(bonus) a set s of ordered pairs of integers is defined recursively by (1) (1,2) ∈ s and (2,1) ∈ s, and (2) if (m, n) ∈ s, then (m+2,n) ∈ s, and (m, n+2) ∈ s, and (m+1,n+1) ∈ s. there is a simple description of the ordered pairs in s. what is it? your description should be good enough so that you can instantly decide if (12236,912242) is in s.

Set $r$ is a set of rectangles such that (1) only the grid points shown here are used as vertices, (2) all sides are vertical or horizontal and (3) no two rectangles in the set are congruent. if $r$ contains the maximum possible number of rectangles given these conditions, what fraction of the rectangles in set $r$ are squares? express your answer as a common fraction.