The group of braids, which can be plaited from n untwisted threads tied together at each end, is examined and its structure is determined. An algorithm is derived for deciding whether or not a given braid can be so plaited and a calculation procedure is described. The problem arises from a process of manufacturing braids by a machine which plaits by passing a shuttle, on which the constructed braid is wound, between the threads, which are supplied from bobbins effectively fixed and maccessible. Every plait on three threads can be constructed in this way. For more than three threads, examples are given both of plaits which can be so constructed and of plaits which cannot.