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**Pn0yS0ld13r** If there are n tiles along the entire edge of a square room, then let x be the number of tiles along one width of the room. Thus there must be x-2 tiles along one length of the room (We can't count a tile twice; draw a picture to help visualize this).

Thus we can obtain an expression for n:

$\displaystyle n = 2x+2(x-2) = 2(x+x-2)=4(x-1)$.

Solving for x in terms of n:

$\displaystyle x=\dfrac{n}{4}+1$.

Since x denotes the number of tiles along the width of the room, and the room is a square, the total number of tiles is $\displaystyle x^{2}$. Hence

$\displaystyle x^{2} = \left(\dfrac{n}{4}+1\right)^{2}$.

Therefore, $\displaystyle S(n) = \left(\dfrac{n}{4}+1\right)^{2}$.

Since the smallest square room possible for n tiles along the edge is n = 4 (a 2x2 tile room), then the domain of $\displaystyle S(n)$ is all $\displaystyle n \ge 4$.