Algorithms of moving of the end of chain to the given point

The problem of chains is investigated. Results on the existence of chains obtained from given chain by moving of the end of the chain to a given point; Bounds of the minimum of the Euclidean distance between chains obtained from each other by moving of the end to a given point; possible number of chains obtained by moving the end to a given point and differing by the minimum number of elements from a given chain; the possible number of chains that are at the minimum distance from the given and obtained by moving the end of the chain to a given point, for $ n = 2 $ and $ n = 3 $. Algorithms for moving the end of a chain to a given point are described: an exponential algorithm that sorts out all possible chains with step $\varepsilon$, a linear algorithm giving an approximate solution for Euclidean distance, and a linear algorithm giving an exact answer for the Hamming distance and approximate for the Euclidean distance.