POW #3 Just Count the Pegs
In the “Just Count The Pages” problem, we studied multiple polygons on a peg board and recorded information on the area, including the different aspects of a polygon that can effect it. From that information, we found small formulas that regarded different cases of polygons including measure them by interior pegs, or border pegs. From all the research we attempted to come up with one superformula that can solve the area of any polygon when given the interior and exterior pegs.
In order to solve this problem, I went through creating many different tables depicting certain circumstances in which I could solve a problem for area. The first tables that I made regarded the areas of polygons in which the interior dots were constant. For example, I would make a table for different shapes with no interior pegs, draw a bunch of those types of shapes, and then record my results on the table.
Through each table I did for the constant of interior dots, I came up with a formula. For example, for a shape with no interior pegs, the area is y=x/2 -1. In this case, x is the amount of dots on the border, and y is the total area. After doing multiple cases for constants of interior, I changed things up and made tables for a constant of pegs on the boundary. For this case, I followed a very similar protocol to the first time I made tables. This time though, I was able to use information from both of these sets of tables to come up with the “superformula”.
The answer I got for the problem consisted of 3 different variables. X which represented boundary/exterior pegs, Y which stood for the area, and Z which was the variable for the interior pegs. From the first set of tables I did where the constant was the interior pegs, I noticed the formula for each started out with x/2, than it subtracted a variable. After looking at the second tables, I realized the formula was y=x/2+(z-1). I tried out this formula on multiple polygons, and it did in fact work. This formula makes sense because it is common sense that the more interior pegs there are on the inside of a polygon, the larger it would be. More interior pegs does not only mean that the area is bigger, but also that the exterior pegs are used more efficiently.
Two Habits of a Mathematician that I displayed are Solve a Simpler Problem, and Be Systematic. I was able to Solve a Simpler problem by not just going straight at trying to find the superformula. If I would have done that, it would have taken ages to come up with a solution. Instead, I made two sets of these tables and solved each set of problems like it was its own problem. I was systematic by following protocols in order to solve the formulas. Instead of just going all crazy with the problem, I thought out the steps I should make in order to solve the big over arching problem. You can't just go all willy nilly on a problem, you must be precise in solving it, not only the solution, but also the strategy.