Putnam 1971

A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $a$ or $b$ points ($a$ and $b$ are positive integers with $a$ greater than $b$), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is 58. Find $a+b$.

A "spherical ellipse" with foci $A, B$ on a given sphere is defined as the set of all points $P$ on the sphere such that $\widehat{PA} + \widehat{PB} = \text{constant}$. Here $\widehat{PA}$ denotes the shortest distance on the sphere between $P$ and $A$. Determine the entire class of real spherical ellipses which are circles by finding the \text{constant} value.