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 the puzzle pieces may be later released for repeat play by using a provided puzzle release mechanism.