Let $G$ be a group and $S\subseteq G$. Consider a $d$-dimensional chessboard of size $n_1\times n_2\times \ldots \times n_d$, where $n_1,n_2,\ldots,n_d\in\mathbb{N}$. Each unit hypercube of the chessboard contains an element of $G$, which is called the label of the hypercube. A move consists of three parts:
(1) selecting an element $s\in S$,
(2) choosing a unit hypercube of the chessboard, say with index $\left(i_1,i_2,\ldots,i_d\right)$, and
(3) multiplying $s$ on the left to the label of every unit hypercube with index $\left(j_1,j_2,\ldots,j_d\right)$ such that $j_\mu\geq i_\mu$ for all $\mu=1,2,\ldots,d$.
The task is to make a finite sequence of moves from an arbitrary initial configuration to a situation in which every label is the identity $1_G$ of the group $G$. Here are the questions.
(a) If $S=G$, then justify that it is always possible to complete the task in $N:=n_1n_2\cdots n_d$ moves.
(b) Assume Part (a). Determine whether or not $N$ is the smallest integer $\nu$ such that, from every initial configuration, it is guaranteed that there exists a sequence of at most $\nu$ moves to complete the required task.
(c) If $S$ generates $G$ as a monoid (namely, for every $g\in G$, there exist $s_1,s_2,\ldots,s_l\in S$ with $l\in\mathbb{N}\cup\{0\}$ such that $g=s_1s_2\cdots s_l$), and there exists a natural number $\ell$ such that every element of $G$ is a product of at most $\ell$ (not necessarily distinct) elements of $S$, then verify that it is always possible to complete the task in $\ell n_1n_2\cdots n_d$ moves.
(d) Assume Part (c). Suppose that $\lambda$ is the smallest possible of such integers $\ell$. Is it true that there is an initial configuration that requires at least $\lambda n_1n_2\cdots n_d$ moves to complete the task?
(e) Suppose that $S$ generates $G$ as a monoid, but the length of an element of $G$ with respect to $S$ can be arbitrary large. (The length of $g\in G$ with respect to $S$ is the smallest integer $l\geq 0$ such that there exist $s_1,s_2,\ldots,s_l\in S$ for which $g=s_1s_2\cdots s_l$.) Prove that it is still possible to complete the task from any initial configuration in finitely many moves.
(f) Assume Part (e). For a given $k\in\mathbb{N}$, does there exist an initial configuration that requires at least $k$ moves to complete the task?
(g) Show that the required task is always possible if and only if $S$ generates $G$ as a monoid.
Here, $1_G$ is considered to be the product of $0$ element of $S$. The answers to Parts (b) and (d) are unknown, but for abelian groups $G$, the answers are both affirmative. This problem is inspired by $n\times n$ chessboard game with coins (in which $d=2$, $n_1=n_2=n$, $G=\left\langle s\,|\,s^2=1_G\right\rangle$, and $S=\{s\}$). There, I also posted a sketch of a proof for the case where $G=\left\langle s\,|\,s^t=1_G\right\rangle$ for a fixed $t\in\mathbb{N}$ and $S=\{s\}$.
This is not an answer, just a group-theoretic reformulation of the question. I'm posting here because I don't want to overload the question with too much text.
If $S$ generates $G$ as a monoid, then let $\lambda_G(S)$ denote the maximum length of elements of $G$ with respect to $S$ (where we set $\lambda_G(S):=\infty$ if there are no longest elements of $G$ with respect to $S$). Denote by $[n]$ the set $\{1,2,\ldots,n\}$ for each $n\in\mathbb{N}$. For $i_1\in\left[n_1\right]$, $i_2\in\left[n_2\right]$, $\ldots$, $i_d\in\left[n_d\right]$, and $g\in G$, define $T_{i_1,i_2,\ldots,i_d}(g)$ to be the element $\left(t_{j_1,j_2,\ldots,j_d}\right)_{j_1\in\left[n_1\right],j_2\in\left[n_2\right],\ldots,j_d\in\left[n_d\right]}$ of $\mathcal{G}:=G^{n_1\times n_2\times\ldots n_d}$ such that $t_{j_1,j_2,\ldots,j_d}=g^{-1}$ if $j_\mu\geq i_\mu$ for all $\mu=1,2,\ldots,d$, whereas $t_{j_1,j_2,\ldots,j_d}=1_G$ otherwise. Write $\tau(S)$ for $$\left\{T_{i_1,i_2,\ldots,i_d}(s)\,|\,s\in S\text{ and }i_\mu\in\left[n_\mu\right]\text{ for all }\mu=1,2,\ldots,d\right\}\,.$$
(a) Justify that $\tau(G)$ generates $\mathcal{G}$ as a monoid and $\lambda_\mathcal{G}\big(\tau(G)\big)\leq n_1n_2\cdots n_d$.
