The question is the following: Can one create two nonidentical loaded 6-sided dice such that when one throws with both dice and sums their values the probability of any sum (from 2 to 12) is the same. I said nonidentical because its easy to verify that with identical loaded dice its not possible.


Formally: Let's say that $q\_{i}$ is the probability that we throw $i$ on the first die and $p\_{i}$ is the same for the second die. $p\_{i},q\_{i} \in [0,1]$ for all $i \in 1\ldots 6$. The question is that with these constraints are there $q\_{i}$s and $p\_{i}$s that satisfy the following equations:
  
$ q\_{1} \cdot p\_{1} = \frac{1}{11}$
  
$ q\_{1} \cdot p\_{2} + q\_{2} \cdot p\_{1} = \frac{1}{11}$
  
$ q\_{1} \cdot p\_{3} + q\_{2} \cdot p\_{2} + q\_{3} \cdot p\_{1} = \frac{1}{11}$
  
$ q\_{1} \cdot p\_{4} + q\_{2} \cdot p\_{3} + q\_{3} \cdot p\_{2} + q\_{4} \cdot p\_{1} = \frac{1}{11}$
  
$ q\_{1} \cdot p\_{5} + q\_{2} \cdot p\_{4} + q\_{3} \cdot p\_{3} + q\_{4} \cdot p\_{2} + q\_{5} \cdot p\_{1} = \frac{1}{11}$
  
$ q\_{1} \cdot p\_{6} + q\_{2} \cdot p\_{5} + q\_{3} \cdot p\_{4} + q\_{4} \cdot p\_{3} + q\_{5} \cdot p\_{2} + q\_{6} \cdot p\_{1} = \frac{1}{11}$
  
$ q\_{2} \cdot p\_{6} + q\_{3} \cdot p\_{5} + q\_{4} \cdot p\_{4} + q\_{5} \cdot p\_{3} + q\_{6} \cdot p\_{2} = \frac{1}{11}$
  
$ q\_{3} \cdot p\_{6} + q\_{4} \cdot p\_{5} + q\_{5} \cdot p\_{4} + q\_{6} \cdot p\_{3} = \frac{1}{11}$
  
$ q\_{4} \cdot p\_{6} + q\_{5} \cdot p\_{5} + q\_{6} \cdot p\_{4} = \frac{1}{11}$
  
$ q\_{5} \cdot p\_{6} + q\_{6} \cdot p\_{5} = \frac{1}{11}$
  
$ q\_{6} \cdot p\_{6} = \frac{1}{11}$
  



I don't really now how to start with this. Any suggestions are welcome.



Any sum of 2 dice with equal probability
You can write this as a single polynomial equation
$$p(x)q(x)=\frac1{11}(x^2+x^3+\cdots+x^{12})$$
where $p(x)=p\_1x+p\_2x^2+\cdots+p\_6x^6$ and similarly for $q(x)$.
So this reduces to the question of factorizing $(x^2+\cdots+x^{12})/11$
where the factors satisfy some extra conditions (coefficients positive,
$p(1)=1$ etc.).


This is a standard method (generating functions).


