2. Due Tuesday, January 26 Consider a pattern of black and white squares in a 3...

  

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2. Due Tuesday, January 26 Consider a pattern of black and white squares in a 3 x 4 rectangle, such as: This naturally breaks up into groups by color, like this: Yielding 4 groups. It can also be broken up into "checkerboard" groups like this: Again yielding 4 groups (which I think is a coincidence). If we define the complexity of a pattern to be the sum of these to numbers (4 + 4 = 8 in this case), what are the largest and smallest possible values for the complexity of a pattern on a 3 x 4 grid? How about a 4 x 4 grid? A 4 x 5 grid? The expectation for Challenge Problems is that you make a significant effort to explore each problem and that you write a clear, organized summary of the process you have followed and conclusions you can draw. On some Challenge Problems, you will not be able to find a definitive "solution" - that's fine; you should still show evidence that you have put significant effort into exploring the problem and you should provide a meaningful write-up. Many Challenge Problems will involve you conducting several trials of a task or activity. Initially, you will probably conduct such trials without a clear sense of what you're trying to accomplish, but you should look for patterns or general principles in the results you get. Subsequent trials should, then, be aimed at exploring "interesting" or "extreme" cases that will help you test the patterns/principles you notice. When writing up your submission, you do not need to include every trial or attempt you complete. For example, if you conduct 27 trials but notice that many of them share a particular structure, just provide one or two examples of each type of particular structure; provide a sufficient description of why you selected the trials you do include. Provide sufficient (but not necessarily lengthy) explanation of trials, calculations, derivations, visualizations, etc. that you include in your write-up. Imagine you will be giving your write-up to one of your fellow MAT 226 students - you should provide enough explanation that they will be able to effectively follow along with your thought process and conclusions. Feel free to develop notation, visualizations, alternative representations, etc. for components of any Challenge Problem as you see fit. This does not mean that every Challenge Problem needs (or would benefit from) notation, but you are welcome to create such notation if you feel it will be helpful. Provide an explanation of notation, etc. that you employ. At any stage of any Challenge Problem, you should feel free to explore any modification/extension of the stated problem you find interested. Even if such modifications/extensions don't lead to a definitive "solution", you should still mention or describe them in your write-up. If you do feel that you have found a definitive "solution" with relatively limited effort, feel compelled to modify/extend/generalize the problem or your solution in an interesting and appropriate way. Write-ups can be typed or hand-written, but they must be organized and clear. 2. Due Tuesday, January 26 Consider a pattern of black and white squares in a 3 x 4 rectangle, such as: This naturally breaks up into groups by color, like this: Yielding 4 groups. It can also be broken up into "checkerboard" groups like this: Again yielding 4 groups (which I think is a coincidence). If we define the complexity of a pattern to be the sum of these to numbers (4 + 4 = 8 in this case), what are the largest and smallest possible values for the complexity of a pattern on a 3 x 4 grid? How about a 4 x 4 grid? A 4 x 5 grid? The expectation for Challenge Problems is that you make a significant effort to explore each problem and that you write a clear, organized summary of the process you have followed and conclusions you can draw. On some Challenge Problems, you will not be able to find a definitive "solution" - that's fine; you should still show evidence that you have put significant effort into exploring the problem and you should provide a meaningful write-up. Many Challenge Problems will involve you conducting several trials of a task or activity. Initially, you will probably conduct such trials without a clear sense of what you're trying to accomplish, but you should look for patterns or general principles in the results you get. Subsequent trials should, then, be aimed at exploring "interesting" or "extreme" cases that will help you test the patterns/principles you notice. When writing up your submission, you do not need to include every trial or attempt you complete. For example, if you conduct 27 trials but notice that many of them share a particular structure, just provide one or two examples of each type of particular structure; provide a sufficient description of why you selected the trials you do include. Provide sufficient (but not necessarily lengthy) explanation of trials, calculations, derivations, visualizations, etc. that you include in your write-up. Imagine you will be giving your write-up to one of your fellow MAT 226 students - you should provide enough explanation that they will be able to effectively follow along with your thought process and conclusions. Feel free to develop notation, visualizations, alternative representations, etc. for components of any Challenge Problem as you see fit. This does not mean that every Challenge Problem needs (or would benefit from) notation, but you are welcome to create such notation if you feel it will be helpful. Provide an explanation of notation, etc. that you employ. At any stage of any Challenge Problem, you should feel free to explore any modification/extension of the stated problem you find interested. Even if such modifications/extensions don't lead to a definitive "solution", you should still mention or describe them in your write-up. If you do feel that you have found a definitive "solution" with relatively limited effort, feel compelled to modify/extend/generalize the problem or your solution in an interesting and appropriate way. Write-ups can be typed or hand-written, but they must be organized and clear.

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Answer given That The Smallest possible largest possible value of ... View the full answer