\newpage \section {Games; Alternation; Exhaustive Search; Time v. Space.}\label{games} \subsection {How to Win.\label{win}} In this section we consider a more interesting {\em provably} intractable problem: playing games with full information, two players and zero sum. We will see that even for some simple games there cannot be a much more efficient algorithm, than exhaustive search through all possible configurations. The rules of an {\em $n$-player game} $G$ are set by families $f,v$ of {\em information} and {\em value} functions and a {\em transition rule} $r$. Each player $i\in I$ at each step participates in transforming a configuration (game position) $x\in C$ into the new configuration $r(x,m),\ m:I\to M$ by choosing a move $m_i= m(i)$ based only on his knowledge $f_i(x)$ of $x$. The game proceeds until a {\em terminal} configurations $t\in T\subset C$ is reached. Then $v_i(t)$ is the loss or gain of the $i$-th player. Our games will have {\em zero sum} $\sum v_i(t)=0$ and {\em full information:} $f_i(x)=x$, \$-|x|