We present new algorithms for conducting transition path sampling (TPS). Permutation shooting rigorously preserves the total energy and momentum of the initial trajectory and is simple to implement even for rigid water molecules. Versions of aimless shooting and permutation shooting that use flexible-length trajectories have simple acceptance criteria and are more computationally efficient than fixed-length versions. Flexible-length permutation shooting and inertial likelihood maximization are used to identify the reaction coordinate for vacancy migration in a two-dimensional trigonal crystal of Lennard-Jones particles. The optimized reaction coordinate eliminates nearly all recrossing of the transition state dividing surface.