what we could tell. The marbles or ball bearings or beads went every which direction, regardless of the speed of the upper and lower centrifuges, and regardless of the timing of the release of the projectiles, just like when we dropped them by hand into the middle of the funnel.

I was disappointed. Bob wasn’t. “I started off looking for randomness. This definitely looks random to me,” he said. “There are patterns in the height of the impact marks, and overall they make a line, but you can’t guess where any one piece will hit.”

He untaped the plastic where it overlapped. “Oh, here’s another innovation I made,” he said. He peeled the plastic off and laid it carefully on a piece of wood. “See? If we spread this out, we can take a look at it and even tack it to a wall. And it’s easy to put a new piece on the machine.”

He dragged a box from the corner and pulled out a long section of plastic wrap, which he stretched around the machine and taped in place.

“Boy, you’ve really got the system down,” I said admiringly. “I wonder if you could use different colors to show the different speeds or sizes.”

Bob snapped his fingers. “Great idea! Probably for the speeds, because the sizes show up by themselves.” He mused. “It’s not showing much about chaos theory, though. I wonder what that would take.”

“Maybe if the machines could interact, where each machine has an impact on the other one, so that the speed is affected or something,” I said. “Especially if you could get the marbles cycling back through so there’s automatic repetition. But there has to be some means by which each part affects the others without it settling into a steady cycle.”

Bob looked at me with more respect than I had ever seen from him. “Mike, you’re onto something. Go home now and let me think. I have a great idea buried somewhere that I need to let come out.”

“Sounds like constipation,” I grumbled. “You need some brain bran.” I straddled my bike and left.

Bob was distracted but cheerful that week, except in gym class when he got his ankles kicked. He spent his afternoons at his shop, tinkering. I went by a couple of times, but he didn’t talk at all, just grunted. He had some complicated wiring running between his machines, and a sort of tray that they sat on that collected the marbles and fed them into some other machine. He wouldn’t show me how it worked. He just kept on fiddling and mumbling to himself and poking through his electronics books and adding and removing pieces from the circuit board he was building. I finally left.

There was a lot of excitement in the air the next week as the deadline for science fair registration approached. Doug had recruited Eddie to help him this time, something about sound waves and an oscilloscope. Nora and Junia had a chemical properties of foods project going. There were the usual soda-and-vinegar volcanoes from the non-honors students, and a fossil collection, leaf collections, bird sightings, bacteria from different ponds, the life-cycle of the frog, Somebody said there was even a toilet that had been split down the middle.

Thursday afternoon as we were leaving, I asked Bob, “Shall I come over so we can write up our proposal?”

“I’m still wrestling with it,” he said. “I should have it figured out by tonight, but it’s going to take me some time to get the wrinkles ironed out. Don’t worry. I’ll have the proposal written by class time.”

I went home, did my homework, and then started fretting. What if Bob didn’t have it written in time? I took out the guidelines for science fair exhibits, and tried to write a proposal using what I knew of Bob’s machine. It looked very lame when I looked at what I wrote, so I tried adding some things about parsimony, minimalism, chaos theory, and the other concepts we had been studying. I looked over this draft, and it seemed kind of wordy and incoherent. I fussed over it until bedtime.

That night I dreamed of fooling around with Bob’s machine in his shop while our classmates were all at the science fair, showing off nice elegant projects on big posterboard displays. Bob was lying on the ground looking at the machine, and I was dodging graphite-covered BBs that the machine was slinging at me. I dreamed that Tony was outside the shop, using a remote control to make the machine sling the BBs straight at me.

I felt worn out and nervous when I got to school. Bob wasn’t there. He finally arrived just as the bell rang. He looked pretty tired himself. “Did you get it written up?” I asked.

He grunted and walked past me. He was so distracted that when Tony ran into his legs with his pack again, all Bob did was grab his ankle and hop on one foot. He didn’t even curse Tony or mutter.

In math class, he didn’t pay any attention to what was going on. He was feverishly writing. He would scribble something, read it over, mutter under his breath, then flip to another page and try again.

Finally, near the end of the class, he pulled some papers out of his pack. I recognized the science fair guidelines. He read them over, then wadded up the papers he had been writing into a huge lump, and tossed it into the waste basket right next to where Mr. Boyce was standing and talking. The lump hit the bottom of the can with a loud thunk.

Mr. Boyce jumped and glared down at Bob.

Bob had bent to write again. After a minute, the sudden silence registered with him, and he glanced up and saw Mr. Boyce looking down at him. “Yes sir?” he asked.

“Bob, what are you doing?”

Bob flushed. “Writing, sir.”

“What have I been talking about today?”

Bob thought for a minute. “Statistics?”

“What about statistics?” Mr. Boyce asked patiently.

Bob reflected. “No clue.”

“Bob, I’ll need you to stay after class until the second bell. Now please pay attention and don’t do any more writing during this class period.”

In science class, everyone was buzzing about their science projects. Bob slid in at the last minute, and immediately began to write, ignoring everything around him.

“All right, class,” said Mrs. Grissom. “Today you’re to turn in your science fair proposals. I would like each of you to come up and present your project to the class. Let’s start with you, Nora.”

Nora and Junia stood up. Nora made a rambling speech about the chemical properties of foods, in particular acidity, and how they were going to analyze them. She used a lot of big words to say it. I kept expecting her to mention that she had seen three Shakespeare plays, but that didn’t come up.

Eddie and Doug went to the front, and Doug talked about sound waves, the hearing spectrum, hearing loss, and some other stuff. Then followed kids talking about bacteria, frogs, bats, birds, how a radio works, and various other projects.

Tony got up. “I’ve designed a robot,” he said.

Someone said, “Oh, cool!” There was a buzz of reaction.

Tony went on to boast about what his robot would be able to do and how it worked and its potential uses.

Suddenly I noticed that Bob had sat up and was staring intently at Tony. Bob looked angry. He raised his hand. “Where did you get your design?” he asked.

Tony said, casually, “I’ve been studying computer construction and programming, and I got the idea for the robot from looking at our vacuum cleaner and watching a TV show about MIT students who design robots for competitions.”

Bob snorted and went back to his writing, a frown on his face.

Mrs. Grissom said, “Thank you, Tony. Bob, you and Mike are the only ones left.”

“Just a second.” Bob frantically wrote a final sentence, then thumped his pen down for the period. He stood up, and I stood beside him.

Bob held up his paper. “We have designed a cybernetic system of three machines that illustrates several scientific principles and theories. The most interesting of these is chaos theory, in particular the complex patterns that result from the interaction of multiple systems, like the factors affecting weather, or the population cycles of different plant and animal groups in a symbiotic relationship. The machines also illustrate random distribution, parsimony, gravity and centrifugal force, and other principles and phenomena.”

Tony said, “Is this your vomit-slinging machine?”

The class broke into guffaws and cries of, “Oh, gross!”

Mrs. Grissom frowned at Tony. Bob calmly said, “The centrifuge is an important part of it, but it includes two other machines. We’re building the circuitry and mechanical connections between the machines, based on an idea of my esteemed colleague’s.” He gestured at me, and I bowed.

Bob went on. “The machines themselves are adapted from existing equipment, but we’ve modified each one, and the way they connect is original.”

Mrs. Grissom thanked us all, and launched into her lesson for the day. Tony was busy drawing something. After a while, he held it up to show us. He had drawn a futuristic robot with a flamethrower, torching our centrifuge which was covered with slime. The caption read, “Failed experiment is prepared for recycling.”

Bob snorted and frowned. I thought the picture was pretty funny, but I didn’t think much of Tony, so I didn’t laugh.

