Glass Half Full. Cork, Bottle, Coin. Globe Traversal.

Glass Half Full (source: [wu:riddles]):
You are in an empty room and you have a transparent glass of water. The glass is a right cylinder, and it looks like it's half full, but you're not sure. How can you accurately figure out whether the glass is half full, more than half full, or less than half full? You have no rulers or writing utensils.

Solution:
Spread your thumb and index finger on one hand to measure the distance between the top of the glass and the water line. Lock your thumb and index finger in that position. Next, move your hand down such that whichever finger was at the top of the glass is now positioned at the water line. Now, if your other finger is exactly at the the bottom of the glass, the glass is half full. If that other finger is below the bottom of the glass, the glass is less than half full. Otherwise (if that other finger is between the water line and the bottom of the glass), the glass is more than half full.

This problem is a good example of simple solutions being overlooked. The first two solutions I came up with--rapidly placing the glass upside-down on the floor; or, raising one edge of the glass (tipping the glass) by a length equal the base's diameter--were both more complex and less accurate.


Cork, Bottle, Coin ([wu:riddles]):
If you were to put a coin into an empty bottle and then insert a cork in the bottle's opening, how could you remove the coin without taking out the cork or breaking the bottle?

Solution:
Push the cork all the way inside the bottle. Then, remove the coin.

Another simple solution that can be overlooked. My guess as to why is that people are accustomed to undoing things by way of the opposite of what they did in the first place (e.g., if the door was opened via a pull, it should be closed via a push). The lesson here is to question your approach--or more broadly, your assumptions.


Globe Traversal (source: [wu:riddles]):
How many places are there on the earth that one could walk one mile south, then one mile west, then one mile north and end up in the same spot? To be precise, let's assume the earth is a solid smooth sphere, so oceans and mountains and other such things do not exist. You can start at any point on the sphere. Also, the rotation of the earth has nothing to do with the solution; you can assume you're walking on a static sphere if that makes the problem less complicated to you.

Solution:
Infinitely many. To see this, temporarily disregard the south and north legs of the journey, as they will eventually cancel each other out. Now, the only way to travel west for some distance and end up in the same place we started is if we circle the globe. Thus we pick a cross-section of the earth perpendicular to its axis, say near the South Pole, which produces a circle having a circumference of one mile. Note that this circle is comprised (at least theoretically) of infinitely many points; and, that walking around the edge of this circle for one mile will always take us back to our original position. Now we factor the south and north legs of the journey back in by simply shifting our infinitely many starting points one mile north.

This is another solution that involves an atypical "undo action." Specifically, typical reasoning suggests that for each unit traveled in one direction, one must travel one unit in the opposite direction in order to return to where to he started.