A particle is initially at the origin in the (X, Y) plane and each successive step it takes is of unit length and parallel either to the X-axis or to the Y-axis. Its path of n steps is called a spiral if (i) the particle never occupies the same position twice, (ii) any turns the path makes are all counter-clockwise or all clockwise and (iii) for every m > n, the path can be continued to m steps without violating (i) or (ii).