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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