by David Lubar
“I’m not sure,” Dr. Thagoras said. “Nobody has ever done this.”
“Trapped,” Cypher said. “You’ll be trapped forever.” He laughed again.
“He’s joking, right?” I asked.
“He’s not programmed to joke,” Dr. Thagoras said.
“Maybe we should just go home,” Benedict said. “I mean, we can do arithmetic. That should be good enough.”
“Remember when you thought we didn’t need any math at all?” I said. “That didn’t work out very well for us, did it?”
“I guess not,” he said.
“Giving up now won’t work either,” I said. “We have to go.”
“You’re right,” he said. “But why worry? There’s nothing that can stump the two of us.” He turned toward the open door and said, “Bring it on, Mobius. We’re ready for you.”
I started to go inside and then looked back. “Any idea how many problems there will be?”
“Well, you had one problem the first time and then two the next time,” Dr. Thagoras said. “What do you think comes next?”
“Three?” I guessed.
“Possibly. But I think four might be more probable,” he said. “Remember, numbers like patterns. Starting with 1, you double it to get 2, and when you double 2, you get 4.”
“It’s a good thing this is the last time,” I said. “I wouldn’t want to go into a place with eight problems.”
Benedict and I entered the loop. “I hope there aren’t any more timers,” he said.
Dr Thagoras closed the door behind us. Numbers and symbols flooded back into my brain, along with other things I couldn’t identify. Some were just squiggles. Others looked like Greek letters. Suddenly, I knew how to solve those problems from the math test about pets, clothing, fences, and all of that stuff. Too bad I wouldn’t get a second chance.
We were in a low, narrow corridor. It seemed pretty square—the floor was as wide as the walls were high. I guess we were inside the ring. There was a steel door blocking the path to the left, so we headed to the right. There was just enough light to see where we were going.
“Do you understand this twice-around thing he told us about?” Benedict asked.
I thought about the video, with the guy drawing a line on the side of the strip. It still didn’t make much sense. “Not really. I hope it isn’t on the test.” I looked down as I walked. “Cool.” My footsteps left green glowing marks on the floor, showing me where I’d been.
There was a slight tilt to the walls and the floor. The farther we walked, the more we leaned sideways. After a while, everything tilted so much, the left wall became the floor and the ceiling became the left wall.
That’s when we reached the first problem. There was a door ahead of us, with a keypad and a screen. Actually, there were two screens. One was lit. The other, above it, was upside down and dark. I read the question on the lit screen:
WHICH HAS MORE VOLUME,
A CUBE 50 INCHES TALL OR
A SPHERE 50 INCHES IN DIAMETER?
The keypad had two keys, labeled Cube and Sphere. We’d learned about area in math and had started to learn about volume. “Do you know how to figure out the volume of a sphere?” I asked Benedict.
“It’s something with pi, right?” Benedict said.
“Yeah. But there has to be another way to figure this out.” I pictured the cube. It didn’t matter whether it was one inch or a million inches. I just had to picture a sphere the same size.
“Got it!” I said as the image in my mind gave me the answer.
“Me too,” Benedict said.
I tapped Cube. The sphere would fit inside the cube, so it had to have less volume. I saw another way to think about it. If I started with a 50-inch-tall cube, I’d have to carve parts of it away to make a 50-inch-tall sphere.
“That was easy. We’re one-fourth finished,” Benedict said.
“Right. But if we miss any of the four, we’re totally finished.” I opened the door and kept walking. Once again, our right wall gradually became our floor. As soon as we got to the point where the floor felt level, we reached the next door.
And once again, there were two screens. I saw a problem on the lower screen and a keypad below it, with numbers, an Enter key, and a % key.
Benedict read the problem out loud:
A COIN WAS TOSSED 5,000 TIMES.
IT LANDED WITH HEADS SHOWING
2,786 TIMES AND TAILS SHOWING 2,214
TIMES. ON THE NEXT TOSS, WHAT IS THE
PROBABILITY OF HEADS?
