Let’s say I have 4 (A,B,C,D) parameters with 3 possible values, also 81 unique combinations. With e.g. orthogonal array, I will end up with 9 test cases, each combining 3 pairs. But that means that if one test case fails, I do not know which combination in the test case is faulty, right? Whether values of A,B or A,C or BC. Just wanted to be sure I understood it well.
edit: Corrected (thanks GlenH7).
Have a look at the Wikipedia article on Orthogonal Array Testing as well as the linked PDF from 51 testing
Some of your numbers are wrong.
Original version: 3 parameters (factors) with 4 values (levels) each, results in 4^3 combinations or 64 unique combinations, not 81.
Updated version: 4 parameters with 3 values each results in 3^4 combinations, or 81 as you asked.
Original: The minimum number of test cases to check all the pairings is factors * levels or 12 runs, not 9. Note that if you had differing number of levels, you would need to calculate that in a different manner. The explanatory PDF goes into more detail.
Update: The minimum number of pairings drops to 9. However, if only one test fails, you’ll still be able to identify the faulty pairing by comparing it against the ones that cleared.
I think that the incorrect numbers you were using threw off your reasoning and your question about knowing which pair failed. With the those 12 minimum runs (also listed below), you will know the pairing that failed.
Original version:
A B C
0 0 0
0 1 1
0 2 2
0 3 3
1 1 0
1 2 1
1 3 2
1 0 3
2 2 0
2 3 1
2 0 2
2 1 3
3 3 0
3 0 1
3 1 2
3 2 3
Updated version:
A B C D
0 2 0 0
0 1 1 1
0 0 2 2
1 0 1 0
1 2 2 1
1 1 0 2
2 1 2 0
2 0 0 1
2 2 1 2
4