Japanese Mathematical Olympiad 1995

The sequence $\{a_1, a_2, a_3, \dots \}$ is defined by $a_{2n + 1} = (-1)^n$ and $a_{2n} = a_n$ . A point $P$ moves on the coordinate plane as follows.

• First $P$ moves from the origin $P_0(0,0)$ to $P_1 (1,0)$ .
• After $P$ has moved to $P_i$ , it turns $90^{\circ}$ to the left and moves forward $1$ unit if $a_i = 1$ , and turns $90^{\circ}$ to the right and moves forward $1$ unit if $a_i = -1$ . Denote this point by $P_{i + 1}$ .

Can the point retrace an edge? That is, can $P_u = P_w$ and $P_{u + 1} = P_{w + 1}$ for some integers $u,w$ ?