Mason couldn’t believe his eyes when Sue Ellen disappeared from the school playground. One second she was there, walking across the balance beam; the next second she was gone. Mason blinked a few times to make sure his eyes were working properly, and then he looked all around at the boys and girls playing on the bars and other equipment. Sue Ellen wasn’t among them, and apparently nobody else had seen her disappear.
Mason ran to the teacher’s aide and blurted out what he had seen—or what he hadn’t seen. The young woman knew about Mason’s inventiveness. She smiled, indulgently, and said, “She’ll turn up. It’s almost time to go back to class.”
Ten minutes later the teacher noticed that Sue Ellen wasn’t in her seat in the classroom. She asked whether anybody had seen her. Mason raised his hand.
“She was walking on the balance beam, and then she just…evaporated.”
“Mason, this is no time for jokes. If nobody knows where she went I’m going to call the office and have them look for her.”
Mason felt like the girl in a very old movie on TV who kept telling her father, “But the horse really did talk.” Adults were too smart to believe in things that couldn’t be logically explained.
By the end of the school day the police arrived. They talked to the teachers and asked the students in Mason’s class whether any of them knew what happened to her. Mason didn’t try to tell his story again. Police were adults. He wouldn’t have any more success with them than he’d had so far.
“Matthew, we have a problem.”
Matthew turned impatiently from the computer where he was deep into Africa on Google Earth.
“Why is it when you have a problem it’s always our problem.”
“Because I need your help.”
Mason hadn’t tried to talk to Matthew on the bus or the walk from the bus to their house. There were too many other children around, and he was tired of having people dismiss him as a jokester or just a kid with a wild imagination. When they arrived home his mom gave him a snack and then went off on some errand. He didn’t try to talk to her; after all, she was an adult. After he finished his snack he followed Matthew upstairs to the computer room where they were now.
Matthew had seen the police at the school and had heard about Sue Ellen’s disappearance. Mason told him his story. Matthew looked skeptical.
“People don’t just disappear.”
“Then why can’t anybody find her?”
Matthew didn’t have an answer for that. Mason remembered something he had forgotten in the confusion. Just after Sue Ellen had evaporated he had spotted a piece of paper on the ground at about the spot where he had last seen her. He had picked it up and stuffed it in his pocket. He retrieved it now. Something was printed on it.
Matthew looked over his shoulder and said, “What’s that?”
“It was on the ground near Sue Ellen.”
Matthew looked at the printing. “That’s a web address—a URL.”
Mason read it. “Kingdomofnim.com. Let’s take a look at the website.”
Matthew typed it in and pressed Enter. The words “Kingdom of Nim” came floating across the screen in different colors. The background was a beautiful vista of rolling hills, green trees, and blue streams.
Words appeared at the bottom of the screen. They read, “You must be 12 or under to access this site. If you are over 12 please EXIT.” There were two buttons: ENTER and EXIT.
“We’re the right age,” Mason said. “Click ENTER.”
Matthew looked dubious. “Maybe we should tell Mom what we’re doing.”
“Mom is over twelve. Go ahead—click ENTER.”
Matthew overcame his hesitation and clicked ENTER. The lovely scenic background remained, but some new words appeared on the screen. They read, “To get into the Kingdom of Nim you must win the following game.”
Four rows of marks were laid out on the screen like this:
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The instructions also appeared on the screen: “The layout for Nim is multiple rows of marks (or toothpicks, checkers, playing cards, coins, etc.). On your turn you may remove one or more marks from any single row up to and including all the marks in that row. You must remove at least one mark on your turn. Take turns with the computer. Whoever is forced to remove the last mark loses.”
“That sounds easy,” Mason said. “Let’s do it.”
“Wait.” Matthew was the cautious one. “It also says that if we lose this game we will never be admitted to the Kingdom of Nim.”
“Oh. We’d better be careful then. I wonder if that’s where Sue Ellen is.”
“It’s just a website, Mason. But let’s make sure we know how to win before we play against the computer.”
Matthew took a piece of paper and a pencil and began examining different moves. Mason helped. After a while they came to a conclusion voiced by Matthew.
“On our first turn we have to remove the whole row of 4.”
They rechecked their work and came to the same answer.
Matthew said, “Okay, here goes.”
Using the mouse, he moved the cursor through the four marks of the largest row, which eliminated them from the game, leaving rows of 3, 2, and 1. He clicked on YOUR TURN.
