The effective resistance between A and B is Z = 1/(1/z + 1/(x + y)).
The effective resistance between B and C is X = 1/(1/x + 1/(y + z)).
The effective resistance between C and A is Y = 1/(1/y + 1/(z + x)).

These effective resistances are the easy ones to measure as it is forbidden to break the connections for the purposes of measuring. When I learned of the algebra problem of solving for x, y and z in terms of X, Y and Z, I indeed failed to find an algebraic solution. Some years later I did find it but just now I have misplaced that solution. A clue is to exploit symmetry.

My father told me that a technition would not make these measurements but would do it the easy way. I thought about that for a while and could not figure out an easy way. When he told me it was indeed easy. Can you see how to learn x, y and z using only elementary cheap stuff, and, of course, what ever it takes to measure resistance? This would not have occurred to me! You need only a bit of high school electricity theory and algebra. See the answer.