  In the framework of the color-magnetic interaction, we systematically
investigate the mass splittings of the $QQ\bar{Q}\bar{q}$ tetraquark states and
estimated their rough masses in this work. These systems include the explicitly
exotic states $cc\bar{b}\bar{q}$ and $bb\bar{c}\bar{q}$ and the hidden exotic
states $cc\bar{c}\bar{q}$, $cb\bar{b}\bar{q}$, $bc\bar{c}\bar{q}$, and
$bb\bar{b}\bar{q}$. If a state around the estimated mass region could be
observed, its nature as a genuine tetraquark is favored. The strong decay
patterns shown here will be helpful to the experimental search for these exotic
states.
