It's daytime when you awaken on the Digits' asteroid (see "The Alien Baseball Field"). Beside you, Anita and Bill are just starting to stir. Most of the Digits are still asleep, but Eight is already up and walking toward you.
"Good morning! I have a story to tell you," Eight says.
Anita, Bill, and you sit down with Eight, facing one another in a square formation.
"Long, long ago in a galaxy far, far away, I lived where everything was perfectly predictable and regular," Eight says. "All of my neighbors were perfect cubes, like myself."
Noticing Bill's puzzled look, he explains. "Eight equals 2 cubed, or 2 ✕ 2 ✕ 2. My next-door neighbor was 64, which is 4 cubed, or 4 ✕ 4 ✕ 4. Each of us lived in a cubic house on a square plot of land, and the streets lined up like a giant checkerboard. The ground was perfectly flat, of course."
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A geometric cube. |
"I live in a town like that on Earth," Anita says."There are no hills in Cedarville, and all the streets form a grid."
"Ah, but when you stand on Earth and look up at the night sky, the stars appear to be scattered randomly, just as they are here in Pi Land," Eight says (see "Pi in the Sky"). "In my native galaxy, all the stars are lined up in the form of a three-dimensional grid, perfectly arranged in evenly spaced rows, columns, and stacks. We call it the Cubic Grid Galaxy."
"I'd like to see that!" exclaims Bill.
"You can. All you need is a laser navigator. When I left the Cubic Grid to seek my fortune in Random Universe, I brought along a bunch of them."
Handing you a sleek black box with a silver button, Eight continues, "This laser navigator will beam you straight to the Cubic Grid Galaxy. When you get there, press the 'zoom' button and it will send you in one of six directions, chosen randomly: left, right, up, down, forward, or backward. You can explore the galaxy by zooming randomly from star to star."
"Sounds awesome," Anita says.
"One warning, though: if your route crosses over and through itself to form a knot, you will enter a strange new world," Eight cautions.
Before you know it, Anita, Bill, and you are back inside your space capsule, speedily approaching the Cubic Grid Galaxy. You can see the galaxy glittering ahead, like a diamond in the sky! Then you enter the giant diamond and begin whizzing by row after row and column after column of brightly shining stars.
When you press the "zoom" button on the laser navigator, your space capsule shoots straight ahead and begins orbiting the nearest star. Then you press "zoom" again, and this time you head for the star to the left.
As you continue zooming, however, you notice that the laser navigator never lets you visit the same star twice.
You are having great fun roaming from star to star when suddenly you recall Digit Eight's warning. What are the chances that your random route will form a knot?
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