(b) Determine whether $\lambda_\mathcal{G}\big(\tau(G)\big)=n_1n_2\cdots n_d$.
(c) If $S$ generates $G$ as a monoid and $\lambda_G(S)<\infty$, verify that $\tau(S)$ generates $\mathcal{G}$ as a monoid and $\lambda_\mathcal{G}\big(\tau(S)\big)\leq \lambda_G(S)\,n_1n_2\cdots n_d$.
(d) Is it true that $\lambda_\mathcal{G}\big(\tau(S)\big)= \lambda_G(S)\,n_1n_2\cdots n_d$, provided that $S$ generates $G$ as a monoid and $\lambda_G(S)<\infty$?
(e) Prove in general that, if $S$ generates $G$ as a monoid, then $\tau(S)$ generates $\mathcal{G}$ as a monoid.
(f) Does it hold that $\lambda_\mathcal{G}\big(\tau(S)\big)=\infty$, given that $S$ generates $G$ as a monoid and $\lambda_G(S)=\infty$?
(g) Show that $\tau(S)$ generates $\mathcal{G}$ as a monoid if and only if $S$ generates $G$ as a monoid.
The answers to Parts (b) and (d) may be "no" (but I believe that the answer to (b) is "yes"). I now believe that (d) is false for most pairs $(G,S)$. A probable counterexample is $d=1$, $n_1=2$, $G=\big\langle a,b\,|\,a^2=1_G,\,b^3=1_G,\text{ and }ba=ab^2\big\rangle$ (i.e., the dihedral group of order $6$), and $S=\left\{a,b\right\}$. I think that $\lambda_\mathcal{G}\big(\tau(S)\big)=4<6=\lambda_G(S)\,n_1$, but I am not certain whether I have made any errors.
If the answer to (d) is indeed negative, it may make sense to ask the following question: asymptotically, how bad does $\lambda_{\mathcal{G}}\big(\tau(S)\big)$ deviates from $\lambda_G(S)$? That is, it may be of interest to compute $$\underline{\lambda}^d_G(S):=\liminf_{\prod\limits_{\mu=1}^d\,n_\mu\,\to\,\infty}\,\frac{\lambda_{\mathcal{G}}\big(\tau(S)\big)}{\prod\limits_{\mu=1}^d\,n_\mu}\text{ and }\overline{\lambda}^d_G(S):=\limsup_{\prod\limits_{\mu=1}^d\,n_\mu\,\to\,\infty}\,\frac{\lambda_{\mathcal{G}}\big(\tau(S)\big)}{\prod\limits_{\mu=1}^d\,n_\mu}\,,$$ or even $\underline{\Lambda}_G(S):=\lim\limits_{d\to\infty}\,\underline{\lambda}^d_G(S)$ and $\overline{\Lambda}_G(S):=\lim\limits_{d\to\infty}\,\overline{\lambda}^d_G(S)$.
Generalization: Each unit hypercube $\left(i_1,i_2,\ldots,i_d\right)\in\left[n_1\right]\times \left[n_2\right]\times\ldots\left[n_d\right]$ is associated to a left $G$-set $M_{i_1,i_2,\ldots,i_d}$ and a label of this hypercube is an element of $M_{i_1,i_2,\ldots,i_d}$. A move is defined as before. Fix $m_{i_1,i_2,\ldots,i_d}\in M_{i_1,i_2,\ldots,i_d}$ for all $i_\mu\in\left[n_\mu\right]$, $\mu=1,2,\ldots,d$. The task is to make a finite sequence of moves from an arbitrary initial configuration to a situation where the label of the unit hypercube $\left(i_1,i_2,\ldots,i_d\right)$ is $m_{i_1,i_2,\ldots,i_d}$ for every indices $\left(i_1,i_2,\ldots,i_d\right)\in\left[n_1\right]\times \left[n_2\right]\times\ldots\left[n_d\right]$. Then, this task is always possible if and only if
(i) $G$ acts transitively on every $M_{i_1,i_2,\ldots,i_d}$, and
(ii) the image of $S$ under the canonical projection $G\,\to \, G/K_{i_1,i_2,\ldots,i_d}$, where $K_{i_1,i_2,\ldots,i_d}$ is the kernel of the $G$-action on $M_{i_1,i_2,\ldots,i_d}$, generates $G/K_{i_1,i_2,\ldots,i_d}$ as a monoid for every $\left(i_1,i_2,\ldots,i_d\right)$ in $\left[n_1\right]\times \left[n_2\right]\times\ldots\left[n_d\right]$.
You can similarly ask questions (a), (b), (c), (d), (e), and (f), but I shall leave them to you.