The semester moved along. Bob’s ankles healed, and he no longer had to wear bandages. We memorized Shakespeare poems and read plays, we gave speeches, we did math homework, and we did calisthenics and played several sports.

Nearly every afternoon, I went over to the shop, where Bob and I slaved over our machine. We had lots of problems with the system to feed the marbles back up, and it took weeks to get it working right. Bob’s circuitry that made the machines affect each other was intricate, and it took a lot of experimenting and modifying to get it to work where it didn’t either lock both machines in their highest speeds or shut them down entirely.

Tony took every opportunity to brag about his robot, and to make jokes about our slime-slinger, cocktail-shaker, reverse toilet, and a whole series of other names. Bob seethed but didn’t say anything.

Finally the science fair was only a week and a half away. I went over to Bob’s on Saturday. “Bob, we don’t have anything to analyze yet,” I said.

Bob shrugged. “We’re just now getting to the point where the machine does what we planned for it to do. We’ll have to run it a lot this week to get the results we need to work with.”

“And we have to analyze them. It’s going to be a tough week,” I said.

“No kidding. Let’s get started.”

He opened the top machine, and dumped graphite into one chamber. Then he took a bottle of blue chalk dust and shook it into another chamber. He shook a different color into a third. “I added this feature on Saturday. I think it will help a lot,” Bob said.

“What’s it for?” I asked.

“The marbles go through one color, then when it runs out, they go through another, then another, then another. It’s a way to see what’s happening over time, so we don’t just have a gray mass that we can’t analyze.”

He dropped several dozen marbles into the hopper and closed the top machine. Then we wrapped fresh plastic around the frame.

I started the main centrifuge first, then the bottom machine that brought the marbles back up. Then Bob started the top machine.

A marble dropped, then another, then another. They spun out, whacked against the plastic, and dropped into the tray. After several had fallen, the weight in the bottom tray activated a rheostat that triggered the first machine to spin faster and drop marbles more rapidly. The bottom machine fed them back up to the top, but after a couple of minutes, the top machine was out of marbles except for the ones coming slowly up from the bottom. The second machine, meanwhile, sped up in reaction to the increased speed of the first one, leaving a line of gray dots higher up on the plastic.

“It’s working!” I exclaimed.

“Hold on,” Bob cautioned. “We have to see if it keeps cycling, and if the cycles vary or just repeat.”

It took a long time. The centrifuge spun fast, then slow, then fast again, then a medium speed. It jumped to slow, then suddenly to fast. The impact of the marbles made neat lines of gray dots, about an inch higher or lower than the first row.

“I made sure it had clear steps between the speeds, so it would show on the plastic,” Bob said. “Otherwise it would just be a scattered mess instead of distinct levels.”

Something clicked on the top machine, and a few seconds later, the marbles were leaving blue dots. They did this for several minutes, and then it switched to red. The last color was black. Meanwhile, the machine was cycling between dropping marbles slowly or quickly, spinning them out slowly or quickly, and pushing them back to the top slowly or quickly.

When the marbles began to hit without leaving any imprint, Bob shut down the system. I helped him unwrap the plastic. It was lined with four rows of dots. Each line had dots of all four colors.

“Now we need to count the dots, and see how many of each color there are in each row and how far apart they are,” Bob said. “But first, I’m going to spray it.”

He got a can of artist’s fixative from his desk. “I was showing my folks one of the plastic sheets, and everything was so smeared it was hard to read. My mom told me about this stuff. She used to spray it over her pastel drawings,” he said. He sprayed the plastic sheet.

We set up the machine again while we waited for the fixative to dry.

“It’s been working reliably since Thursday,” Bob said. He showed me a couple of plastic sheets tacked on the wall. They were a little different from each other. One had a heavy line at the lowest level, and the other two lines were light. On the other one, the lines were pretty even.

“How come these are different?” I asked.

Bob grinned. “I’ll show you in a minute. Let’s analyze this one, and then we’ll run it again.”

We painstakingly counted the impacts of each color on each row, and measured the distance between the dots. The last measurement was from the first and last dots to the line that showed where the plastic had overlapped when it was wrapped. The measuring took a couple of hours.

Bob entered the distances in a chart, added them up, and calculated the averages. He highlighted the smallest and largest distances and a couple of groups where the dots were a little closer or a little more scattered than the average. That took another hour.

Bob sighed. “I don’t like computers much, but I’m going to have to use one to figure the statistics out on this stuff. My uncle said he’d help me run the programs and he’d show me what the numbers mean.”

“What is it with you and computers?” I said. “You’re the biggest nerd in the school, and you love gadgets, but you have this prejudice against computers.”

Bob frowned. “I like the idea of computers,” he said, “but there’s so much tedious work just to get the simplest results. Maybe in a few years they’ll be more advanced. I mean, look at Tony’s famous robot. He copied pages and pages of code just to make something that runs around on the floor and backs up if it hits something. It doesn’t even do anything interesting. I want to build things that actually do something, pretty quickly. Like the sub, or the winch, or my bike.”

We tacked the plastic on the wall and crouched back by the machine. Bob opened the top, took most of the marbles, and dropped them into the bottom. He closed it up again, and we started the motors all at once.

The upper machine spun furiously for a few seconds, but had nothing to spin out. It slowed to the lowest speed. Then the first marbles began to go through it. The centrifuge spun them out at the low speed. As the lowest level began to empty, the upper one sped up again, and then the middle one cycled through the middle speed and then began to spin rapidly. They moved through their various speeds as the colors changed, but the amount of time they spent on each speed seemed to be different from the previous time. The impact lines certainly looked different.

Bob was ecstatic. “Whoa! That doesn’t look like either of the others!” he exulted.

When the colors were gone, we spread this sheet and sprayed it. Bob said, “Starting with all the marbles at the bottom instead of the top makes a big difference. See? The middle line has the most hits, especially at the end with the black chalk. The lowest is darkest at the beginning. That’s the gray. But in the middle it’s evenly spread between the middle and upper lines, and hardly any down below.”

We measured and began entering figures. It was already past lunchtime. I was bummed. “This is going to be real tedious,” I predicted. “We need several more of these for our work to be worth anything, and we still haven’t analyzed the results. It’s going to kill us. And I’m starving.”

Bob said, “Let’s go get some sandwiches. I can’t think when you’re hungry.”

We spent the afternoon and evening working, and only got a total of four plastic sheets done and entered into tables. Two of them were very similar, but the others were pretty different. We had started the machine with all the marbles in the top, all in the bottom, half up and half down, and three quarters up and one quarter down.

As I got ready to go home that evening, Bob said, “I’m going to have my uncle teach me to run the computer statistics program at the university, so I can fool around with these numbers and see if anything jumps out. And I’ll ask Mr. Boyce for help with the statistics if my uncle is busy.”

“Good idea,” I said. “We’re going to need all the help we can get.”

On Monday, I asked Bob, “So, should I come over so we can crank out another sheet or two?”

“No,” he said. “I got three done yesterday. I think we need to take what we have over to the university and get it analyzed. I told my uncle I’d be by after school. Want to come?”

“Sure,” I said.

After classes ended, we got on our bikes and headed for the university. It was on the other end of town from the high school, but there were good biking roads the whole way.

Bob’s uncle was in the sociology department, which was in a shabby old building that had been temporarily thrown together during World War II, and never torn down.

Uncle Milt was an interesting guy. He was completely bald, but had a long gray beard. His eyes were blue and very lively. He beamed and squeezed Bob’s shoulder. “So, how’s the brainiac? Are you going to make it to the state science fair this year?”

Bob shrugged. “We’re trying, if we can put our display together this week. There are so many numbers, though, and we can’t tell if they mean anything.”

“All right. Show me what you have.”

