# Intersecting Squares and Circles

Two players play a game using the interval $$[0,33]$$ on the $$x$$-axis. The first player randomly chooses a square of side length $$s \in \mathbb{Z}^+$$, which has a side that lies entirely on the interval. The second player randomly chooses a circle with radius $$r \in \mathbb{Z}^+$$, which has a diameter that lies entirely on the interval. After repeating choosing random squares and circles in this fashion, the players realize that the probability that the circle and square intersect is $$\frac{1}{2}$$. Let $$S = \{(s,r) : \mbox{probability of intersection is } \frac{1}{2}\}$$. Determine $$\sum\limits_{(s,r) \in S} (s + r).$$

Details and assumptions

Clarification of notation: The set $$S$$ is the set of all ordered pairs of integers, $$(s, r)$$, such that the probability that a square of side length $$s$$ and a circle of radius $$r$$ will intersect is $$\frac {1}{2}$$.

The notation $$\mathbb{Z}^+$$ denotes positive integers.

Clarification: A shape consists of the perimeter and the interior. 2 figures can intersect in their interior without intersecting on their perimeter.

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