We have systematically investigated the mass spectrum and rearrangement decay properties of the exotic tetraquark states with four different flavors using a color-magnetic interaction model. Their masses are estimated by assuming that the $X\left(4140\right)$ is a $cs\overline{c}\overline{s}$ tetraquark state and their decay widths are obtained by assuming that the Hamiltonian for decay is a constant. According to the adopted method, we find that the most stable states are probably the isoscalar $bs\overline{u}\overline{d}$ and $cs\overline{u}\overline{d}$ with $J^P=0^{+}$ and $1^{+}$. The width for most unstable tetraquarks is about tens of MeVs, but that for unstable $cu\overline{s}\overline{d}$ and $cs\overline{u}\overline{d}$ can be around 100 MeV. For the $X\left(5568\right)$, our method cannot give consistent mass and width if it is a $bu\overline{s}\overline{d}$ tetraquark state. For the $I\left(J^P\right)=0\left(0^{+}\right),0\left(1^{+}\right)$ double-heavy $T_{bc}=bc\overline{u}\overline{d}$ states, their widths can be several MeVs.