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A stair stringer is to be cut to reach a $2^{nd}$ story floor in a house with nine foot ceilings. The distance from the $1^{n1}$ to the $2^{nd}$ floor can be calculated by adding up the height of each material. Bottom plate $1\frac{1}{2},$ stud $104\frac{1}{4},$ two top plates $1\frac{1}{2}$ each, floor joist $11\frac{7}{8},$ and subfloor $\frac{3^7}{4}.$ Construction code stipulates that each rise cannot exceed eight inches and must be equal so that the stair case will not be a trip hazard. Calculate the number of rises and the height of each rise $($rounded to the nearest $16$ of an inch$)$ for all possible designs where the rise is between six and eight inches.

A stair stringer is to be cut to reach a $2^{nd}$ story floor in a house with nine foot ceilings. The distance from the $1^{n1}$ to the $2^{nd}$ floor can be calculated by adding up the height of each material. Bottom plate $1\frac{1}{2},$ stud $104\frac{1}{4},$ two top plates $1\frac{1}{2}$ each, floor joist $11\frac{7}{8},$ and subfloor $\frac{3^7}{4}.$ Construction code stipulates that each rise cannot exceed eight inches and must be equal so that the stair case will not be a trip hazard. Calculate the number of rises and the height of each rise $($rounded to the nearest $16$ of an inch$)$ for all possible designs where the rise is between six and eight inches.