The puzzle with attachable and releasable pieces includes shaped puzzle pieces and a two-part base, which includes a front base section and a back baseplate. The puzzle pieces have rear-facing engaging mechanisms that are attachable to forward-facing male studs carried by the back baseplate. The puzzle pieces each have a particular outer shape determining placement. The puzzle may be stored with the puzzle pieces engaged to the back baseplate, but may be later released for repeat play by using a provided puzzle release mechanism.