The computer removed the mark from the row of 1, leaving 3, 2. Matthew removed one from the row of 3, leaving 2, 2. The computer removed one from one of the rows, leaving, 2, 1.
“We win!” Matthew cried, removing the row of 2 and forcing the computer to take the last mark.
They weren’t prepared for what happened next. The scene on the computer screen suddenly became three dimensional, as if they were looking through a window. They felt themselves being pulled into the screen by an irresistible force. Mason grabbed Matthew’s arm and yelled as he tried to resist, but Matthew was being pulled in also.
The next few seconds were swirling confusion. They appeared to be flying through the air on a hurricane. Objects went by them, but they couldn’t see clearly what they were. Then they were dumped unceremoniously onto the ground.
“Ouch.” Mason spoke before he realized that he wasn’t really hurt. In fact, the landing was so soft that he didn’t feel it.
Matthew sat beside him on grass that was unbelievably green. They looked around at the hills dotted with multi-colored flowers and shapely trees. A couple of butterflies flitted by and a bird chirped somewhere. The sun shone in a blue sky that contained a few puffy white clouds. Matthew noticed that he could look directly at the sun without being blinded.
“Where are we?” Mason asked.
“This must be the Kingdom of Nim.”
“It’s a pretty place.”
“Yes, but why are we here? We need to get back home.”
“Wait. We’re here for a reason. We’re trying to find Sue Ellen, remember?”
Matthew scowled. “You have absolutely no reason to believe that Sue Ellen is here.”
“I think she’s here.” Mason nodded his head, trying to convince himself. The more Matthew said she wasn’t here the more Mason was positive she was.
“If she’s here, how are we going to find her?”
“We need to ask somebody.”
“Who, the bird in that tree or a butterfly?”
There was a distinct absence of people in the Kingdom of Nim. Mason stood up. Or rather he popped up. It didn’t take any effort on his part. He slowly turned completely around, scouring the countryside. In one direction he saw what looked liked the tops of buildings. He pointed in that direction.
“I think I see a city.”
Matthew stood up and looked. “Those buildings are miles away. We’ll never get there before dinner.”
“We don’t know how to get home. Do you have a better plan?”
Mason started walking on a dirt path that headed in the general direction of the city. It was strewn with pebbles and meandered all over the place. Matthew watched him go, convinced that this was a waste of energy. In spite of the fact that he didn’t seem to be walking fast, Mason was putting a lot of distance between them. He would soon be out of sight over that hill. Matthew began to panic. He knew they shouldn’t get separated.
“Wait,” he called. He ran after Mason. He was surprised at how fast he could go. He caught Mason within a few seconds without even breathing hard.
“Rule number one,” Matthew said. “We have to stay together.”
They walked side by side toward the city whose buildings were rapidly getting larger. Matthew realized that it wouldn’t take them long to get there. He had been wrong. The laws here were different than what he was used to.
A person came into view. It must be a person because it was walking on two legs, on a route intersecting theirs. The person was dressed all in red—red shoes, red pants, red short-sleeved shirt, red wide-brimmed hat—and appeared to have purple hair hanging down from under the hat. The reason Matthew and Mason weren’t positive it was a real person was because its face and arms looked as if they were blue
“Is that a boy or a girl?” Mason asked as they approached a meeting point.
“Or a man or a woman?”
“It couldn’t be a man or a woman. Adults aren’t allowed here, remember?”
If they were indeed in the Kingdom of Nim. It did appear that the person was probably a child.
As they came together, Matthew said, “Hello.”
The person stopped and looked at them. “Hello, Joe, what do ya know?”
“Not Joe, Matthew. And this is Mason. What’s your name?”
“What’s my name? Let’s play a game.”
Matthew was getting irritated. “If you won’t tell us, we’ll give you a name. Are you a girl or a boy?”
Mason said, “We’ll call you Joy because you’re so joyful.” He snickered.
The person said, “Girl or boy, my name is Joy.”
“Good,” Matthew said, “that’s settled. All right, Joy, where are we?”
“Just like your whim, in the Kingdom of Nim.”
Matthew shook his head. “Not my whim. All right, how do we get out of here?”
“Not so fast,” Mason said. “We’re looking for someone. A girl.”
“Looking for a girl, as priceless as a pearl.” Joy grinned at them.
“Her name is Sue Ellen,” Matthew said.
“Sue, Sue Ellen, Sue, cooked in a pot with mulligan stew.” Joy laughed.
