Suppose that $U$ and $U'$ are normal ultrafilters associated with some supercompact cardinal. How may we compare $U$ and $U'$? In what ways are they similar, and in what ways are they different? Partial answers are given in [1], [2], [3], [5], [6], and [7]. In this paper, we continue this study. In [6], Menas introduced a combinatorial principle $\chi(U)$ of normal ultrafilters $U$ associated with supercompact cardinals, and showed that normal ultrafilters satisfying this property also satisfying this property also satisfy a partition property. In [5], Kunen and Pelletier showed that this partition property for $U$ does not imply $\chi (U)$. Using results from [3], we present a different method of finding such normal ultrafilters which satisfy the partition property but do not satisfy $\chi (U)$. Our method yields a large collection of such normal ultrafilters.