A Very Small Collection of Geometry Puzzles

1. Imagine a standard cylindrical can that holds three tennis balls. Cut a piece of string that wraps exactly once around the can. Now straighten it out and hold it up against the side of the can. Will it be shorter, the same size, or longer than the can? First guess the answer, then prove it.
2. Imagine a ribbon circling the equator of a sphere the size of the sun. Now imagine increasing the length of the ribbon by 1 foot so that now the ribbon circles the sphere with a constant gap between the ribbon and the sphere. Is the gap large enough to slide a penny through? Push a basketball through? Walk through? First guess the answer, then prove it.
3. It is easy to dissect a circle (with its interior) into a finite number of congruent pieces, each of which touches the center of the circle. Can you dissect the circle into a finite number of congruent pieces such that not all of them touch the center?
4. A round hole is bored through the center of a solid sphere. This amounts to removing two caps and a cylinder from the sphere. If the length of the cylinder (excluding the caps) is 6 inches, what is the volume of the remaining part of the sphere?
5. You have a square chocolate cake with chocolate frosting that is spread on both the top and the sides. How can you cut the cake with straight cuts into seven pieces so that each piece has the same amount of cake and each piece also has the same amount of frosting?
6. Find a way to cut a file card into a finite number of pieces that can be reassembled to make a square.
7. Find a way to dissect an equilateral triangle into a finite number of pieces that can be reassembled to make a square.