Solution for #1
Push the cork into the bottle and shake the coin out.
Solution for #2
Put the valuable object into the box, secure it with one of your locks, and send the box to your friend. Your friend should then attach one of his own locks and return it. When you receive it again, remove your lock and send it back. Now your friend can unlock his own lock and retrieve the object.
Solution for #3
After you draw one of the papers, swallow it. The jailer will be forced to check the remaining paper to determine what the one you drew said. The jailer will of course see a paper with "DEATH" written on it, assume you drew the one with "LIFE" written on it, and set you free.
Solution for #4
You can make 120 cubes (10 full trays) in the time it takes to freeze two trays. First, fill four of the trays with water and turn the other three upside down and use them to space the four apart. That gives you 48 cubes. Next, empty the four trays and put two ice cubes in diagonally opposed corners of each of six of the trays. Fill the remaining holes -- and the entire seventh tray -- with water. Using the ice cubes to hold the trays apart, stack all seven (the seventh tray should go on top), and freeze them. You'll get an additional 72 cubes. You can get 72 cubes for every batch except the first, for which spacer ice cubes are not yet available.
Solution for #5
A square manhole cover can fall into the hole on the diagonal; a round manhole cover cannot be dropped into the hole.
Solution for #6
Turn on switches 3 and 4 and wait fifteen minutes or so. Then turn switch 3 off, turn switch 2 on, and enter the room. If the bulb is dark and cool, switch 1 controls it. If the bulb is bright and cool, switch 2 controls it. If the bulb is dark and warm, switch 3 controls it. If the bulb is bright and warm, switch 4 controls it.
Solution for #7
The first surgeon operates with the first glove (glove A) inside the second glove (glove B). The second surgeon operates using just glove B. The third surgeon operates using glove A, turned inside out, inside glove B.
Solution for #8
Alternate Solution #1
Almost all of it. First tie the ropes together at the ends. then climb one of the ropes. Tie a loop in the rope as close to the ceiling as possible. Hang from the loop, then cut the rope just below it (but don't drop it, or you'll be stuck on the ceiling). Run the rope through the loop and tie it to your waist. Swing over to the other rope and pull the rope that's going through the loop tight. Cut the other rope as close as possible to the ceiling, holding tight to the end of it. Once the rope is loose, you will swing down below the loop. Let yourself down to the ground by letting the rope out. At the bottom, untie yourself and pull the rope completely through the loop. At this point you'll have all of the rope except what was used to tie the loop.
Alternate Solution #2
All the rope, provided that the knife is such that: it can support your weight when half the blade is stuck in ceiling, and it can bend at the blade-handle junction without breaking. Climb to top of the first rope, bringing other rope along. At the top, stretch the second rope toward you and circle it around your waist. Cut the first rope at the ceilling, hang on, and swing down on the second rope. Climb back up to the ceiling. Pull up the other end of the second rope, and tie it to the end of the first rope. Loop the tied ropes around the handle of the knife. In one (quick) motion, cut the second rope at the ceiling while pushing the knife into the ceiling. Bend the handle of the knife enough so that the rope won't slip out. Go down, hanging on the double rope with both hands. At the floor, jerk hard to release the knife and the rest of the rope.
Alternate Solution #3
Climb up the first rope, carrying up the end of the second with you. Cut the first rope, flush with the ceiling, swinging down on the second. Climb up the second and cut that flush with the ceiling too, but only part way, until it can just barely hold the rest of your weight. Slowly climb down until you're just short of the bottom. Tie a small loop here and climb up about two feet. Secure the loop about your feet and fall to the ground (roughly three feet). Just before your feet hit the ground the force of the fall will snap the rope at the top.
Solution for #9
One is in Eastern Oregon (Mountain time); the other in Western Florida (Central time), and the phone call takes place on daylight-savings-time changeover day at 1:30am.
Solution for #10
Turn on the first switch and wait five minutes. Turn off the first switch, turn on the second switch, and run up the stairs. If the light is on, it's the second switch. If the light is off but hot to the touch, it's the first switch. If the light is off and cool, it's the third switch.