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Scott Zweig
Jennifer Tamez
Assignment #4
2/2/04

1. One important concept is that we are able to recognize, understand and describe complex systems because of a hierarchic and nearly decomposable structure. The other concept is that complex systems can be broken down using a hierarchic system which shows the subsystems and makes present the order of the levels - when the lowest level is reached.

1.1 Scott and I consider the first concept important because it deals with the theme and the third section of the chapter - hierarchically organized systems can be decomposed into subsystems in order to analyze their behavior. Let's look at the biological systems. We are able to recognize, understand and describe this systems because of a hierarchical structure, i.e. cells organized into tissues, tissues into organs, organs into systems. And again, the cell - nucleus, cell membrane, microsomes, mitochondria. Because of hierarchy, we are able to "see", understand and describe systems which we can then analyze.

The second concept ties into the first concept, but also includes the fourth section (the relation between complex systems and their descriptions). The first concept discusses how we are able to understand systems better when broken down - the second concept also implies this but shows that we can better understand systems depending on how they are described. The description of a structure can make it a complex or a simple structure. The hierarchy and decomposition of systems can be broken down and analyzed, but how hard or easy this shall be can be dependent upon the way the system is described.

1.2 Yes, these concepts are important because it can help make many things become understandable depending on how we choose to look at it. We are both in the TAM program. When a program becomes difficult, looking at the hiearchy of the system can help make it more understandable. The big picture can be seen as well as the steps to get there. Not only in our work, but in other problems does it help to break things down into levels and a hierarchic system in order to understand things better.

2. Mutilated Matrix Problem

Q: Can you cover the mutilated 8x8 matrix with 31 "2-piece" domino blocks?

A: The answer is YES, but there needs to be an exception. It is not physically possible to cover a 62-square "mutilated matrix 8x8" with 31 2-square domino blocks as long as the squares must be physically connected. The exception must allow me to separate two of the squares from a connected pair. IF you disconnect one pair of blocks from the 31 sets of two and make two singles, then you can cover all of the appropriate boxes. In the example the 8x8 matrix has two opposing corners removed. If you placed 2-square blocks on every space on the matrix you would come up with two single square gaps somewhere in the middle. This is where the separate single squares come in handy to complete the puzzle. While even though 31 fit evenly into 62 on paper, it can't be done in this scenario without overlapping or disconnecting two squares from a designated block. It is possible to cover the amount of squares in the mutilated matrix with only 31 2-square blocks, but it has to be arranged differently. If the two empty spaces are located adjacent to one another then the matrix could be completed with only 31 2-square blocks. (P.S. If, you are not allowed to cut apart the two squares, then the answer is NO, and you will not be able to cover all the spaces on the mutilated matrix.)

1.) I tried to find a solution to this problem by printing out a number of these "mutilated matrices" and drawing on them with marker. After conducting multiple trials I finally came up with an answer that suited me.
2.) The resources that I used to solve this problem were my own knowledge of concept and space and "packing". I also worked out possible scenarios of the matrix with my teammate, Jen.
3.) The process I used was called trial-and-error. It is often used when performing scientific experiments when the potential outcomes are unknown/unpredictable.
4.) I guess I used the practice of hands-on technique because I printed out copies and drew on them and even cut out squares.
5.) Computers could have been useful to solve this problem, in letís say a virtual environment, but it would not have made the process go any faster. In fact using a computer to display my findings is taking me almost as long as it did to figure out the problem. Sometimes computers slow things down.
6.) In solving this problem I learned that sometimes you have to bend the rules in order to accomplish your intended goal, and that's o.k... I also learned that when all else fails, turn to trial and error, the old fashioned way with paper and a pencil always works.
7.) In not solving the problem I learned that sometimes rules are put into place solely for the purpose of restricting where you want to go or what you want to do.

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