  In this work, we study systematically the mass splittings of the
$qq\bar{Q}\bar{Q}$ ($q=u$, $d$, $s$ and $Q=c$, $b$) tetraquark states with the
color-magnetic interaction by considering color mixing effects and estimate
roughly their masses. We find that the color mixing effect is relatively
important for the $J^P=0^+$ states and possible stable tetraquarks exist in the
$nn\bar{Q}\bar{Q}$ ($n=u$, $d$) and $ns\bar{Q}\bar{Q}$ systems either with
$J=0$ or with $J=1$. Possible decay patterns of the tetraquarks are briefly
discussed.
