IMP 2 POW 8 Essay -- Math Mathmatics Problems

Problem StatementMy task was to find 3 equations, that would give me an answer, if I had certain information. The set-back was to find one that if you knew that in that location were four pegs on the boundary, and none on the interior, you could get the area. The second was if you knew that in that location were 4 pegs on the boundary, and you knew how many were on the interior, you could get the area. And last, if you had the number on the interior, and the number on the boundary, you could get the area.ProcessThe first two equations, were a preparation for the final, building up towards the complete idea. This helped, because I could complete the first two pretty quickly. For Freddie I draw a 3 column T-Table, with a drawing of the figure, the number of Pegs (in), and the Area (out). I looked for a approach pattern between the in and the out, and quickly found one that made sense, and I worked it into a formula. I got X/2-1=Y. Where X is IN (number of pegs) and Y is taboo (A rea). This works in all shapes with no interior pegs, like Freddie described. I attached this T-Table.For Sally I followed my luck of the 3 column T-Table, and drew a nonher with the same guidelines. The figure, the interior pegs (in), and the area (out). After I fill in a few figures, and their properties, I noticed a pattern, and not long after, a formula, which worked for them. It was X+1=Y. This T-Table is also attached.Now...the next was not so easy. Frashys required a long thought process, an...