Matthew was trying to keep his temper. He phrased his next question carefully.
“Do you know where Sue Ellen is?”
Joy didn’t say anything for a few seconds. She appeared to be thinking. Then she spoke.
“She’s very pretty, and in the city.”
Mason turned to Matthew, feeling excited. “See, she’s here. In the city.”
“If we can trust her.” Matthew wasn’t convinced. But what choice did they have? “Will you guide us there?”
Joy nodded. “Walk with me, talk with me.”
Joy set out toward the city. Matthew and Mason walked on either side of her, determined not to let her get away. They asked her questions, which she answered in rhyme, but her range of knowledge seemed to be limited.
In answer to a question about what she had been doing, she said, “Mining for red, in a color bed.”
When Matthew asked her again how to get out of Nim, she ignored the question. She apparently didn’t know where Sue Ellen was located within the city. Matthew took a different approach, asking her about the government of Nim, but she completely blanked on that.
They rapidly approached the city. It was surrounded by a very high stone wall, like that of a medieval castle. The tops of the buildings were visible above the wall, soaring to fabulous heights, but not always in a straight line.
Mason spotted a door in the wall that appeared to be locked. He asked Joy whether they could go in through the door.
Joy fished around in a pocket of her pants and said, “I’ve got a key, it works for me.”
She pulled out a very large and ancient looking key that she inserted in the keyhole of the door. The door creaked open and Joy went through the doorway. Matthew and Mason tried to follow her, but the door closed behind Joy and their attempts to hold it open failed. Matthew jerked his arm out of the rapidly closing gap, afraid that it would be crunched by the door.
Mason looked despondent. “What do we do now?”
Matthew didn’t know, but he was older and couldn’t show it. “We’ll think of something.”
Mason was studying the stone wall. “Maybe we can climb the wall.”
“It’s got a few footholds and handholds, but it’s very high.”
“Still, things are different here.”
Mason put a hand on the wall and tried to find a foothold. Both his hand and his foot slipped off the wall. He tried and failed several more times.
Matthew touched the wall and his hand also slipped off.
“The wall doesn’t have any friction. There’s no way to get a grip on it. Here’s another idea. You’ve noticed that we can cover more ground here faster and with less effort than at home. Maybe we can jump to the top of the wall.”
He bent his legs and tried to leap upward, but he found that he couldn’t even get off the ground. One foot always stayed in contact with the path. Frustrated, they stood and stared at the wall, as if they could disintegrate it with the evil eye. But that didn’t work either. After a few minutes, as they grew more and more disheartened, out of desperation they started walking around the wall, looking for a way inside.
Mason wondered what his brother was seeing. “What’s strange?”
“There, on the wall. It looks like telephone poles make up a section of the wall, instead of rocks.”
At least they looked like telephone poles. They must have come from thick and very straight tree trunks. Matthew and Mason walked up to them and touched them. Their surfaces were slippery and couldn’t be climbed, any more than the rocks could. There were small gaps between them, tantalizing, but much too small for even Mason to squeeze through, although he tried.
Matthew stood back and studied the layout for several minutes. It wasn’t just one line of telephone poles; it was multiple lines—or rows—of poles.
“Mason, what does this remind you of?”
“A bunch of telephone poles crammed together.”
“Come on, you’re the one with the imagination. Doesn’t this look like the layout of the Nim game we had to play to get in here?”
After a moment’s reflection, Mason had to admit that it did, on a much larger scale. There were four rows of poles. The row that was part of the wall had the most poles. Each row going out from the wall contained fewer poles; the last row had a single pole. Although the poles were too close together to count them all directly, Matthew was quite sure the layout was 7, 5, 3, 1.
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“So what?” Mason asked when Matthew told him that. “What are you going to do, make the poles disappear by playing Nim?”
“Exactly. We just have to find out the mechanics of playing the game. Look for something unusual close by.”
Matthew couldn’t tell Mason what to look for, but they started searching. Agonizing minutes went by before Mason called out.
“Here’s a button on the wall.”
Matthew came running over. “Push it. We don’t have anything to lose.”
Mason pushed the button. There was a soft scraping sound and a panel lifted up, revealing a computer sitting on a shelf. On the computer screen was a game of Nim with the layout 7, 5, 3, 1.
“That’s it,” Matthew said. “This is where we play the game.”
He glanced quickly over the rules. It was indeed the same game.
