Consider the plane symmetric random walk on a square lattice: a particle is initially at the origin in the xy-plane, it makes n consecutive steps of unit length, and each step is made with the probability 1/4 in each one of the four directions parallel to the axes. We call the path of the particle a spiral if the following conditions are met: a) the particle never occupies the same position twice, b) the path of the particle, whenever it turns, either turns always clockwise or always counter-clockwise throughout the path, and c) for every m > n the given n-step path can be continued in atleast one way to give an m-step path meeting the conditions.