Numbed! by David Lubar

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.
    â€œIt’s something with pi, right?” Benedict said.
    â€œYeah. 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.
    â€œGot it!” I said as the image in my mind gave me the answer.
    â€œMe 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.
    â€œThat was easy. We’re one-fourth finished,” Benedict said.
    â€œRight. 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?
    â€œHow 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.”
    â€œI 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.”
    â€œThat’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?”
    â€œIt’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.”
    â€œYeah, 50 percent. Go ahead. You do it.”
    Benedict put in the answer. “That’s two,” he said. “We’re at 50 percent.”
    â€œJust like the coin.” I opened the door and walked through.
    â€œWow,” Benedict said. “Look at that.” He pointed at the ceiling.
    â€œYeah, 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