“It’s a different configuration than we played before,” Mason said. “How do we know how to win it?”
“We have to figure it out. Collect sixteen pebbles from the path.”
Matthew arranged sixteen pebbles in the same configuration as the computer screen. He began trying out various moves.
“This is a more complicated game. It’s going to take time.”
They worked at it, using a pebble to scratch combinations that showed promise into the dirt of the path. Matthew’s problem was that no matter how he made his first move, he was able to figure out a way in which the computer would win. He threw up his hands in frustration.
“I can’t find a winning move.”
“I have an idea,” Mason said, a little timidly. After all, his brother was older and knew more about math.
“Yeah, what is it?”
“Let the computer go first.”
Matthew stared at him. “But…but…”
The idea actually made sense. Why hadn’t he thought of that? Matthew went over to the computer and clicked on the YOUR TURN button, wondering whether it would tell him that he hadn’t made a move. Instead it eliminated one from the row of 5.
“Look.” Mason sounded excited.
Matthew glanced up in time to see one of the tree trunks in the row of 5 evaporate.
“Wow,” Mason said. “That’s the way Sue Ellen disappeared.”
It was working. Matthew eliminated one from the row of 7. The corresponding tree vanished. Mason ran over to the gap that appeared in the wall and tried to get through, but he was repelled by an invisible force.
Matthew shook his head. “We have to win the game before we can get do that.”
The computer took one from the row of 3. Matthew eliminated the tree from the row of 1. The layout was now 6, 4, 2.
“Take your best shot,” Matthew said, challenging the computer.
The computer’s best shot was to remove one from the row of six, leaving 5, 4, 2. Matthew made it 5, 4, 1. The computer went to 4, 4, 1.”
“It’s a snap now,” Matthew said, snapping his fingers and making it 4, 4. “Just keep it even.”
The computer removed one tree at a time. In turn, Matthew made it 3, 3 and then 2, 2, at which point Mason recognized the winning combination from the game they had played before. He let out a cheer.
As the last tree trunk evaporated, Matthew and Mason ran through the gap in the wall. As soon as they reached the other side, the trees reappeared behind them. Yes, they were inside, but now were they trapped here?
Suddenly, they were in the midst of a tremendous amount of hustle and bustle. People rushed past them in every direction, seemingly intent on important errands. They wore bright-colored clothes and their skin varied in color, but none of the colors resembled those the boys associated with the skin of human beings.
“This looks like a computer game,” Mason said.
Matthew thought about that. “Yes, it does. In fact, the whole Kingdom of Nim looks like a scene in a computer game.”
“Since we got to it through a computer, that makes sense.”
They saw crisscrossing streets with houses and other buildings on them. The streets were paved with what looked like cobblestones from old pictures. Matthew thought the uneven surface would be difficult to walk on, but when he tried he found that it was as if his body were weightless and he sort of skimmed along the top. He was about to peer inside a shop that had a split door, the top half of which was open, when Mason spoke.
“How are we going to find Sue Ellen in all this confusion?”
“We’ll have to ask somebody.”
A person dressed in orange with green skin came bouncing down the street toward them like a rubber ball.
“Excuse me,” Matthew said, trying to get the attention of what he guessed was a boy. “Can you help me?”
The boy ignored him and walked—or rather bounced—right past them, finally turning a corner and disappearing behind a building. Mason tried to stop somebody else with the same result. Both boys attempted to intercept other people, even standing in their way to block them, but they maneuvered around the pair and kept going.
Mason slumped against the brick wall of a house. “This isn’t working.”
He wondered if they were ever going to find Sue Ellen. For the first time, he also wondered how they were going to get home. What if they were trapped in this unfriendly place? Maybe Matthew’s caution had been a good thing.
Matthew was also thinking. “If this is really a computer game, it means these people are programmed to ignore us. We need to find somebody we can interact with.”
“Like Joy. She left us stranded at the wall, but at least she talked to us. She should be here somewhere.”
“Right.” Matthew could see the inside of the wall, looming up behind the houses. “If we walk back toward where she came through the wall, we might find her.”
They started walking along a street, keeping the wall on their left. They peered through windows and open doorways, looking for Joy. They checked out all the people, but none of them was dressed in quite the same red color that Joy wore.
The stores had signs over their doors with quaint names such as Literate Letters and Naughty Numbers. Mason saw a sign on a building that said Red Dye Works. He grabbed Matthew’s arm.
