  Define the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ so
$D(x_1,x_2, ...,x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m,
..., x_n+x_1 \; \text{mod} \; m)$. In this paper, we will prove that for $n$
odd and $m=2^lm_1$ where $m_1$ is odd, then the largest the length of a tuple
can be is $l$ and $K(\mathbb{Z}_m^n)=\{(x_1, x_2, ..., x_n) \in \mathbb{Z}_m^n
\; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$.
