The snake-in-the-box problem is concerned with finding the longest induced path in a hypercube Qn. Similarly, the coil-in-the-box problem is concerned with finding the longest induced cycles in Qn. We consider a generalization of these problems, first studied in [Sin66], that considers paths and cycles where each pair of vertices at distance at least k in the path or cycle are also at distance at least k in Qn. We call these paths k-snakes and the cycles k-coils. We present two algorithms for exhaustively searching for the longest k-snakes and k-coils: the first requires O(nk−1) time per recursive call and the second requires O(t) time per recursive call where t is the current length of the snake being searched. Then for coils, we consider two optimization strategies: the first considers a connectivity constraint and the second takes advantage of rotational equivalence. For rotational equivalence, several heuristics are considered with varying amounts of overhead. Finally, using a cluster of processors we apply a parallel approach to find many new bounds for the longest k-snakes and k-coils. Using our best optimization we improve the exhaustive search for k = 2 and n = 7 by a factor of 19.