KöMaL 2016 A679

Let $n = 2^{128}$, $M = \{1, 2, 3, 4\}$, and let $M^n$ denote the set of sequences of length $n$ and consisting of elements of $M$. Decide whether there exist functions $f_1,\ldots,f_n$ and $g_1,\ldots,g_n\colon M^n\to M$ such that for every pair of sequences $(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in M^n,$ at least one of the following statements holds: $\bullet$ $f_i(y_1,\ldots,y_n) = x_i$ for some index $i$ with $1\le i\le n$; $\bullet$ $g_j(x_1,\ldots,x_n) = y_j$ for some index $j$ with $1\le j\le n$.