The question of whether there exist bound states of two heavy quarks Q=(c,b) and antiquarks Q =(c,b ), distinct from a pair of quark-antiquark mesons, has been debated for more than forty years. We estimate masses of Q1Q2Q 3Q4 resonant states XQ1Q2Q 3Q4 and suggest a means of producing and observing them. We concentrate on the ccc c channel which is most easily produced and the bbb b channel which has a better chance of being relatively narrow. We obtain M(Xccc c)=6,192±25 MeV and M(Xbbb b)=18,826±25 MeV, for the JPC=0++ states involving charmed and bottom tetraquarks, respectively. An experimental search for these states in the predicted mass range is highly desirable.