For Freddie I drew a 3 column T-Table, with a drawing of the figure, the number of Pegs (in), and the Area (out). I looked for a 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 OUT (Area). 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 another with the same guidelines. The figure, the interior pegs (in), and the area (out). After I filled 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. Frashy's required a long thought process, and several hours thinking it over, logically. I thought that this next equation would be a combination of the two, it would have to incorporate what I had found out from both of the above. Especially the first. So I thought to myself what this equation, or formula, would have to include. And realized there wasn't 1 variable, but 2. Because it has the variable from the first, and the second problem. 1: The number of pegs on the border, and 2: The number of pegs on the interior. So this means that there are 2 IN's.
For Freddie I drew a 3 column T-Table, with a drawing of the figure, the number of Pegs (in), and the Area (out). I looked for a 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 OUT (Area). 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 another with the same guidelines. The figure, the interior pegs (in), and the area (out). After I filled 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. Frashy's required a long thought process, and several hours thinking it over, logically. I thought that this next equation would be a combination of the two, it would have to incorporate what I had found out from both of the above. Especially the first. So I thought to myself what this equation, or formula, would have to include. And realized there wasn't 1 variable, but 2. Because it has the variable from the first, and the second problem. 1: The number of pegs on the border, and 2: The number of pegs on the interior. So this means that there are 2 IN's.