Fuse Puzzle

OK, this should take you more than a few seconds, it's a tough puzzle, but give it a go:

I'm quoting verbatim from the Web page: My Favorite Puzzles:

"You have two fuses that both last one hour, and you have no other ways of telling time. The fuses may be thicker at some points, so in half an hour, the amount of fuse that has burned may or may not be half the length of the whole fuse. How do you measure 45 minutes worth of time?"

Source: Tonya Khovanova

159 match sticks

I have 159 match sticks that got wet and are now useless as matches. I want to cut the heads off of them and make a mathematical sculpture, but I haven't decided what to make yet. What would you make with 159 match sticks? If you have a good idea please share it.

More Fundamental Mathematics

Topology is like geometry, but with a little less structure. (Don't confuse topology with topography. They are not the same thing.) In topology you don't care about actual distances, just a more abstract notion of "nearness". Planar topology is sometimes called rubber-sheet geometry, because two-dimensional figures can be deformed without their topology changing (though their geometry certainly changes) as if they were drawn on a rubber sheet. Topology developed as a mathematical subject much later than geometry, probably because it is more abstract and it's ideas are more slippery, but topology can be thought of as a stripped down version of geometry. Today topology itself is a full-blown branch of mathematics.

I know that you probably won't come up with a whole new branch of mathematics in the next few seconds, but try anyway. Think of a branch of mathematics that you know (maybe number-theory or sequence patterns) and try to strip it down to something more fundamental and abstract. Don't stress. Success is not expected. It's the journey that is important.

Base Three

Computers are designed around the fact that it is easy to store and manipulate bits with components that have two states (e.g. 1 or 0, on or off, high voltage or low voltage). It is just possible that at some point it will become more efficient to work with components that naturally have three states. Then computers will work base three (or ternary). If that happens then the sequence of numbers starting with three and continuing by multiplying by three to get bigger and bigger numbers will become very important to computer scientists:
3 -> 9 -> 27 -> 81 -> etc.
Continue this sequence as far as you can mentally, or as far as you have time on paper.

Powers of Two

One plus one is two. Two plus two is four. Four plus four is eight. Continue this sequence as far as you can mentally.

Three-Dimensional Forms

You know some solids (or three-dimensional figures) such as cubes, spheres, pyramids. Name as many solids as you can think of. Think of some that don't have names (and think up your own names for them).

Kaprekar Routine

Start with any whole number (but it should have more than one digit. Four digits works well.) I'll start with 5820. Make a new number by writing the digits in descending order:
8520
and another by writing them in ascending order:
0258 = 258
Now subtract the two:
8520-258 = 8262
This process is called the Kaprekar routine. Notice that you start with one number and end up with a new number. From there you can repeat the process if you wish.
Here's the sequence I get:
8520 -> 8262 -> 6354 -> 3087 -> 8325 -> etc.
If you continue long enough what do you notice?

Source: Wolfram MathWorld:Kaprekar Routine