  The question of whether there exist bound states of two heavy quarks
$Q=(c,b)$ and antiquarks $\bar Q = (\bar c, \bar b)$, distinct from a pair of
quark-antiquark mesons, has been debated for more than forty years. We suggest
some means of producing and observing $Q_1 Q_2 \bar Q_3 \bar Q_4$ resonant
states, concentrating on the $c c \bar c \bar c$ channel which is most easily
produced and the $b b \bar b \bar b$ channel which has a better chance of being
relatively narrow. We obtain $M_{(cc)(\bar c \bar c)} = 6,192 \pm 25$ MeV and
$M_{(bb)(\bar b \bar b)} = 18,826 \pm 25$ MeV, for the charmed and bottom
tetraquarks, respectively. Experimental search for these states in the relevant
mass region is highly desirable.