“How are we supposed to figure that out?” I said.
Benedict pulled a coin from his pocket. “I’ll start tossing. You keep track until we hit 5,000. It’s a good thing there’s no time limit.”
“I don’t think that’s how we find a solution,” I said.
Benedict flipped the coin and let it land in his open palm. “I guess you’re right. Besides, it’s not the same coin as in the problem.”
“That’s it!” I grabbed his wrist and pointed at the coin. “You’re right—it isn’t the same coin.”
He stared at his palm. “It doesn’t matter which coin we toss, does it?”
“It’s a brand-new toss. That’s the answer. It has nothing to do with what happened before. Five thousand tosses, five million, it’s the same. All we need to know is the chance of heads on the next toss.”
Benedict turned the coin over. “Two sides. Two ways it can land. So it’s one out of two.”
“Yeah, 50 percent. Go ahead. You do it.”
Benedict put in the answer. “That’s two,” he said. “We’re at 50 percent.”
“Just like the coin.” I opened the door and walked through.
“Wow,” Benedict said. “Look at that.” He pointed at the ceiling.
“Yeah, wow.” I saw footprints up there, leading away. We’d started our loop on the other side of this door. Except the ceiling had become the floor as we moved through the twist. One more loop and we’d be back at this door for the final problem. I was beginning to understand what was so special about this Mobius loop.
Once again, we went halfway around before we came to the next door. Actually, I realized, it was the lower half of the first door we’d come to in the loop. The part that had been the ceiling then, halfway through our first time around, was now the floor, halfway through our second time around. Every time we went halfway around the loop, the walls and floor made a quarter turn.
The keypad under the lit screen had two buttons. One was marked with an A. The other had a B. I read the problem on the screen:
WHICH WOULD BE BETTER TO GET?
A. $1 A DAY FOR A WHOLE YEAR
B. 1 PENNY THE FIRST WEEK,
2 PENNIES THE SECOND WEEK,
4 PENNIES THE THIRD WEEK,
AND SO ON, DOUBLING THE AMOUNT
EACH WEEK FOR A YEAR
“That sounds like an easy choice,” Benedict said. “I mean, one penny to start. How much could that end up being?”
“It could be a trick,” I said.
“Or a double trick,” Benedict said. “They make it sound wrong so you’ll think it’s right, but it’s really wrong.”
“It could be a triple trick,” I said. “Or even a quadruple one.”
“Oh, no, I hadn’t thought of that.” Benedict backed away from the door.
I didn’t really think it could be a triple trick, but I couldn’t resist seeing how Benedict would react. “Come on—let’s stop guessing and figure it out.”
I wondered how hard it would be to keep track of doubling the money and also adding in the total for each week. “I’ll do the doubling, and you add it. Okay?”
“Go for it.”
“Week one, 1 cent.”
“One,” Benedict said.
“Week two, add another 2 cents.”
“And 1 + 2 = 3,” Benedict said. “This is easy.”
“Week three, add 4 cents.”
“Then 3 + 4 = 7.”
“Week fo
ur, add 8 cents.”
“Then 7 + 8 = 15. It’s almost a month, and it’s nowhere near a dollar, yet,” Benedict said. “And the other way, we’d get $365. That has to be the right answer.”
“Hold on. I want to be sure. Where was I? Oh yeah. Week five, add 16 cents.”
“That’s … I lost track,” Benedict said. “Let’s start over.”
“No. I don’t think we have to. I guess 15 cents isn’t very much, after four weeks. But it’s still fifteen times more than we started with. I think the amount is going to grow so big that the answer will be obvious, even if we don’t bother adding the totals. Let’s start doubling, and see what we get.”
I held up my fingers, one at a time, to help me keep track: “1, 2, 4, 8, 16, 32, 64, 128.” I looked at my hands. We’d finally passed a dollar on the eighth week. I kept going.
