Fool's solitaire on joins and Cartesian products of graphs

Beeler and Rodriguez proposed a variant where we instead want to maximize the number of pegs remaining when no more jumps can be made. Maximizing over all initial locations of a single hole, the maximum number of pegs left on a graph G when no jumps remain is the fool's solitaire number F(G). We determine the fool's solitaire number for the join of any graphs G and H. For the Cartesian product, we determine F(G□Kk) when k≥3 and G is connected and show why our argument fails when k=2. Finally, we give conditions on graphs G and H that imply F(G□H)≥F(G)F(H).