Below is my attempt, but I failed to continue. The 2 dices are not necessarily identical but they are independent.
Let $A, B$ denote the outcome of 2 dices rolling, then the pmf of $A, B$ would be $P(A_i) = p_i, P(B_i) = q_i, i = 1, 2, 3, 4, 5, 6$ Then I write all of pmf of $S$. However, there are 12 unknowns and only 11 equations. Furthermore, there is no way for me to solve it by hands.
Any hint or help is welcome, thanks a lot!
Let $X$ the random variable recording the value of die $A$ when it is rolled, and $Y$ the random variable recording the value of die $B$, then the pmf of $X$ and $Y$ would be $P(X=i)=a_i$ and $P(Y=i)=b_i$, so using independence
\begin{eqnarray} P(X+Y=s)&=&\sum_{k=1}^6P(X=s-k, Y=k)\\&=&\sum_{k=1}^6P(X=s-k)P(Y=k)\\ &=&\sum_{k=1}^6a_{s-k}b_k \end{eqnarray} So for unfairs dice contructed such a way that $a_{s-x}b_x=1/11$ we get the desired distribution for $S$.
For $S=2$ and $S=3$ we get $a_1b_1=a_1b_2=a_2b_1=1/11$, so $b_1=b_2$ and $a_1=a_2$. Using induction it is easy to prove that $a_{s-x}b_x$ implies $a_1=a_2=\ldots=a_6$ and $b_1=b_2=\ldots=b_6$, wich is a contradiction with the fact that the dice are unfair. So it is impossible to construct such a dice.