Bob fished the tables of figures out of his pack. His uncle whistled. “That is a lot of data. What are the variables?”

Bob explained to him how our machine worked, with the three machines and their interaction and the different colors of powder. His uncle whistled again. “Pretty complicated stuff. Well, let’s start with the simplest analysis and work up from there. Once you get all this data entered, then we can use the computer to play around with it.”

He led us to the sociology computer room. There were several computer terminals with green screens, and some card punching machines. “You can enter the data on cards if you want to, but you may as well just use a terminal and store it right in the computer,” Uncle Milt said. “They’ll be trashing the card machines pretty soon, but some people still have their projects on cards.”

He sat down at a terminal, logged into the computer, and set up a table for Bob. He showed us how to enter the data in columns and rows. “When you’re done with one set, save it under a new name, and open the original blank table again,” he said. He showed us how to save and open files.

Bob started typing. Uncle Milt looked at me. “Shall we log you in on a different terminal? That way you can both work at the same time,” he said. “If things get busy around here, you’ll have to log off, but in the meantime, it’ll go a lot faster.”

“Sure,” I said.

He logged in under a different username, and I started entering another table of figures.

“I’ll be in my office,” said Uncle Milt. “Look for me if you need anything.”

Bob and I typed furiously for an hour and a half, and then double-checked each other’s work. After correcting a few mistakes, we saved everything with filenames that showed which table was which. Bob went and got Uncle Milt.

Uncle Milt said, “Well, now we’ll tell the computer what the columns and rows are called, and what kind of analysis to do on them. It’s going to take a few hours, so we’ll do a very small analysis first to make sure we’ve defined things correctly, and then we’ll do the whole thing.”

He entered a string of commands. The screen sat empty for a long time. Then the words “Job done” appeared in the command line. Uncle Milt typed “Display results”, and the screen filled up with tables of figures and words.

Uncle Milt looked it over and shook his head. “I did something wrong,” he said. “None of this makes any sense.”

Bob scrutinized the display. “It’s only looking at two columns,” he said.

Milt snapped his fingers. “That’s it! I defined only two instead of three. Let’s do this again.” He began typing furiously, then hit “Enter,” and the screen again went blank.

A few minutes later, “Job done” appeared. Milt typed “Display results”, and the tables appeared again. Milt beamed. “There we go! Here’s your averages, your median, your mean, and all those kinds of things. Now that we have the bugs worked out, let’s print this stuff, and then run the real analysis.”

He typed a print command, and then began to enter a long series of commands. He hit “Enter,” and sat back.

“There we go,” he said. “That should work. If you come by tomorrow, the results should be ready, and I’ll go over them with you and see what we can do with them.”

He took us over to the window, where a student gave us the printout.

“Thanks, Uncle Milt,” said Bob. “Say, do you have a book on the statistics program? That way I can run the program myself and you don’t have to spend as much time helping us.”

“Sure,” said Uncle Milt. He took us to the office and handed Bob a fat, very intimidating-looking paperback volume. “You’ll probably need this, too,” he said, pulling a manual of statistics off the shelf.

Bob glanced at it. “I checked this one out of the library several weeks ago. I didn’t understand a lot of it, but I didn’t have any data yet,” he said.

We lugged our stack of paper and the two books back to the bikes. “Your uncle is pretty cool,” I said.

Bob said, “Yeah, he’s a lot of fun. He’s always after my dad to go back to school and get a serious job instead of running the junkyard. Dad says he makes more money with junk than Uncle Milt does with all his brains and studies, so why should he go back to school?”

We went back to the university on Tuesday. Bob had read a lot of the statistics book the night before and during classes that morning, and he claimed he understood a lot of it, although I had my doubts.

Uncle Milt took us to the lab again, and we anxiously waited while he logged in and opened the results. Milt beamed again. “Oh, these are great results! You should be able to do something with these. See, there are significant patterns within each table, and then some cool things going on between the tables.” He jabbered on using a lot of statistical jargon that meant nothing to me.

Bob looked like he understood. “Well, I guess we need to get this printed out and go over it,” he said.

Milt typed in a print command, then continued talking about the statistical patterns. Then he went over to the window to get the printout.

I looked at Bob blankly. He looked back at me and grinned.

“Do you know what he’s talking about?” I asked.

“No. Well, I recognized a few words from the manual. I’ll get him to explain it to me, and try to figure out the rest of it. But it sounds really good.”

Uncle Milt came back with an armful of paper and said, “Let’s go up to a classroom so we don’t tie up the terminal.” He logged out on the terminal, then led us to a classroom near his office. He began flipping through the printout, and tore it into several sections.

“Okay. This section here is the most important,” he said. He launched into a technical explanation that was completely beyond me. Bob interrupted him numerous times with questions, and made notes in the margins of the printout and on a notebook.

I sat back and listened, but zoned out after a while. I pulled a notepad out of my pack and started listing the things our display was supposed to illustrate: random distribution, chaos theory, minimalism, parsimony, cybernetics....

When there was a pause in the conversation, I said, “Bob, I’m going over to the university library to do some research. I’ll meet you back over here at 5:30.”

He nodded and continued talking. I walked over to the library building and inside. There was a reference desk, with a woman who looked like the stereotypical librarian: reading glasses low on her nose, gray hair in a bun, a dark dress, and lots of wrinkles in her forehead. I hesitated, half-afraid she would shush me if I said anything, but she looked up and smiled and said, “How can I help you?”

“Uh, I need to learn about cybernetics,” I said. “Just the basics.”

She said, “Well, the encyclopedia is a good place to start. There are several sets right over there.” She pointed to a door to her right. “You can also look in the stacks for a good introduction. Let me show you how to use the card file.”

She helped me sort through the cards in the cybernetics section, and even led me into the stacks to find the books I had chosen. Then we went back to pick up a couple of encyclopedia volumes. After I was set up at a study table, she smiled again and said, “If you need anything else, just ask!”

I read the encyclopedia articles first. From what I could see, Bob’s design with the interacting machines was a cybernetic system. I photocopied the articles, and then opened the books.

One was very easy to read. I read the first two chapters, and skimmed through the rest. There were charts and diagrams all the way through, all done using a standard format. The other book, which was much more technical, used the same kind of diagramming. I copied the first two chapters and several diagrams, and then it was 5:30 and I had to hurry back over to the sociology building.

Bob and his uncle were deep in conversation. I said, “Bob, I have to go home now. Are you coming?”

He reluctantly gathered up the printouts and his notes. Uncle Milt said, “I will be here tomorrow afternoon. I don’t think we’ll need to run any more computer analyses. Between what we did yesterday and today, you should have more than enough. If you want, you can bring your paper by after you’ve written it for me to look over.”

We thanked him and left.

Bob said, “Wow. It was like drinking from a firehose. I hope I can hang onto it all. It sounds so simple when he explains it.”

“Bob, we’ve got to get this thing done this week!” I said. “We need a plan. It looks to me like I can’t do much with the statistics, because they’re over my head. I can make the actual display, and I can write short articles about the scientific stuff we’re illustrating. You’ll have to write the technical part about the numbers, and we’ll work together on tying your stuff to my stuff.”

“Good plan,” said Bob. “I’m going to go home and get working on it right now.”

Bob worked on his section and I worked on mine the whole rest of the week. I bought a big set of poster boards and built a three-walled display that would sit on a table. I wrote short articles about each of the scientific concepts.

The toughest and most satisfying part was making a technical diagram of Bob’s system of machines. I had to go over it with him several times before I had it right. Bob was very impressed with it. I had drawn it using stencils from an engineer’s kit.

“That is so cool! I need to learn to draw like this,” he said. “It would help me a lot in designing my inventions. I know how to draw circuitry, but this shows more than just electronic components. It shows process and flow.”