“Matthew, look at that sign. Didn’t Joy say she was mining for red? Maybe that’s where she works.”
“It’s worth a try.”
The building didn’t seem to have any windows and the door was closed. However, it opened to Matthew’s touch on the knob. They walked inside. Although there was no visible source of light, they could see just as well inside the building as outside in the sunlight. They were in a spacious room with plank walls and large, wooden beams above them. Several people were working at something. It appeared that they might be dyeing clothes the same red color that Joy wore. But where was she?
Mason was about ready to risk further rejection by asking one of the workers where Joy was when Matthew spoke.
“There she is.”
Joy had just come in through a back door, lugging a large vat. Matthew and Mason rushed over to her. She saw them and smiled.
“The wall opened wide, and you came inside.”
Matthew was about to add, “No thanks to you,” but he decided that wasn’t a good idea. They needed her help. Instead, he said, “Yes, we got inside, but we still need to find Sue Ellen.”
Joy put down her vat and appeared to think. Mason wondered whether computer characters could actually think. At least she was doing a good imitation. After a few seconds she spoke.
“I know where, I’ll take you there.”
Mason felt the best he’d felt since entering the Kingdom of Nim. He high-fived Matthew. They followed Joy out the door and along one of the twisting streets. She walked rapidly, and although they didn’t have any trouble keeping up with her they had to make sure that none of the many people on the street got between them and blocked their view of her. They didn’t want to lose her.
She kept turning corners, which complicated matters, but they stayed as close to her as they could. After what seemed like about their twentieth turn, Matthew wondered whether they were walking in circles. He was about to ask Joy about that when she stopped abruptly in front of a large building, apparently made out of the same kind of stone with which the city wall was built.
Joy pointed at the building. “Sue Ellen, Sue, is here for you.”
Mason went over to a door set in the stone and tried to turn the knob. He couldn’t budge it. Matthew tried also, with no better success.
“How do we get in?” Matthew asked Joy.
“Oh woe, oh woe, I just don’t know.”
Mason stifled a stronger exclamation and looked for another door or a window—anything—but the side of the building was completely blank except for the door. Matthew went to the corner of the building and scanned the adjacent side. That side didn’t even have a door.
Joy, who had been standing with a smile on her face, as if things were just fine, turned to leave. Mason knew that if she left they would be all alone again in this strange world.
“Don’t go, Joy. We need your help.”
She turned toward them. “I will stay with you today.”
Matthew thanked her and also felt relieved. He tried to think of another question he could ask her about getting into the building, but he suspected that she had nothing more to tell them. Just having her around helped his spirits. It was up to him to get them inside. What was the logical way to do that? This was apparently a computer-generated land, and computers were nothing if not logical.
“Do you think we can get in the same way we got through the wall?” Mason asked.
“I was just about to say that.” Matthew easily convinced himself that he would have come up with the idea. “Let’s look for a computer or a button or something.”
Mason started walking along the wall, examining it closely. Matthew went back to the door and looked at it again. Just above the knob where he had assumed was some sort of a lock, he now saw small metal pins sticking out from the door. They were in a familiar configuration:
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“Mason,” Matthew called, and Mason came running back to the door. “The reason I didn’t see this before is because it’s so small. A little different than the tree trunks in the city wall.”
“Where’s the computer?” Mason took the glasses he wore at school out of the case in his pocket and peered at the pins. “Look, there’s a button underneath that says, “YOUR TURN.”
“It’s the old familiar 7, 5, 3, 1 configuration, so we’ll let the computer start.”
Matthew pressed the button. He was feeling very in control. Three of the pins immediately disappeared from the row of 5, leaving 7, 2, 3, 1.
“Nice try. The computer’s attempting to be clever, but I’m cleverer. Notice that if I remove the row of 7, it will leave 2, 3, 1, which is the same as 3, 2, 1. Get out of the way.”
“I’m reading the rules.” Mason had his head close to the door and was reading from a small plaque.
“We already know the rules.” Matthew reached past Mason’s head and found that he could pull the pins from the row of 7 out of the door. When he did they immediately vaporized. He pressed YOUR TURN.
“Wait!” Mason’s voice sounded frantic.
“What’s the matter now?”
“The rules have changed.”
Matthew had a moment of panic. What was Mason talking about?
“What do you mean, the rules have changed?”
“Instead of the player who takes the last one losing, that player is now the winner.”
