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Assignment 6, Phong

Questions about Reading Assignment:

Name the two most important things/concepts which you learned from the reading the chapter “The Architecture of Complexity”

1. give a one paragraph explanation why you consider these concepts important
2. are the concepts relevant to your work, to your interest, …. – if yes, why?

Simon's discussion of the architecture of complex system and problem solving in this context were important. with respect to structure, Simon's idea of hierarchy allows us to treat subsystems in the abstract and enable us to better reason about complex systems. This is particularly relevant for me when thinking about developing and designing complex software system. Although most of my programming assignments do not approach the complexity of multi-billion dollar systems, using the modular, object-oriented approach has been particularly useful. I also found resonanance in the application of this idea of hierarchy in the work of sociologists such as Michel Foucault where society is modeled as numerous subsystem with many foci of power and myriad interactions. Second, I also found Simon's discussion of the description of complexity, particularly the characterization of problem solving as "the continual translation between the state and process description of the same complexity" useful. This is useful to me because it provides insight into a problem solving strategies that I've taught (e.g. scientific process / critical thinking skills, etc) but whose processes I've never thought about.

Questions about The Importance of Representations in Design — The Mutilated “8x8” Matrix

The Problem: The associated PDF file shows you a mutilated “8x8” matrix (the two opposing corners cut out) and a domino block. One domino block covers exactly two fields of the “8x8” matrix. Note: It is straightforward that one can use 32 domino blocks to cover a complete “8x8” matrix. Question: Can one cover the mutilated “8x8” matrix with 31 domino blocks? Remark: the major objective of this assignment is that you spend some effort trying to solve this problem and answering the questions below — it is not so important that you will succeed solving the problem! Also: engage in some collaborative efforts solving it

Please do the following (please structure your answer accordingly — thanks):

1. try to find an answer to this problem! ‡ document briefly your thinking — including all the important intermediate steps and failing attempts (i.e., create a “think-aloud protocol”)

I began by looking at the graphical diagram included with the problem description. I then tried to solve the problem by looking at a smaller version of the matrix (i.e. 4x4 and 6x6). In the 4x4 matrix with opposing corners missing I tried to cover the matrix with 7 domino blocks. I first tried filing the matrix by working from the center of the matrix outward. I tried different combinations of the blocks by lining them up vertically to each other on the inner rows of matrix and filling the outer blocks with different combination of horizontal and vertical orientation of the blocks. That didn't work. Then I tried lining the blocks in from the outside of the matrix and working toward the center with a similar strategy. That didn't work either.

Although I didn't exhaust all possible combinations for the 4x4 matrix, I switched to the 6x6 matrix and tried the same set of possible combination using the same steps outlined above. I had hoped a larger matrix may differ from the smaller matrix. I got the same failures. I then switched to the 8x8 matrix to try the same strategy. But I was not able to find any combination that would work.

2. which resources did you use to solve the problem? I used pen and papers to sketch out the various combinations.

3. which process did you use? In trying to solve this problem I used the strategy of looking at a simpler case. I had hoped that once I found solution for simpler case I then would "scale it up" for a larger case. Implicit in this strategy was the hope that the mutilated larger matrix would decompose into smaller subcomponents of un-mutilated matrices and smaller rows or sections that contain the missing square. I wanted to isolate the portion that was mutilated and concentrate on that section.

4. which practice (of you or others) did you use? I'm not sure I understand this question. But if the question refers to collaboration I did work with a friend to figure out different approaches to this problem. We both thought of different variations of the same approach of trying to reduce the problem to a simpler case to see what might work before considering the problem of the full 8x8 matrix.

5. could computers be useful to solve this problem? A computer would be useful in the brute force approach of finding all possible solution. But it might not be realistic given the tremendous number of combination of solution that it needs to check.

6. what have you learned solving the problem: in general and for our course? I was never able to solve the problem. See the question below.

7. what have you learned not being able to solve the problem: in general and for our course? With respect to my own problem solving approach I discovered that I implicitly assumed that the problem is hierarchical. That is, I assumed that if I break the problem down to a simpler version that any solutions found would scale up to the larger problem. The process that I used, whether looking at a simpler or larger version, implicitly used the state description and process description stages where I described the initial state (mutilated matrix), desired outcome (fitting the domino blocks on the mutilated matrix), and couching the problem as finding the process that would lead from the initial to the desired outcome. I found that I also used natural selection implicitly (i.e. placing blocks on paper until I can determine that the solution does not work) in trying to find a solution. I use this trial and error heuristic for all types of problems. Indeed, most – if not all – of the problem solving approach that I've learned are variant of this process. Are there other approaches that I should learn? A corollary question for me is: did I use this process because it was taught to me? Or was because the problem description was organized this way so I follow this particular structure trying to find a solution. That is does framing a problem a certain way force you to follow a certain process in trying to find the solution. If so then perhaps finding a solution is really finding the right representation of the problem.

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