That night he whipped out a schematic of his circuitry. He gave it to me the next morning, and I pasted it onto the side of the sheet with my diagram.

Bob was totally absorbed in his numbers. He kept talking about chi squares and t-values and r-values and other things that meant nothing to me, and he walked around bent double because he carried his printouts in his pack and fished through them every chance he got.

On Saturday, I went over to his shop. We lay the posterboard display on a plastic dropcloth on the floor, so it wouldn’t get dirty, and scattered our papers around it. Bob was still struggling with his numbers. I was beginning to get desperate.

“Bob, we have the display, these diagrams, and my articles, but without your part, it doesn’t mean much,” I said.

He shrugged. “I’m working on it. I’ll have it done.”

“Yeah, that’s what you said about our proposal, and you wrote that exactly three seconds before you got up and read it,” I complained.

We laid out what I had. We finally settled on an arrangement of the articles on the posterboard, with the diagram as the centerpiece, the other papers scattered around it, and the plastic sheets hanging below in layers. Each sheet now had a label that described its initial conditions. Bob was supposed to write the analysis, and add another page to the chaos theory paper to show how his system illustrated it.

I set to work attaching the papers that were done.

“Uncle Milt said we needed to be sure to copy everything,” Bob commented. “He said the computer printout was saved on the mainframe, but I should still copy the main results pages. I have the copies in my room.”

“I copied everything I typed,” I said. “Remember last year when that volcano erupted all over my stuff while I was setting up?”

“Yeah. Maybe we should spread plastic over the front of everything,” Bob said, half-joking. He had statistical papers scattered all over his desk and table.

I got the display arranged to my liking, and stood it up in its three parts. I stood back. “How does it look?” I asked.

“Lovely,” said Bob, not turning around. He was writing furiously now. “Just leave it all there nd I’ll put my papers on it when I’m done typing. My mom can bring it to school on Monday morning in the station wagon.”

Monday morning was hectic. The halls were crowded with kids carrying cardboard displays, boxes, models, trays of plants, all kinds of stuff. A section of the gym was set aside for the science fair.

I found Bob struggling to stand up our display. “Did you get it done?” I asked, taking the other side and standing it up.

He grunted. His eyes had dark bags under them, and were bloodshot. After we had the boards up and connected, I scanned the papers on the front. They were the same ones I had pasted up on Saturday!

“You didn’t get them done!” I exclaimed. “What are we going to do?”

“Patience, you idiot!” growled Bob. “Just give me time.”

He fished in his pack and brought out a grubby folder. My heart sank. However, the papers he pulled out were clean and neatly typed. He began to attach them to the board. I reached out to help, and he snapped, “I’ll take care of it.”

Hurt, I wandered off to see what the other displays looked like. Most were the usual kind, with diagrams, series of pictures, handwritten or typed signs and explanations. Nora and Junia had done theirs in bright colors, and had several platters of food and some Petri dishes and a mortar and pestle on the table in front of their poster board with its charts. Doug and Eddie had an oscilloscope set up, and printouts of sound waves. One kid was singing into their microphone and watching the waves of his voice on the display, while a group of other kids watched and laughed and joked.

The toilet display was close by. It was the most interesting, because there was a full-sized toilet cut right down the middle so you could see how it all worked, and there was a miniature toilet that actually worked. The explanation talked about water and air pressure, hydrodynamics, turbulence, bacteria, toilet paper.... It was really cool.

Tony was running his robot on the floor in front of his display, boasting loudly about how much work it had been, how well it worked, and all the things it could do. The little machine darted back and forth, bumping into people’s feet, backing off, and foraying in other directions.

I wandered back and found that Bob was just finishing his work. He grinned sheepishly. “Sorry I was so grumpy. I finished at 6:45 this morning, and then when I came to school, I ran into Tony at the door.”

I glanced at what he had posted. There were brief summary pages, with much longer reports behind them, pinned to the board. He had stretched red thread (dental floss, he told me later) from points in his report to the articles I had written, and had highlighted phrases and paragraphs here and there. There were threads running down to the plastic sheets, as well.

The machine was set up on the table in front of it, with a fresh plastic sheet around it and a few marbles showing in the tray at the bottom. “I added a hidden switch for all the power,” Bob said in a low voice. “I don’t want anyone starting it up or messing with it.” He showed me where the switch was.

Then class bell rang, and we trooped off to math. As we entered the classroom, Mr. Boyce looked at Bob’s face and laughed. “Are you going to survive, Bob, or do I need to send you to the nurse?”

Bob grunted and sat down. Mr. Boyce said, “I hope our conversation last night was helpful. But please don’t call me after 10:00 again.”

Bob flushed. “I appreciated your help, sir. It made a big difference.”

We limped through math class. Everyone was pretty distracted.

The science classes all met in the gym today, for the beginning of the science fair. Gym class was cancelled, so that each display could be looked at by the science teachers and explained by the students.

We waited patiently while the teachers made their rounds. I wandered over to see the more interesting displays and hear their presenters. The toilet one was really cool. The presenter, whose dad was a plumber, actually flushed the little toilet with a big wad of toilet paper in it so it clogged, then whipped out a tiny plunger to clear it out. The kid’s name was Mac. I had seen him in gym class, but had never talked to him. He had his talk all memorized, and wove in a lot of jokes.

Doug talked about sound waves, and Eddie illustrated with the oscilloscope, either with his own voice or that of one of the spectators. He also played some recordings through the machine. They had practiced a lot, and it was a great presentation.

Most of the presentations weren’t that interesting. Nora and Junia’s was okay, but I didn’t care enough about food to pay attention. They used a lot of big chemical and scientific words, and had molecule diagrams and illustrated formulas, and when they dripped chemicals on the foods in the Petri dishes, things foamed or changed color.

Tony’s presentation was just plain annoying. He talked loudly about the language the programming was written in, and how it made the robot move forward until it hit an obstacle and then change direction, and how this could be applied to run a vacuum cleaner or lawn mower. He said he had bought the parts at a couple of different electronics shops and a hardware store and a hobby shop, and had come up with the plans from looking at several existing machines and designs.

Behind me, Bob snorted. “He has no clue how it works, actually,” he whispered.

The teachers asked Tony several questions. He did know his stuff, I had to hand it to him. He knew the programming language and how the program worked, and had a diagram of the electronic circuit board and could explain it.

“I plan to write more maneuvers into the program soon,” he said. “I want it to be able to run some simple errands around the house. I’d like to add sound to it. It has a speaker, but I haven’t had time to put it to use. Right now it just alerts me if it’s stuck or if the battery is low.”

Everyone seemed to be quite impressed, other than Bob. Bob snorted again and went back to our display.

When our turn came, I talked about the scientific principles we wanted to illustrate, in particular chaos theory. I only mentioned briefly that it was cybernetic, thinking I would get back to it, only I forgot. Then Bob showed them the system, how each machine worked, and how they interacted. He showed them the plastic sheets, and how each one was different, and then he zoomed through the statistics to show how it illustrated chaos.

Mrs. Grissom smiled broadly. I think it went over the heads of most of the other teachers, although they were obviously impressed with how elaborate and ambitious it was. Mrs. G asked some questions about the statistics, which Bob was able to answer articulately, and then she asked me about cybernetics. I explained what a cybernetic system was, and showed how the machines worked together and affected and regulated each other.

It took a long time for us to get through our stuff, and the other kids got kind of edgy. Finally Bob fired up the three machines, and they got to watch the marbles whiz and whack the plastic, drop to the tray, and get fed back up to the top, and hear the changing speeds of the three machines. They thought that was pretty cool, but Tony said, “Seems like a pretty worthless invention. What does it do? Makes noise and slings dusty marbles.”

