mathematical puzzles

At the Game Developers Conference i ran across a small booth run by the small press publishers A K Peters. I'd come to the booth for a particular computer book, but i stayed for the esoteric math stuff! Books on origami (both representational and non-), straight-up math books, math-history books, math puzzle books, wow. In addition to a bunch of other regular computer books. They were all quite good. So: A K Peters, right on.
Anyhow,
so i picked up Peter Winkler's Mathematical Puzzles - A Connoisseur's Collection. And wow, am i out of my league. These are all puzzles in the Martin Gardner tradition, but i only managed to get i think about half of one right in the whole book. (Except the section on geography)

Here's a relatively straight-forward one:

Three coins are put into a bag: a coin with both sides "heads", a coin with both sides "tails" and a regular "heads/tails" coin.
You reach into the bag and bring out a coin at random and flip it.
If it comes up heads, what's the probability that if you turn it over, the other side is also heads ?

here's a harder one:

Associated with each face of a solid convex polyhedron (such as a cube, a pyramid, ehatever) is a bug which crawls along the perimiter of the face, at varying speed, but only in the clockwise direction. Prove that no schedule will permit all the bugs to circumnavigate their faces and return to their initial positions without a collision.

- i mean, jeeze.

anyhow, that's all.

Oh i should also add that i thought the puzzles and solutions could have been presented more clearly. I often found myself not knowing what was being asked, or in the few cases where i went ahead and did some work to determine an answer, i discovered that i'd misunderstood the question.