“Then 256.” I paused for a second, wondering whether I could do the next one in my head. But I saw that I could ignore the 6 at first and just double the 250. That was easy—250 doubled was 500. Double the 6 and add it back in, and the result was 512.
I kept going. “Next is 512.” I had all ten fingers up now. I nodded at Benedict, who took over with his fingers as I counted. But I didn’t think we’d need to go much further. We were already up to $5.12 on the tenth week.
“Then 1,024. That’s $10.24. Then 2,048. That’s $20.48.”
“Wow, that’s already way more than $7 a week, which you’d get if you took a dollar a day,” Benedict said. “We’re only at the twelfth week, and we’re not even adding up the total from each week. You’re right—it’s going to get huge.”
I could have stopped right there, but I was curious. Rounding the $20.48 to $20, just to make the math easier, the weekly payment would grow to $40, $80, $160, and so on. And that was just the fifteenth week. I had no idea how much it would be at the end of a year, but I had a feeling it would take a long time to even write the number.
“I guess it really was a double trick,” I said.
“Yeah, the trick is to try to double your money as often as you can,” Benedict said.
I pressed B on the keypad.
The door swung open.
“Last one’s up ahead,” I said.
“We’re ready.”
“I hope so.” I was proud of how well we’d done, but I had a feeling the final problem might be four times harder than any of the others.
We walked along the second half of our second pass through the Mobius loop. We were now looking at the bottom half of the second door. We’d already done the problem on the top half during our first pass through the loop, when we’d been walking on the ceiling.
There was a screen. But there was no problem. Instead of a keypad, there was a whole keyboard. I hadn’t noticed that before, in the darkness.
“Welcome to infinity,” a familiar voice said. “You have this much time to solve the problem.”
Numbers started streaming across the screen, flowing faster and faster until they were a blur.
“I’m starting to not like this,” Benedict said.
I didn’t like it, either. I sort of knew what infinity was, but I didn’t think I really understood it.
“You can solve the problem or remain in this infinite loop,” the voice said.
“What’s the problem?” I asked.
“A hotel has infinite rooms, each of which can hold only one guest. Every room is filled. So the hotel also has infinite guests. Do you understand, so far?”
“Yeah,” I said. That didn’t seem too hard to imagine, as long as I didn’t think about it too closely. The rooms went on and on forever.
“I’m getting an infinite headache,” Benedict said.
I shushed him and waited for the rest of the problem.
“Infinite more guests show up. How can you find rooms for all of them without asking any of the current guests to leave?”
“What?” I asked.
The voice repeated the question. Then it added, “Take your time. You are allowed to make infinite guesses.”
Benedict stepped past me and typed: Build more rooms.
“Wrong,” the voice said.
Benedict stepped away from the keyboard. “Well, that was the best I could do. Any ideas?”
“Not yet. But we’ve solved every problem. And we’ve done it using things we’ve known. We have to be able to do this one. What do we know about infinity?”
“It’s big?” Benedict guessed.
“Yeah. But it’s more than big. It’s endless.”
“Let’s try that,” Benedict said. He typed: Put all the new people at the end, after the guests who are already there.
“Wrong,” the voice said. “There is no end to an infinite number of rooms already filled with an infinite number of guests. Nice try. Go on. You have infinite guesses remaining.”
“Forget it!’ Benedict said. “This is a trap. Dr. Thagoras and his robot want the two of us to stay here forever. I never did trust the two of them.”
“Two …” I said. I thought about everything I’d learned or discovered about numbers and patterns and binary robots. “I have an idea.” I wanted to think about it for a minute, because I wasn’t sure how to write it out.
“Oh, man,” Benedict said. “Having infinite time is worse than having a limit. The other way, it’s over quickly. That’s why they yank loose teeth. This way, you can suffer forever.”