One of the teachers scolded him, and several kids snickered. Mrs. G congratulated us, and they moved on to the other displays.

We found some empty chairs and sat down. I tried to talk about the displays, but Bob lay his head on his arms and closed his eyes. Next thing I knew, he was asleep.

During our last class period, the results of the fair were announced. Five exhibits qualified for regional, and they were, in reverse order: Nora and Junia’s food project, Doug and Eddie’s sound waves, Mac’s toilet, our cybernetic chaotic system, and Tony’s robot.

Tony stood up and pumped his fists, then bowed around the speech classroom. A few kids clapped. Bob slumped in disgust. “I can’t believe they gave that bozo first place!” he hissed.

After school, Bob zipped off by himself. He was in a black mood, and he needed rest anyway. I went home, did my homework, and took it easy.

The science fair was on display all week. I enjoyed walking through and looking at the different displays. Bob showed up once or twice when we had to be there to answer questions from parents or students, but otherwise he stayed away. He was particularly angry at Tony, although I couldn’t figure out why.

It was a relief to have only the normal load of classes to deal with. I took a couple of afternoons to goof off by myself, and then went by Bob’s shop on Thursday after school. Bob was still in a bad mood. He was tinkering with a tiny cassette recorder on his workbench, and answered me only with grunts and monosyllables. There were several other pieces of electronic junk around. I picked up one interesting-looking little device.

“What’s this?” I asked.

“Came out of a doll,” Bob said. He clicked the record button on the recorder and tapped on the mike a couple of times. Then he rewound it and hit Play. Nothing happened, and Bob hissed and banged the recorder on the workbench in disgust.

I examined the little thing I had picked up. It was a little flat box with some electronic circuitry and a tiny speaker. There didn’t seem to be any buttons, though, and Bob was obviously in no mood to explain it to me. I set it down and went home.

The regional science fair was the next week. The school lent us a van, and the science teachers helped us load our displays and take them to a municipal auditorium in the next town, which was the county seat. We got a day off of school to do it. That was nice.

Bob and I set up our display. As soon as we were done and had checked everything, Bob sat down by the table and ostentatiously pulled out an issue of Technogeek magazine to read. It was from several months earlier. I wondered why he brought that one instead of the latest issue.

I headed off to see the other displays. They were quite varied, but a lot of them were very cool. One on leverage featured all kinds of equipment: levers, wedges, block and tackle, even a car transmission cut open to show the gears. Someone had a display on animals with mutations, covered with photos of animals with extra limbs, missing limbs, two heads, multiple tails.

After I had wandered through a good part of the auditorium, I made my way back to our table.

There were science teachers from a lot of other junior highs, as well as some high school and college teachers who were volunteer judges. One, from the university, knew Bob’s Uncle Milt, and came over to look our display over. He leafed through the printouts, read every word of Bob’s reports, looked at all the plastic sheets, watched gravely as Bob put the machines through their paces, and asked a lot of probing questions. Bob answered most of them, but I talked a fair amount, too. Afterwards, he shook hands with us and went off to talk with another couple of judges.

Tony’s display was near ours. He ran his robot whenever anyone came near, but he looked very nervous and not cocky like he had back at school. He kept a close eye on Bob, who ignored him and kept his nose in his magazine.

The presentations started at 9:00, in one corner of the room. As in our school, the judges went from display to display, and the students talked about what they had studied. They weren’t anywhere near done by noon. We were given an hour for lunch.

Bob and I went to a fast food place around the corner and bought burgers. Bob was in a better mood. “Doctor Gibbs really likes our project,” he said. “He was interested in every detail. After he talked to that other professor, that guy came and looked everything over, too. I think we’ll do better here. At school, no one understood what we were doing except Mrs. Grissom.”

We walked around the block, and then went back into the auditorium. When we reached our display, Bob said, “My magazine is gone.”

“Where?” I asked.

“I left it right here, on the corner of the table. It’s gone. I know where it went, too,” he said, angrily staring at Tony, who was now studiously ignoring him.

Bob kept staring at Tony and fuming. The judges were about to start with more presentations several rows away from where we were. Finally Bob walked over to Tony and said something.

Tony answered, “I don’t have your stupid magazine!”

Bob said something else, and they talked for several minutes in low voices. I caught an occasional word or phrase, something about Technogeek magazine and the principal, Mr. Bosnick, and Tony saying, “No way! I’d have to pull my display! You’re crazy!”

It looked very tense. I was tense, because the judges were done with the first presentation, which was very brief, and it looked like they were heading our way.

The judges stopped to listen to a presentation at one of the plant displays, and I breathed easier. I didn’t want Bob to be in a bad mood when they came, because it might make our presentation not be as good.

Bob finally left Tony and came back to our table, still breathing fire. “Says he didn’t touch it. Had lunch with Doug and Eddie. Pfaah! Right,” he muttered. He straightened up one of our reports and checked the plugs on the equipment.

I relaxed. The judges were still intently listening to the plant girls talking. Nora and Junia were in the crowd around them, I noticed. I hadn’t seen Doug and Eddie, nor Mac. I think they were way on the other side of the room.

When the judges finished asking their questions and examining the plants, they came to us. I couldn’t figure out why they skipped a couple of tables; I found out later it had to do with what grades the presenters were in. In any case, suddenly we were next.

Bob still looked grumpy, but he pulled himself together. Once the judges were in place, Dr. Gibbs said, “All right, boys. You may begin.”

I launched into my spiel about scientific principles and how our network of machines would illustrate them. Our speech teacher had helped me polish it up, and I had it mostly memorized. Bob talked, then, explaining the equipment. He did pretty well, although he didn’t sound very cheerful.

Finally the moment came to turn on the machine. We turned on the bottom machine. Then Bob flipped the switch for the middle one, and chaos ensued.

The plastic from the surrounding cage whipped loose and wrapped itself around the centrifuge, making the cage itself bang into the side of the centrifuge and almost knock everything over. At the same time, black ink came flying out of the funnel, liberally spattering our display, our faces, and our clothes. Bob got a big smear right in the middle of his school t-shirt, as well as spatters above and below. I was a couple feet away, so I got a pattern of blotches across the lower part of my shirt and a lot of little dots above and below. Mr. Gibbs had black speckles on his beard, glasses, and tie, among other places. Most of the other judges were further away and didn’t get hit.

Bob quickly turned everything off, and we stared around in disbelief. “Sabotage,” Bob said. He was surprisingly quiet.

Mr. Gibbs said, “Well. I take it that was not a part of your exhibition?”

“No, sir,” I said. “Someone poured ink into our machine. All we use is graphite and chalk dust.”

“I know who did it,” Bob muttered. He looked in Tony’s direction. Tony was at the back of the crowd, looking as shocked as everyone else, from what I could see.

One of the lady judges ordered someone to go for cleaning supplies, and Bob and I began to clean up as best we could. Our posterboard was heavily spattered with ink. The reports had whole sections blotted out. Trickles of ink ran down the big plastic sheets spread across the bottom and onto the table. The table had a plastic cloth on it, fortunately.

As we straightened up the machine and were untangling the plastic from the centrifuge, Bob said, “Aha! Here it is.”

He pointed to a length of clear strapping tape, now wrapped around the narrow part of the funnel. The plastic was attached to it.

“Someone,” Bob cleared his throat, “someone ran tape from our plastic screen to the centrifuge. Then when we started it up, it pulled the plastic loose so the ink could go everywhere.”

He looked in Tony’s direction. Tony was talking to some kid from another school. Bob looked fiercely angry. Then I saw him take a couple of deep breaths and relax. He muttered something to himself.