“Let me see.”
Matthew elbowed Mason out of the way and read the rules. Mason was right. What had he done? Had his impatience lost them the opportunity to enter the building? He looked at the remaining configuration, which was 1, 3, 1. The computer had just removed one from the row of 2. His turn. He needed to be able to remove the last pin. How could he do it? After a bad minute or so, Matthew saw the answer. He breathed a silent sigh of relief.
“It’s okay. I’ve got it under control.”
Matthew removed the whole row of 3. The computer had to remove one pin and Matthew removed the last one. The door clicked open.
Mason gave his brother a hug. Matthew shrugged him off, realizing how close they had come to disaster. They went into the building, making sure that Joy followed. When the three were inside, the door clicked shut behind them. Neither of the boys tried to open it again. They didn’t want to verify that they were now locked inside a building inside the wall inside the Kingdom of Nim.
They were in a large, empty room, emphasis on empty. The ceiling was so high they couldn’t see it without looking straight up. Mason turned to Joy.
“Where’s Sue Ellen?”
“She’s with her cares, up those stairs.”
Joy meant a spiral staircase in the corner of the room. A very tall spiral staircase. They started climbing the metal stairs. Round and round. It seemed to go on forever. Don’t look down. Mason was getting dizzy, but, strangely enough, not tired. Finally, they came out onto the roof of the building.
Matthew arrived first and glanced quickly around. “This must be the tallest building in the city. We’ve got a 360-degree view.”
A view of not only the city but the city walls and the countryside beyond. It was a landscape worthy of a king. Mason came clambering up the stairs next. He was more interested in finding Sue Ellen. He spotted some children at the other end of the roof and raced toward them. He heard Sue Ellen’s voice before he recognized her.
“Mason. What are you doing here?”
She stood up and hugged him and wouldn’t let go.
She asked, “Did they take you too?”
Mason stepped back, embarrassed by her affection. He wasn’t sure who “they” were.
He said, “We came looking for you.”
He recognized the others, two boys and two girls, huddled together. They were all from his school.
Sue Ellen said, “They arrived after I did. They were taken when they were getting on the school buses to go home.”
Matthew had crossed the roof. He knew Sue Ellen slightly through Mason. Her long, red hair was a mess and it looked as if she had been crying. He had questions.
“Are you captives?”
Sue Ellen said, “We were kidnapped and brought here. We can’t get away. I guess that makes us captives.”
“Who kidnapped you?”
The boys and girls looked at each other. Sue Ellen spoke for them.
“We think we were kidnapped by computers.”
It was Matthew’s and Mason’s turn to look at each other. Computers? Matthew turned to Joy who had joined them.
“Why are they here?”
Joy looked thoughtful, if someone with a blue face can be said to look thoughtful. When she spoke it was with more than her usual animation.
“With flags unfurled, we’ll rule the world.”
Mason couldn’t believe what he was hearing.
“Computers will rule the world?”
Matthew said, “Computers don’t have feelings. Without feelings they can’t have ambition. Therefore, a computer can’t have the desire to rule the world. They don’t think the way we think. A computer can do calculations very rapidly and make you think it’s thinking, but it isn’t.”
“All that thinking about thinking is blowing my mind.”
Mason’s statement brought smiles from the children. Even Matthew smiled. He decided he’d better explain better.
“For example, a computer can play a good game of chess. But it doesn’t figure out moves the same way a person does. It relies on speed to try out millions of combinations.”
Sue Ellen said, “I know you’re older and understand more things than we do. But the fact remains that we were brought here using technology run by computers. We hadn’t seen any human beings until you arrived. Only computer-generated images like that.” See jabbed her finger at Joy.
For Mason, it all came down to one question. “Why do the computers want you here? They should be capturing presidents and other important people.”
“Because they want to pick our brains. Pick them apart. Find out how they work. Then they want to duplicate them.”
“That’s scary.” Mason shuddered.
“You bet it’s scary. Why do you think we’re so scared?”
Matthew was still trying to figure this out. “There must be some human intelligence behind this. Computers were created by human beings.”
Sue Ellen pointed to a computer that Matthew hadn’t seen before.
“That thing has been running all day, printing out tons of reports about their plans. At first there was an audio telling us about the plans, but the voice was so spooky and inhuman we shut it off. Of course you’re right that this whole thing was started by a person. But we think whoever it was is no longer connected with what’s going on.”
