Fix a prime number $p$ and consider the affine space $V = \mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, \ldots, V_n \subseteq V$ of dimension $k$, and take $v_i \notin V_i$. Do there always exist subspaces $W_i \supseteq V_i$ of codimension $1$ and $t_1, \ldots, t_n \in V$ such that $v_i \notin W_i$ and $$t_1 + W_1, \ldots, t_n + W_n$$ is a cover of $V$ by affine hyperplanes? Is this true for at least some $n$?
Note that:
- Comparing cardinalities, one needs $n \geq p$.
- It suffices to do the case where $k = \lfloor \frac{n-1}2 \rfloor$ is the largest integer strictly less than $n/2$ (we can always replace the $V_i$ by larger subspaces)
I'm especially interested if there exists an even $n > p$ for which this is true. Also, I don't mind assuming that $p$ belongs to some infinite set.
Motivation: If it is true for some even $n$, this would imply that the answer to this question is 'yes': Hat 'trick': Can one of them guess right?
