  We investigate both analytically and numerically the ground-state and
thermodynamic properties of the quantum mixed spin-1/2-1/2-1-1 chain described
by the Hamiltonian $H=\sum_{\ell=1}^{N/4} (J_1\vecs_{4\ell-3}\cdot
\vecs_{4\ell-2}+J_2\vecs_{4\ell-2}\cdot\vecS_{4\ell-1}+J_3\vecS_{4\ell-1}\cdot
\vecS_{4\ell}+J_2\vecS_{4\ell}\cdot\vecs_{4\ell+1})$, where two $S=1/2$ spins
($\vecs_{4\ell-3}$ and $\vecs_{4\ell-2}$) and two S=1 spins ($\vecS_{4\ell-1}$
and $\vecS_{4\ell}$) are arranged alternatively. Inseveral limiting cases of
$J_1$, $J_2$, and $J_3$ we apply the Wigner-Eckart theorem and carry out a
perturbation calculation to examine the behavior of the massless lines where
the energy gap vanishes. Performing a quantum Monte Carlo calculation without
global flips at a sufficiently low temperature for the case where $J_1=J_3=1.0$
and $J_2>0$, we find that the ground state of the present system in this case
undergoes a second-order phase transition accompanying the vanishing of the
energy gap at $J_2=J_{2c$ with $J_{2 c} = 0.77 \pm 0.01$. We also find that the
ground states for both $J_2 < J_{2 c$ and $J_2 > J_{2c$ can be understood by
means of the valence-bond-solid picture. A quantum Monte Carlo calculation
which takes the global flips along the Trotter direction into account is
carried out to elucidate the temperature dependences of the specific heat and
the magnetic susceptibility. In particular, it is found that the susceptibility
per unit cell for $J_2=0.77$ with $J_1=J_3=1.0$ takes a finite value at
absolute zero temperature and that the specific heat per unit cell versus
temperature curve for $J_2=5.0$ with $J_1=J_3=1.0$ has a double peak.
