So if the system is consistent, following logic brings us a conclusion that

C did it.

How? that's simple

A says: (a) I did not do it. (b) B did it.
Assume that A(a) is TRUE and A(b) is FALSE


B says: (a)I did not do it. (b)I know that C did it.
being consistent with A(b), we've to assume that B(a) is TRUE, which means that B(b) is FALSE


C says: (a)I did not do it. (b)B does not know who it was.
being consistent with B(b), C(b) is TRUE since they both state same thing, which means that C(a) is FALSE.


so C did it.

So if the system is consistent, following logic brings us a conclusion that

C did it.

How? that's simple

A says: (a) I did not do it. (b) B did it.
Assume that A(a) is TRUE and A(b) is FALSE


B says: (a)I did not do it. (b)I know that C did it.
being consistent with A(b), we know that that B(a) is TRUE, which means that B(b) is FALSE


C says: (a)I did not do it. (b)B does not know who it was.
being consistent with B(b), C(b) is TRUE since !B(b) and C(b) state same thing, which means that C(a) is FALSE.


so C did it.