V_2-V-1 is determined by the charge q_1,

V_3-V_2 is determined by q_1+q_2

V_2-V-1 is determined by the charge q_1,

V_3-V_2 is determined by q_1+q_2
Where does q_3 figure then?

q_3 matters only for the field outside the three shells

$V_2-V_1=\frac{q_1}{4\pi\epsilon}(1/r-1/2r)$
$V_3-V_2=\frac{q_1+q_2}{4\pi\epsilon}(1/2r-1/3r)$
Yes! Now set V_3-V_1 equal to zero...

$V_2-V_1=\frac{q_1}{4\pi\epsilon}(1/r-1/2r)$
$V_3-V_2=\frac{q_1+q_2}{4\pi\epsilon}(1/2r-1/3r)$

$V_2-V_1=\frac{q_1}{4\pi\epsilon}(1/r-1/2r)$
$V_3-V_2=\frac{q_1+q_2}{4\pi\epsilon}(1/2r-1/3r)$

$V_3-V_1=\frac{q_1}{4\pi\epsilon r}+\frac{q_2}{4\pi\epsilon 6r}=0?$
$V_3-V_1=\frac{q_1}{4\pi\epsilon r}+\frac{q_2}{4\pi\epsilon 6r}=0?$