Rolling Around 1. The portrait faces left, just as it began. The quarter rotates 180 degrees for rolling a distance of half its circumference, plus 180 degrees for rolling halfway around a stationary circle, for a total of 360 degrees.
2. If the diameter of the stationary quarter is twice that of the rolling quarter, the quarter rotates 360 degrees for rolling a distance of one full circumference, plus 180 degrees for rolling halfway around a stationary circle, for a total of 540 degrees. So the quarter is upside down. If the stationary quarter is half the size of the rolling quarter, the quarter rotates 90 degrees for rolling a distance of one quarter its circumference, plus 180 degrees for rolling halfway around a stationary circle, for a total of 270 degrees. So the portrait faces up.

3. The north pole of the rolling globe will be touching the south pole of the stationary globe. In fact, the rolling globe always ends in the same orientation as it began, no matter what path it takes. To see why, imagine stacking up a stationary globe, a plate of glass, and an upsidedown mirrorimage globe, with its north pole pointing down, as shown above. The two globes appear to be reflected in the piece of glass. As one globe rolls around the other, imagine that the glass tips so it stays sandwiched at the kissing point of the two globes. If you were to hold the glass still and roll the globes instead, you would see that the two globes are always mirror images of each other. The mirrorimage relationship remains true if you hold one globe still and let the glass and the other globe move. So when the rolling globe reaches the bottom, it will continue to be a mirror image of the stationary globe, which puts it in the same orientation as it began.

4. If you roll the smaller globe down the larger globe along a single line of longitude, then the north pole of the smaller globe is pointing down when the smaller globe reaches the larger globe's south pole. But by rolling the smaller globe differently, you can force it to any desired orientation when it reaches the bottom of the larger globe. First, roll the small globe down to the bottom of the big globe, as shown above. The north pole is now pointing down. Second, roll the small globe up one line of longitude to the equator, turn to roll a distance around the equator, then turn again to continue down to the bottom. By choosing the correct lines of longitude and distance around the equator, you can make any point of the rolling globe touch the south pole. You can also "twist" the orientation of the smaller globe while keeping the same point touching the south pole by rolling it back to the top along one line of longitude, then back down along a different one. By using both operations— changing the point at which the smaller globe touches the south pole and "twisting" about the point by bringing the ball up and down along different lines of longitude— you can vary the smaller globe's orientation.
Round Trip There are many answers to these problems if you allow such devices as treadmills, continental drift, and wormholes. Here are answers that require only spherical planets.
1. The explorer started at the north pole. The explorer could also have started 10 miles north of a ring around the south pole that has a circumference of 10 miles (or any other distance that divides evenly into 10 miles).
2. The explorer started 5 miles north of the south pole. Note the difference between walking in a direction 10 miles and turning to face a direction and walking forward 10 miles.
3. The explorer starts from any point other than the south pole on an asteroid with a circumference of 10 miles (or any circumference that divides evenly into 10 miles).
Around the World 1. Most people guess the string would be raised an imperceptible amount, but in fact, increasing the circumference of any circle by 20 feet increases the diameter by 20/
. Therefore the circle's radius is increased by half that amount, or a little more than 3 feet.
2. The pole would be approximately 1,332 feet tall. Because the string is 20 feet longer than the circumference of the Earth, all the difference in length occurs where the string leaves the Earth (see the diagram above). Therefore, the string segment CD (from the top of the pole to the point where the string leaves the Earth) should be 10 feet longer than the arc BD. We need to compute CD and BD in terms of the angle F. The algebra to solve for F exactly is tricky. To find a value for length BC, try using a spreadsheet program or sweat it out doing successive approximations.
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