When a proof falls apart in the inductive step, the first recourse is
to look for a \textit{stronger} induction hypothesis.  Specifically,
we need to find some property of the puzzle that is \textit{always}
maintained, but that the transpose configuration lacks.  If we could
find such a property, then we would know that the theorem is true.  In
general, any property of a system that is always maintained is called
an \term{invariant}.  Let's consider some candidate properties: