After reading this page, it seems that pairwise testing requires a set of test cases in which every pair of values from any two of the n categories occurs at least once among the test case n-tuples. In the present case, the problem is to find a minimal subset of the 6x6x6 = 216 total triples (a,b,c) such that
each pair of values for a and b
occurs at least once, i.e. (a,b,*),
each pair of a and c values occurs
at least once, i.e. (a,*,c)
each pair of b and c values occurs
at least once, i.e. (*,b,c)
Any subset satisfying these requirements must have at least 36 elements just to satisfy the (a,b,*) requirement. In the present case I think 36 test cases are also sufficient, as in the following set of triples:
(1, 1, 1), (1, 2, 2), (1, 3, 3), (1, 4, 4), (1, 5, 5), (1, 6, 6)
(2, 1, 6), (2, 2, 1), (2, 3, 2), (2, 4, 3), (2, 5, 4), (2, 6, 5)
(3, 1, 5), (3, 2, 6), (3, 3, 1), (3, 4, 2), (3, 5, 3), (3, 6, 4)
(4, 1, 4), (4, 2, 5), (4, 3, 6), (4, 4, 1), (4, 5, 2), (4, 6, 3)
(5, 1, 3), (5, 2, 4), (5, 3, 5), (5, 4, 6), (5, 5, 1), (5, 6, 2)
(6, 1, 2), (6, 2, 3), (6, 3, 4), (6, 4, 5), (6, 5, 6), (6, 6, 1)
In this example each of the three kinds of pairs occurs once and only once, i.e. there is no overlap. I don't think this will be possible in general, so it might not always be easy to come up with minimal subsets that cover all the cases.