# IMP 2 POW 8 Just count the pegs Essays and Papers

This essay has a total of 1175 words and 5 pages.

IMP 2 POW 8 Just count the pegs

Problem Statement
My task was to find 3 equations, that would give me an answer, if I had certain
information. The first was to find one that if you knew that there were four pegs on the
boundary, and none on the interior, you could get the area. The second was if you knew
that there 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.

Process
The 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 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. And operations on the
two variables will give me my out. So then I went to the T-Table, the perfect tool for
seeing patterns. This time I had 4 columns and extra for my extra variable. I had the
figure in one column, then the pegs on the interior, pegs on the boundary, and finally,
the area. I started drawing shapes, and then filling in their properties, as I did
earlier. Until I had about 5 or 6. I then started to look at the numbers, hoping to find a
pattern of some sort. I didn't. I then put the table entries I had, into order, by the
area. This helped a lot, so I could see how the different shapes related. I noticed
patterns in groups, I saw a pattern if there were 0 pegs on the interior, or 1 peg on the
interior, but not one that flowed through-out them all. It was very frustrating.

Then I tried to work with the equations I already had, from the other two. I started with
the first, X/2-1=Y and I thought, if I were to add the other variable (which I called I,

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