Matthew had a vision. “Somebody turned on the switch and then went away and forgot about it. The program will keep running until—”
“Until somebody turns off the switch,” Mason finished.
“How do we do that?”
Nobody answered. Everybody looked very gloomy. Matthew turned to Joy.
“How can we stop this craziness?” Joy wouldn’t understand that. He tried again. “How can we stop computers from taking over the world?”
Joy spoke without hesitation. “With vigor and vim, and a game of Nim.”
Matthew almost smiled. “Of course.”
“What does that mean?” Sue Ellen asked.
Mason explained. “The way we got here was by winning three games of Nim. It’s a mathematical game. It’s a good thing Matthew is a whiz at math.”
“Mason helped,” Matthew said, trying to sound modest. “Computers are good at mathematical games. But, fortunately, they’re also programmed to follow the rules. So they can be beaten.”
He went over to the computer and searched through its program files until he found what he was looking for. He brought up a screen that showed a Nim game that looked like this:
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Mason looked over his shoulder. “That’s by far the most complicated game we’ve seen so far.”
Matthew nodded sadly. “We won’t solve this one by trying out combinations of pebbles.”
Sue Ellen and the other children crowded around the screen.
One boy said, “You mean you can get us home by winning that game?”
Matthew nodded. He was reading the text that accompanied the game. It appeared that this was so. The children were beginning to sound excited. He didn’t want to give them false hope.
“The problem is, if we play and lose, that’s it. We’re here forever. Or until the computers dispose of us.”
Everybody groaned. Sue Ellen hugged Mason again.
“You risked your life by coming here. I won’t forget that.”
It was small compensation. Mason wanted to go home. There must be a way to figure this out. He turned to Matthew.
“Since people created computers, people must be able to think like computers.”
“As I said, computers can’t really think. In fact, they’re so elementary they don’t even use our number system. We use what’s called base ten or decimal. We have ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Computers only use two digits: 0 and 1. That’s base two or binary.”
Sue Ellen scratched her head. “But computers can work with numbers in what you call the base ten.”
“They do calculations in binary and then convert the answer to decimal using a table. I’ll show you.”
At least it would take their minds off their problems for a while. Matthew counted the marks in the rows of the Nim puzzle and came up with 19, 18, 17, 16, 15. He borrowed the piece of paper Mason still carried with the web address that had gotten them here, and scrounged a pencil from one of the girls.
He listed the numbers vertically on the paper and said, “I’ll put the binary equivalent next to them.”
19 – 1 0 0 1 1
18 – 1 0 0 1 0
17 – 1 0 0 0 1
16 – 1 0 0 0 0
15 – 1 1 1 1
Matthew was a born teacher. He said, “In our decimal or base ten system, each time you move a position to the left you have a number ten times as large. For example, 10 equals 10 times 1. 100 equals 10 times 10. In the binary system each time you move a position to the left you have a number two times as large. So 1-0 equals 2, 1-0-0 equals 4, 1-0-0-0 equals 8, and so on.
“In the Nim game, 15 is composed of a 1, a 2, a 4, and an 8. Add them up and you get 15. The next binary position to the left is 16, which is why the 16 is composed of a 1 and four zeroes.”
There was some oohing and aahing as several of the children understood. Mason was catching on, but he felt it would be easier if Matthew simplified it a little.
He said, “Matthew, show the previous Nim puzzle in binary.”
Matthew listed the rows in decimal and binary:
7 – 1 1 1
5 – 1 0 1
3 – 1 1
1 – 1
He explained how each decimal number was converted to binary. For example, 7 equals 1 + 2 + 4. Mason was studying the binary numbers. He spotted something very interesting. He grabbed Matthew’s arm to get his attention.
“Notice that there is an even number of ones in each column.”
Matthew was about to shrug this off when he had what he would later call a Eureka moment. He studied the configuration as his excitement grew. He could no longer contain himself.
“That configuration, 7, 5, 3, 1, is what I call a winning combination, because if you leave it for your opponent, there’s no move he can make that will prevent him from losing. It appears that all winning combinations have an even number of binary digits in each column.”
They tried some other winning combinations to make sure, such as 6, 4, 2 and 5, 4, 1 and 3, 2, 1 and 4, 4 and 3, 3 and 2, 2. The theory worked every time.
Mason voiced their thought.
“We know how to win that game.”
They looked at the configuration again.
