You may define the problem as you wish
as long as […] the solutions are the same, no more and no less
Can you clarify what do mean by “solution” and what is allowed and what is not in problem definitions?
Consider the example from the problem statement. Let me denote values of cells as x0, …, x3. The unique solution for the example is x0 = 2, x1 = 1, x2 = 1, x3 = 3
.
Can I define following mappings between qubits and the problem domain? And if not, why?

I have only one qubit
q0
and I map its measured value to cells as such:x0 = 2, x1 = q0, x2 = 1, x3 = 3
, and the problem in my “definition” is whetherq0 = 1
. This problem has the same solutions as the original, hasn’t it?
So what does mean “solution” and why this “definition” is incorrect?
Do you mean that the set of possible puzzle states should be the same in the real problem and in my defintion? 
By subtracting eq. 5 from eq. 7 you can get that
x2 = x1
. Can I use the same qubits for values ofx1
andx2
? 
Suppose that using some optimization technique I get that
x2 < 2
. Can I use only one bit for the value ofx2
?