The table above is from the Japanese game of Janken as presented in the Art of Strategy book.

a$)$ First, explain how the game of Janken is typically played.

b$)$ The optimal strategy for this game turns out to be very different from the $\frac{1}{3},\frac{1}{3},\frac{1}{3}$ random mix of

traditional rock$-$paper$-$scissors. Without delving into a formal mathematical explanation, why do you

think the best strategy turns out to be randomly play scissors $\frac{5}{8}$ ths of the time, and paper only $\frac{1}{8}$ ths of

the time? $($Helpful hint: you might want to play against someone to get a better feel for what is

happening.$)$