Let $G = ({\Large\ast}^n\mathbb{Z})/K$ be a group, and for each $g \in G$ define $l(g)$ as the smallest positive integer $m$ such that $g = g_1 \ldots g_m$, where each $g_i$ is a generator of $G$. Now let $H < G$. The problem is finding $\operatorname{arg\,min}_{g \in H \setminus \{1\}} l(g)$.
My motivation is the commutator-based magic' puzzle posted by Gil Kalai here, which boils down to finding a non-trivial element of $\bigcap_{i=1}^n \operatorname{ker}p_i$, where $p_i: {\Large\ast}_{j = 1}^n \mathbb{Z}a_j \to {\Large\ast}_{j \neq i}\mathbb{Z}a_j$ is defined by $p_i(a_j) = a_j$ when $i \neq j$ and $p_i(a_i) = 1$. One non-trivial element is $[\ldots[a_1, a_2], a_3], \ldots], a_n]$, but I heard people complaining that it's too long and the corresponding loop will be too hard to draw, so I'm now interested in finding a smaller solution, if it exists.
Are there any known theorems concerning this kind of problems or any techniques I could try or any references I could read?