The relative growth rate for a certain type of fungi is 70% per hour. A small culture has formed and in just 6 hours the count shows to be 54,273 fungi in the culture. What is the initial number of fungi in the culture? Use a formula with e.

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.