You have an unlimited supply of stamps in each denomination from eight cents to the number of cents equal to the answer to last week’s Duplicate Division problem. However, there is an unusual rule when you apply postage to a package: the number of stamps of a given denomination must be the denomination of a different kind of stamp on the package, and the denomination of every stamp must be the number of stamps of some different denomination. For example, the postage on a package could consist of nine eight-cent stamps and eight nine-cent stamps
What is the maximum number of cents of postage you can apply to a single package?
Figure P is a spiral that starts at the point (1,0) and spirals outward clockwise making eight revolutions around the origin to reach the point (10,0). Figure R is a different spiral that starts at (1,0) and spirals outward counterclockwise making seven revolutions around the origin to reach (10,0).
Not counting either (1,0) or (10,0), what is the maximum number of times that figures P and R can intersect?