How to solve for unknown symbols in a base-3 number system?

Question

Imagine that someone has a base-3 number system which is represented by A, B, and C. A, B, and C correspond to our usual 0, 1, and 2, but you do not know which is 0, which is 1, and which is 2. Assuming you can ask questions in terms of A, B, and C (such as “what is A + B + C,”), and you receive the answer in terms of the ABC base-3 number system, what is the fewest questions needed to solve for all of the unknowns?

Edit: There is no limit to the questions asked. The questions may include addition, subtraction, multiplication, division, and the modulus operator. The only restriction is that it is in terms of A, B, and C. Thanks again.

Answer 1

One question: what is $((A+B+C)A + B)(A+B+C)+C$?

Because $A+B+C$ is the base of the number system the result will be $ABC_{base}$, representing $A(base)^2 + B(base) + C$.

Answer 2

I start with $A+A$. If the sum is $A,$ then $A=0$. If the sum is $B$, we have $A=1,B=2,C=0$. If the sum is $BB$, we have $A=2,B=1,C=0$. The other cases interchange $B$ and $C$. If we just got $A=0,$ we need one more question, which can be $B+B$.

If we are allowed more complicated problems, I ask $AAABBBCCC+AAABBBCCC$. If the sum ends in $C$ we have $C=0$, otherwise $0$ is the letter not in the ones place. If $C$ is not $0$, we can identify whether it is $1$ or $2$ by whether $C+C$ carries into the threes place. It will be obvious whether $A$ or $B$ is $0$.

Answer 3

An alternative to others, what is $A+A+A+A+A+A+A+A+A+B+B+B+C$?