Weighted sum of two dice such that the result is a random integer between $0$ and $35$ 
The numbers shown by 2 dice are labelled $d$ and $e$. $A, B$ and $C$ are constants, giving a score $S=Ad + Be + C$. Find $A, B$ and $C$ such that the range of possible values for $S$ covers all integers from $0$ to $35$, with an equal probability of each score. [Oxford PAT exam, 2011]

I have tried to find equations in terms of $A, B, C$ for different values of $S$.

The values of $S$ can be represented as $6a+b$ where $a$ and $b$ range from 0 to 5; there is a one-to-one matching between pairs $(a,b)$ and numbers in $S$. Accordingly, $6(a-1)+(b-1)=6a+b-7$ yields the same pairing, but for $a$ and $b$ from 1 to 6 – the actual faces of the dice. Therefore, to have each score appear with equal probability we should have
$$A=6,B=1,C=-7$$
