  A knot k is called ``strongly (n-1)-trivial.'' if there exists a projection
of k, such that one can choose n crossings of the projection with the property
that making the crossing changes corresponding to any of the $2^{n}-1$
nontrivial combinations of the selected crossings turns the original knot into
the unknot. We prove that given any non-trivial knot k of genus g, k fails to
be strongly n-trivial for all $n, n \geq 3g-1$.