19 – 1 0 0 1 1
18 – 1 0 0 1 0
17 – 1 0 0 0 1
16 – 1 0 0 0 0
15 – 1 1 1 1
Again, Mason beat Matthew to the punch.
“Erase the row of 15 and you’ll leave a winning combination.”
Matthew’s hands were shaking as he operated the computer. He very carefully made his move. The computer responded. Slowly, carefully, he continued moving, always checking with Mason first to make sure he was leaving a winning combination. The rules on this computer said that the player who removed the last mark won.
It finally got to the point that Matthew left a 1, 1 for the computer. The computer had no choice but to remove one of them. All Matthew had to do was to remove the remaining one. He turned to the children.
“Everybody hold hands and get prepared. We’re going for a ride.”
They were all tense with excitement. Matthew held Mason’s hand with one of his hands. With the other he operated the mouse to remove the last mark. As he did so everything started to shake.
“Where are we?”
The question, voiced by one of the girls, was logical considering that they had just been whirling through space—or at least that was what it felt like. Matthew opened his eyes and saw green grass—not the perfect artificial green grass generated by a computer, but real grass with all its blemishes and brown spots. It looked like heaven. When he raised his eyes he knew exactly where they were.
“We’re at the school.”
They all let out a cheer. Within seconds, adults began to converge on them, including parents, teachers, and police. They had been using the school as a command post to search for them.
Mason turned to Matthew, amid the confusion, and said, “Mom’s not here. She doesn’t even know we’re missing.”
“Good,” Matthew said. “That means she’s not worried about us. Anyway, we’ll be home for dinner.”
Winning the Big Nim Game
Matthew and Mason find out that the “Secret of Nim” is to convert the number of marks in each row of the Nim configuration to binary and make sure each binary column contains an even number of ones (or none at all). While playing the game that will get them home they start with a configuration of 19, 18, 17, 16, 15. Their first move is to eliminate the row of 15, leaving the following configuration:
19 – 1 0 0 1 1
18 – 1 0 0 1 0
17 – 1 0 0 0 1
16 – 1 0 0 0 0
Note that the total number of ones in each binary column is even or zero. What move should the computer make? Since this is what Matthew calls a winning combination, the computer can’t make a move that will produce another winning combination. However, that doesn’t mean it can’t muddy the waters. For example, it can take one from the row of 16, leaving the following configuration:
19 – 1 0 0 1 1
18 – 1 0 0 1 0
17 – 1 0 0 0 1
15 – 1 1 1 1
Now all the binary columns have an odd number of ones. Should Matthew and Mason panic? Of course not. They have several options. For example, they can take 3 marks from the row of 17, leaving the following configuration:
19 – 1 0 0 1 1
18 – 1 0 0 1 0
14 – 1 1 1 0
15 – 1 1 1 1
The computer, if it could show emotion, would be gnashing its teeth about now, but it’s not finished yet. It can take 5 marks from the row of 18. (If it reduces the row of 18 to either 14 or 15 the solution becomes trivial, because all Matthew and Mason have to do is balance the rows. For example, if the computer leaves 19, 15, 14, 15 M & M make it 14, 15, 14, 15. After that, regardless of what move the computer makes they keep the configuration balanced by leaving 2 pairs of rows of equal length.)
But let’s assume the computer takes 5 marks from the row of 18, leaving the following configuration:
19 – 1 0 0 1 1
13 – 1 1 0 1
14 – 1 1 1 0
15 – 1 1 1 1
The computer has once again managed to create a configuration with an odd number of ones in each binary column. By now, M & M are ready for this. They take 7 from the row of 19, leaving the following configuration:
12 – 1 1 0 0
13 – 1 1 0 1
14 – 1 1 1 0
15 – 1 1 1 1
Thus they can counter whatever move the computer makes.
Changing the Rules
In the middle of the story the computer changes the rules in regard to what constitutes winning. At first, the winner is the player who isn’t forced to remove the last mark. But then the rule changes and the player removing the last mark wins. Matthew manages to allow for the rule change and win the game. In fact, the rule change doesn’t change the strategy at all. The only difference occurs right at the end of a game. If the player who removes the last mark loses, the winning player has to be careful to leave 1, 1, 1 instead of either 1, 1 or 1, 1, 1, 1. You’ll note that this is a violation of the binary strategy, which is geared to the player who removes the last mark winning. Regardless of what constitutes a victory, leaving 2, 2 is a sound strategy, since it gives the opponent no good option.