Now that we have covered algorithms in class I have started to think of a problem which I could try to develop an algorithm for. I worked through the paper folding exercise given and got some practice working with the patterns and trying different approaches to the same problem. What I liked in particular about the paper folding algorithm was that it mapped a natural number n to a pattern. That one single number can contain not only the number of up folds and down folds, but the order that these folds appeared in as well, that really impressed upon me how powerful algorithms can be and given on of these 'recipes' how much information can be stored in a single number.
A problem that I would like to solve myself goes as follows:
I want to arrange bottle caps into pyramids. I will start by making an equilateral triangle of bottle caps with side length n (where n is some natural number of bottles caps, it would be tough to use a fraction of one or a negative number of caps) then I will make another such triangle with side length n-1, then stack this smaller triangle onto the larger one, and repeat until I get a triangle of side 1 on the top (not much of a triangle I know but none the less). I want to know if I start with a side of n caps, how many bottle caps will it take to finish the pyramid.
I will probably try and solve in by considering the total number of bottle caps in each level and then add the levels. Maybe I could come up with equations analogous to area and volume for the triangles and pyramids respectively. Also I could make a chart of different values of n and the corresponding values for the total number of caps and even some other values might be helpful. I will work on this here and there and see if I am any nearer to a solution next week.
Theo
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