“Are you sure it’s not your own tape that somehow got stuck to the wrong place?” asked Mrs. Simms, the judge who was ordering people around.

I held up a roll of masking tape. “This is all we use, because we have to take the plastic off to analyze it after we’ve run the machine,” I said. “We don’t even have clear tape.”

The cleaning supplies arrived, and we set to cleaning. The judges had a quick conference, and then Dr. Gibbs said, “If you can get things repaired by 4:30, you can have another chance to give your presentation. I’m afraid that’s all we can do, because your division is judged tonight. Can you do it?”

I looked at Bob. He said, “I have extra copies of my reports in my pack, just in case. How about you?”

“My copies are at home,” I said. “I’ll have to call my mom.”

“We’ll do what we can, sir,” Bob said.

Dr. Gibbs nodded and said, “Good luck.”

The judges ambled over to Nora and Junia’s display, then, and we heard them give their animated talk as we cleaned.

I said, “I can have my mom bring an extra shirt if you like.”

Bob didn’t say anything, and then he said, “Let’s keep wearing these clothes. I have an idea on how to fit them in. We’ll clean up the floor and get a new table cloth, and we’ll hang new reports on the board, but we’ll leave the ink on our clothes and on the posterboard. It’ll add to what we’re illustrating.”

“The front plastic piece is the only one that was ruined,” I pointed out. “If we pull it off, the others should still be good.”

I found an office and called home. Fortunately, my mom was home. I told her what had happened, and where the reports were. She said she would bring them as soon as she could.

When I got back, Bob had gotten a new tablecloth. He was fuming. “Tony just gave his presentation. That fraud! He was all cocky and proud of his precious robot. He doesn’t know a thing about robotics, or computers, for that matter.”

After we had done what we could with the table, floor, and our display, we went to the bathroom and washed ink from our faces and hands. It came off with a lot of soap and scrubbing, but I still had some in the pores of my cheek and my arm that I just couldn’t get out. Bob had a splotch over his eye that gave him a sort of piratical air. The ink on the front of his shirt had run together into a sort of star shape.

“Looks like you took a cannonball in the chest, Cap’n Kidd,” I said.

“Avast, you scurvy scalawags,” he growled. “Man the mizzenmast. Prepare to repel boarders.”

My mom showed up not long after. She had found my reports. She was upset when she saw my clothes. “Why didn’t you tell me? I could have brought clean clothes for both of you!” she said.

“Mom, don’t worry,” I said. “We’re going to use the ink stains as part of our presentation.”

She looked over our display. “The spatters add a lot of texture, don’t they? It looks very interesting,” she commented.

I taped up my reports, and we ran fresh red string and threw away the stained floss. Bob stood back to admire it. “It’s actually better than it was,” he commented.

After making sure the ink was all cleaned out of the funnel, we set the machine up with a new surround, and ran it briefly to make sure it worked. Everything seemed fine.

Bob made notes and muttered to himself, working on his speech. He got his highlighter out and redid the highlighting on my reports.

I went over to the other side of the auditorium, where the judges were just getting to Mac and his toilets. He had woven more jokes into his spiel, and the audience obviously enjoyed his display.

Doug and Eddie were in the crowd. Eddie looked at my shirt. “Whoa, they got you good, didn’t they?” he said.

“Sure did,” I said. “We have a pretty good idea of who did it, but I don’t know if we can prove anything. What did you do for lunch?”

“We went down to the chicken place. Mac was there, and then Tony showed up and joined us.”

“He didn’t walk down there with you?” I asked.

Eddie frowned. “No. He came just a few minutes later, right after we sat down with our food. You think he did it?”

“I don’t know who else it would be,” I said.

“I heard the judges really liked his robot,” Doug interjected.

“I wasn’t there, but I’m not surprised. It is pretty cool, and he knows his stuff,” I said.

The judges went on to several more displays, and then Dr. Gibbs said, “Well, I guess now we go back to the chaos and mayhem exhibit.”

I scurried back to make sure it was ready and to warn Bob. We looked everything over one more time, and then the judges were there.

“All right, boys. Let’s try this again,” Dr. Gibbs said. “Were you able to repair the damage? I see your reports have been reprinted.”

“Yes, sir. We’re ready,” Bob said.

I went through my speech again, and then Bob talked. He was inspired. He talked about how the casual use of the word “chaos” differed from chaos theory, and about randomness and the idea that much of what we considered random was predictable if we had access to all the factors. He contrasted the random spatters on our clothes and board with the patterns on our plastic screens, and pointed out how they looked random if you looked at them horizontally, but vertically they showed a pattern based on centrifugal force. Then he talked about cybernetics and how the machines formed a self-regulating and modifying system.

When we moved to fire up the machines, the judges and spectators quickly stepped back. However, this time the machines whirred, the marbles spun out and whacked into the plastic and dropped into the tray, the motor speeds and chalk colors changed, and we ended up with a beautiful display on the plastic when we unwrapped it. Bob and I took turns showing how the impact marks on the plastic showed what could be predicted and what was effectively random, and how the system was chaotic in its intricacy, as shown by how the patterns differed from the sheets for which we had started the machines at different initial states.

It was a long presentation, but the judges paid close attention and seemed to understand what we were talking about. One guy had a lot of questions for Bob about the statistics, which Bob answered pretty well.

I heard the guy muttered to another judge, “I can’t believe these kids worked up these statistics themselves. I know a lot of graduate students who can’t do it.” A woman next to him nodded and said something I didn’t hear.

When they were done with their questions, Dr. Gibbs thanked us and congratulated us on an excellent display and a successful recovery from disaster.

“The results of the judging will be available late tonight,” he said. “They will be released to your schools in the morning. The displays will stay set up until Thursday, for public viewing.”

We gathered up our packs and got ready to leave. My mom had stayed around to watch our presentation, which had impressed her greatly, so we rode with her instead of going in the school van.

“I still wish you had worn clean clothes,” Mom said. “And gotten your face cleaner. But it was a really fine presentation. You’ve worked very hard on it.”

Bob was satisfied. He didn’t say much on the way back to school. I felt drained. We stopped at school, and I loaded my bike into the back of the car. Bob hopped on his bike, flipped it on, and bounced out onto the street. My mom shook her head as she watched him bounce from foot to foot before heading off up the street.

“That boy is going to get himself killed one of these days,” she said.

The next morning at school, I ran into Eddie at the bike rack. “Any word on the judging?” I asked.

He shook his head. “I went in and asked, and they said it was still being decided.”

Bob rode up, and I relayed the news. He nodded and began to unload his stuff.

Hiroshi burst out of the front door of the school. He came over to us, grinning. “I heard someone got you real good yesterday,” he said. “Ink everywhere.”

Eddie said, “You should have seen them. They looked like Rorschach blots. ‘What does this image make you think of?’”

Hiroshi said, “I asked, and the secretary told me that there was a question about the originality of one of the top projects, so they were still looking into it.”

Bob’s eyebrows went up. “So they caught on! I wondered if they would.”

Hiroshi looked puzzled at the comment. “What are you talking about?” he asked.

Eddie said, “He thinks Tony was the one who sabotaged his display yesterday.” Eddie’s brow furrowed. “But what does that have to do with originality?”

“Never mind,” Bob said, as the bell rang.

We went in to math class and sat down. As class began, there was an overhead announcement asking Bob and me to go to the principal’s office.

“What’s this about?” I wondered aloud.

“Probably something about the science fair,” Bob said.

We sat outside Mr. Bosnick’s office for a couple of minutes, and then he called us in. We sat in front of his desk, and he looked at us with his piercing blue eyes. “Well, boys, I got a call from the judges at the regional science fair. It appears that some of them were skeptical as to whether you did all the work on your project yourselves.”