I went to the keyboard. Even though I was still sort of fuzzy about infinity, I felt I had a good idea of how to solve this problem. There was someone in every room. If we moved everyone up to the next room, putting the person in room #1 into room #2, putting the person in room #2 into room #3, and so on, it would just empty the first room. That wouldn’t give us infinite rooms. But if we moved everyone at once, that would do the trick.
I typed my answer: Move each current guest into a room with twice as large a room number.
“Huh?” Benedict said, looking over my shoulder.
“You put the person in room #1 into room #2, the person in room #2 goes into #4, the person in #3 goes into #6, and so on. You free up infinite rooms.”
I could actually picture it. The infinite current guests would go into infinite even-numbered rooms. The infinite new guests would get the infinite odd-numbered rooms.
Guest in room: 1 2 3 4 5 …
Goes into room: 2 4 6 8 10 …
Freeing up room: 1 3 5 7 9 …
“That would never work,” Benedict said.
Before I could argue my point, the voice said, “Correct.”
The door swung open. We walked through and saw the very start of our footstep trail right in front of us. “We did it,” I said to Benedict. “We finished the Mobius loop.”
“Yeah, we did.” Benedict looked like he couldn’t believe it. “What are the odds of that?”
I shook my head. “I don’t even want to think about it.” I opened the outer door.
“Congratulations!” Dr. Thagoras said. “That should fix the problem for good, unless you find another way to get numbed.”
“Thanks,” I said. “I don’t think you have to worry about that. We aren’t planning to have any more conversations with robots.”
Benedict walked over to Cypher. “Hold on. I have one more thing to say to you.” He poked the robot in the chest. “Numbers are—”
“No!” I shouted.
“Awesome,” Benedict said. “I love numbers.” He leaned over and gave Cypher a hug.
Cypher said, “One plus one is two.”
I dragged Benedict away from his new pal, and we headed out of the museum.
“Now, we just have to survive one more thing,” he said.
“What’s that?” I asked.
“Getting our math tests back tomorrow. That’s not going to be fun.”
“Not fun at all,” I said. I could already imagine the look of disappointment in Ms. Fractalli’s eyes.
CHAPTER
52 ÷ 4
“You reall
y better watch where she puts her key,” Benedict said when we got to the classroom. “We’re going to need every little bit of help we can get.”
“For sure. And she’s not the only one who isn’t going to be happy.” I looked around at a room full of classmates who were probably already trying to decide whether to go for hot fudge or butterscotch. “Everyone’s going to be angry with us when they figure out who ruined the average.”
My hopes drooped even lower when Ms. Fractalli walked into the classroom. She unlocked the cabinet, but then she took the old lock off and put another one in its place. The new one looked like some kind of combination lock with a row of buttons on the front.
“That’s the end of that,” I said to Benedict.
“All we can do is wait for our doom,” he said.
We waited all morning, wondering when she’d tell everyone the bad news. But she didn’t say a word.
Finally, when the class headed out to recess and lunch, Ms. Fractalli said, “Logan, Benedict, I need to speak with you.”
Oh no. I looked over at Benedict, who was looking at me. I think we both gulped, trying to swallow. We walked up to Ms. Fractalli’s desk.
“I don’t understand your tests,” she said. “You both did perfectly on the first part and terribly on the last. The word problems counted for half the grade. Your scores dragged the whole class down below 85 percent. Can you explain this?”
Again, I looked at Benedict and he looked at me. I waited. He was really good at coming up with excuses.
“No explanation?” she asked. “I’d love to give you another chance, but I can’t without a good reason.”
“My mind sort of went blank,” I said. “I lost my math skills. But I think they came back.”
“Mine too,” Benedict said.
“I guess I’ll have to wait until the next test to find out,” Ms. Fractalli said.
She got up from her desk and went to the closet. I turned toward the door. “We’re doomed,” I said to Benedict.
“Everyone is going to be angry,” he said. “We might have to move. I have an uncle in Argentina. I’ve heard they have good steaks down there. You can come. He has lots of room.”