“Our project?” exploded Bob. “They called you about our project?”

Mr. Bosnick nodded. “Yes. In particular, they thought it very unlikely that an eighth grader would be able to do the kind of statistical analysis that your report includes, or to design the circuitry that ties your machine together.”

He paused. Bob looked like he was about to erupt. “I worked day and night on those calculations!” he shouted. “My uncle helped me run the computer programs, and he and Mr. Boyce helped me work through the figures, but it’s all our data, and I wrote every word of that report! I waded through two statistics books! You can ask my uncle! Ask Mr. Boyce!”

“It’s true, sir,” I said. “We had help from his uncle, but he didn’t write any of the paper. He helped us, just like Mac’s dad helped him with his toilets. And I know Bob did the circuit board, because I helped him with the soldering. We had to make lots of changes as we went along.”

Mr. Bosnick nodded again. “I don’t doubt it. I told the judges that I knew you two to be excellent and very creative” (he cleared his throat) “students, and that you had never been known to cheat here at school. I asked them to talk with Mr. Boyce and Mrs. Grissom. I also told them that Bob has a reputation as something of an inventor and had shown some of his creations in school in the past. They did say that your paper had in fact referred to a local professor who is a colleague of one of the judges, and that they would be interviewing him. You understand, they didn’t formally accuse you of cheating. It was just that some of the judges were skeptical. I’m sure it will all work out fine. I just wanted to explain the delay to you.”

He dismissed us then, and we went back to class. Bob was seething. “Us cheating! I can’t believe it! Someone’s head is screwed on backwards!”

We muddled through our classes, somewhat distracted. Tony looked even more uncomfortable. He was sweating and tense.

Finally, just before lunch, there was an announcement on the overhead. “Attention. We have the results of the regional science fair. We are proud to announce that Nora Slatten and Junia Schmidt received an honorable mention for their project on the chemical properties of food.”

The announcement paused to allow students to clap. Then it continued.

“The displays of Mac Jones, Doug Wiggins, and Eddie Martin received recognitions of excellence, and were rated among the best at the fair.”

More cheering. Doug and Eddie high-fived each other and shook hands with me and Bob.

Then the announcer continued. “Bob Nelson and Mike Smith received second place at the regional science fair, and will be going on to the state science fair!”

Our class burst into cheers. Bob sat up and looked happier than I had seen him in quite some time. Tony was slumped down in his chair, looking ill.

After a few seconds, the announcer said, “First place at the regional science fair goes to Tony Barzini and his robot! Tony will also be going to state!”

The class burst into cheering again, except for me and Bob. Bob looked like a thundercloud had suddenly appeared over his head.

Tony looked startled, then relieved. Then it seemed to sink in, and he was suddenly exhilarated. He stood up and pumped his fists, then bowed and shook hands with everyone in reach.

The announcer said, “Thursday afternoon, the displays will be brought back to the school and displayed with their trophies in the science lab. On Monday, they will be exhibited during assembly, and each of the winners will present their project.”

I looked over at Bob. Bob was still stewing, but as I watched, a calm settled over him, and his face relaxed into a smile. He glanced at me. There was a glint in his eye that I had seen before. He reached over to shake my hand. “We’re going to state,” he said.

“We’re going to state,” I said, wondering about the glint.

After school, Bob announced he was going to the library. I tagged along. Bob lost himself in a stack of computer books and magazines, so I poked around and found a good spy novel. Bob wasn’t in a talkative mood, and it was obvious that he was going to be engrossed in his reading until time for supper, so I went home after a while.

I spent the rest of the week catching up on subjects that I had neglected for the science fair. Bob was in his own world, reading computer and electronics books and tinkering in his shop, and didn’t respond much to my attempts at conversation. When I asked about going over to his shop, he said, “Probably better not this week.”

Friday he came to school with his pack even more bulky than usual. He stuffed it in his locker, pulled out his books, and went to class. He was absorbed in thought throughout the day. During the walk from English to speech class, he pulled me aside. “Could you ride my bike home today?”

“Sure! It’s always fun to ride your bike,” I said. “Why?”

“I’ll bring you yours tomorrow,” he said, ignoring my question. He handed me the key to his bike lock. I wrote down my bike lock’s combination and gave it to him.

After class, we went to the science room and moved our display to the auditorium. The other kids were moving their displays as well. We set them up in a row on the platform. Tony couldn’t stop bragging to the other guys about winning, going to state, blah blah blah. He studiously ignored me and Bob, and left pretty quickly after setting up.

As we made our way to the front door to leave, Bob and I stopped at our lockers. I slung my pack on my back. Bob said, “I’m going to stop at the men’s room. Don’t wait for me. See you tomorrow.” He ducked into the restroom.

I went out and got his bike. The other kids had been picked up or left. I rode home, enjoying the power of a motorized bike that went 12 ½ miles an hour.

Saturday morning I had chores to do. Bob came by after lunch on my bike. “Thanks,” he said as he handed it to me.

“What did you need it for?” I asked.

“To get home,” he said. “And not to look like mine. You want to play some ping-pong?”

“Sure,” I said. We went into our garage, moved some stuff around, and opened the table.

Bob was an okay ping-pong player, and he really got into it. We played a dozen hard-fought games, laughing and hassling each other. It was the first time we’d just goofed off together all semester, it seemed like.

Afterwards we drank root beer on the porch, enjoying the brisk autumn air. “You looking forward to state science fair?” I asked.

Bob nodded. “We came so close last year. Boy, we worked  hard for it this year! That was a lot of work.” He was silent a minute, then he said, “You know, Tony was right. What does our machine really do besides sling dusty marbles? It’s really pretty worthless.”

“Well, that’s true,” I said. “But then again, what does his robot do except run into things and back up?”

Bob chuckled, then started shaking with laughter. Next thing I knew, he was laughing and rolling around on the floor.

“What’s so funny?” I asked.

He just howled harder. He rolled off the steps and into the prickly brown grass. He sat up, giggling.

“Man, I got to pee!” he said. He got to his feet and staggered into the house.

I followed him down the hall. “But what’s the big joke?” I asked.

He said, “Can’t tell you,” and slammed the bathroom door shut behind him.

I think he threw cold water on his face to calm himself down, because he came back out a while later with his shirt and hair all wet.

“I need to go home,” he said. “But I was going to tell you, I think when we show off our display on Monday morning, we need to be really brief, because the students aren’t going to understand anything about chaos theory or any of that stuff. To them, it’s just a machine that slings marbles. So we’ll just give a quick overview and then get out of the way. All right?”

“Sure,” I said. “Most of them saw it during the science fair anyway.”

“Okay,” he said. “Got to go. See you Monday.”

He ran off to his bike, giggling again.

Monday morning, I arrived at school early to be ready for assembly. Bob showed up a few minutes after I did. We went into the auditorium and onto the stage, to make sure everything was plugged in and working fine. I glanced into the funnel and the marble feeder to make sure they didn’t have any foreign substances. Tony had his robot zipping around on the floor. Mac flushed his toilet a couple of times. The other kids were tinkering with their displays or going over notes.

The auditorium filled up quickly once the bell rang. After the usual announcements, Mr. Bosnick gave a little speech honoring the scientific achievements of the honors students (and Mac, who wasn’t in honors). Then he had the girls present their chemical composition of foods display.

Bob wanted to stand on the side of the stage instead of by our display, by the nook where we’d left our backpacks. Doug and Eddie didn’t know what to do, so they stood with us. Mac had been by his display, but when he saw us congregated there, he sidled over and joined us. Tony stood by his robot in the middle of the stage.

Doug and Eddie were called next, so they went out and did a few things with the oscilloscope. Eddie got nervous and started giggling, which showed as ripply lines on the oscilloscope and got the whole auditorium laughing. Then Doug snickered while he was giving his next spiel, and Eddie started laughing harder. They were barely able to finish, and they staggered back over to where we were.

When Mac got up and started showing off his toilets, some kids started giggling immediately. He glared at them, and continued talking about air and water pressure and turbulence and buoyancy.

Someone yelled, “Buoyancy of what?” A bunch of kids laughed, and Mr. Bosnick stepped onto the stage and shushed everyone.

Mac got through his presentation, which like I said, was very good and pretty funny in places. He got a good round of applause.

Then it was our turn.

I talked first, and just as I got into it, Bob snickered. I faltered, and continued, with my voice kind of squeaky because I was trying to keep from laughing. Then I handed the mike to Bob. Bob gave a quick version of his talk. He looked like he was holding back a big laugh, and he talked faster and faster to finish before it escaped. Finally he erupted in giggles, and handed me back the mike.

I took it and said, “These three machines stacked here are a cybernetic system. What that means is they affect each other and respond to each other. But basically what you’re going to see is that this middle one spins out marbles and they whack against the plastic and drop into the tray at the bottom, and sometimes the speed changes to slower or faster, depending on what they’re telling each other.”

Bob hit the switches, and the machine started whirring and flipping marbles against the plastic. I held the mike near it so they could hear the motors slow down or speed up. The marbles sounded really loud on the PA system. One time I held the mike too close to the plastic, and a marble actually hit it and made a really loud “pop!” on the speakers.

Bob cut off the machine then, and waved to the audience as he walked away. I handed the mike to Tony and followed Bob. Bob put his notes into his backpack. I turned to watch Tony’s presentation.

Tony talked about computer languages, and circuitry, and the parts of a machine like his, and what it could do, and what its potential uses would be. It was pretty much the same speech he usually gave, although he didn’t sound quite as arrogant as he had at first.

Then he put the robot down on the ground and turned it on. “It’s going to go forward now, until it hits something. When it does, it’ll back up and go a different direction,” Tony explained.

The machine did go forward, until it hit Tony’s feet. It backed away, but instead of stopping right away and turning, it backed up about fifteen feet really fast, and then suddenly there was music playing, and the robot spun one way, then the other. The music sounded like it was playing over the PA system. The robot went forward in several jerky moves in time with the music, spun around, zipped off a few feet, spun one way, spun another, and moved in the jerky moves again. It was pretty cute.

Tony was staring at it with his jaw almost lying on the floor. The kids in the auditorium were very impressed. I heard exclamations of, “Awesome!” “That’s wicked!” “Whoa! Cool!” as the robot gyrated and zipped.

Then the music stopped, and so did the robot. Suddenly, over the PA system, we could hear Tony’s voice: “No way I’m telling Mr. Bosnick the plans came from Technogeek! Mr. Bosnick is a caveman. He’s probably never even heard of Technogeek!”

Tony jerked out of his stunned posture and ran towards his robot. The recording stopped, and the robot zoomed away from Tony, yelling, “Mama! Mama!” and sounding just like a talking doll.

Tony got within a couple feet of it and bent to pick it up, but it changed direction and zipped towards the front of the stage. As it turned to go across the front of the stage, it started zig-zagging, and quacking like a duck: “Quack! Quack! Quack! Quack!”

Tony chased it partway across the stage, then stopped, realizing he looked pretty foolish. The robot stopped and turned to face him. It lunged toward him a couple of feet, barking like a dog. “Woof! Woof!” Then it snarled.

Tony actually backed up a step. The robot cackled like Popeye: “Ah ga ga ga ga ga!”

Then it spun in circles like it was chasing its tail. “The plans came from Technogeek. Mr. Bosnick is a caveman! The plans came from Technogeek. Mr. Bosnick is a caveman!” it yelled over and over in Tony’s voice.

Tony lunged forward and managed to snag the robot as it turned to move away. As he picked it up, it began yelling, “Oh, Popeye! Save me!” in Olive Oyl’s voice. “Help! Save me!” it cried, as Tony escaped off the back of the stage. Just after he disappeared, we heard the Popeye laugh again: “Ah ga ga ga ga ga!”

The auditorium was in an uproar. I turned to look at Bob. Bob had picked up his pack and was zipping it up. He was laughing silently, tears streaming down his face. Eddie, Doug, and Mac looked puzzled but absolutely delighted. Nora was frowning, and Junia had her hand over her mouth to cover her giggles.

Mr. Bosnick came out on stage with a microphone and tried to shush the crowd. It was impossible.

“Mr. Bosnick is a caveman!” yelled one of the football players, and then several guys began chanting together, “Mr. Bosnick is a caveman! Mr. Bosnick is a caveman!”

Several teachers stood up and turned to quiet down the kids, but it didn’t work, especially when Mr. Bosnick started cracking up himself. He tried to make shushing sounds into the mike, but suddenly a guffaw escaped him, and he doubled over in laughter.

The assistant principal, Mrs. Ablington, came onto the stage and took the mike from Mr. Bosnick’s limp hand. She began to scold the crowd, and managed to get some of the kids to quiet down. As things settled, Mr. Bosnick pulled himself together.

Finally he took the mike back and said, “I have two things to say before I release you to your classes. The first is,” he turned toward those of us on stage, “Bob Nelson, I will see you in my office immediately. The second is for the rest of you.” He turned back to the front, hunched over, messed up his hair, swung one arm like an ape, and said, “Ooga! Ooga! Grog! You’re dismissed.”

I picked up my pack and turned to Bob. “I guess it’s the firing squad for you, old man,” I said gravely, and shook his hand. “Jolly good show. Stout fellow.”

He saluted me, and said, “So long, captain. Tell my dear Adelaide that I carry her lovely image to the grave, and that my only regret is that I have but one life to give for my country.”

Mac came over and shook Bob’s hand. “Awesome, dude. Awesome. I’ll come see you in juvenile detention.”

Doug and Eddie were right behind him. “I’ll remember that the rest of my life, man,” gushed Eddie. Doug said nothing, but shook Bob’s hand and nodded.

Mr. Bosnick collected Bob and they headed off for the office.

Bob didn’t come to any classes that day, nor did Tony. The rest of us buzzed all day about what had happened in the auditorium. As soon as school was out, I raced home and called Bob’s house.

Bob answered. “Yo, Mike. What’s up?”

“So, what happened? Did you get expelled?”

Bob laughed. “No, Mr. Bosnick was real cool. He asked me what that was all about, so I told him he needed to look at the Technogeek magazine from July. So right away he had the secretary go get it from the library. He read over the article, and kept nodding and muttering to himself, and then he said, ‘Well, I guess we had one pulled over on us that time.’ Then he told me I had to go home for the day for the sake of appearances, and next time I knew someone was cheating, to talk to him or to Mrs. Grissom or someone instead of putting on a public display like that.”

“Do we still get to go to state?” I asked.

Bob chuckled. “As far as I know. But Mr. Bosnick wouldn’t say anything to me about what would happen to Tony. He said it wasn’t my business, that he would handle it from here. So I got a note from him for my parents, and came home. I’ll be back at school in the morning.”

“So how did you do all that?” I asked. “Let me guess. The electronic works from that doll came into it, right? And your little cassette recorder? And those programming books?”

“Yeah. It was pretty complicated. I also went to a hobby shop and bought a couple of remote controls,” Bob said. “Best thirty bucks I ever spent. They’ve probably already found where I plugged my recorder into the sound system, so I probably lost that, too. But, man! It was worth it!” He started laughing again.

“Well, I have just one thing to say to you,” I said. “Ooga! Ooga! Grog.”