<s>[INST] The chart-problem; problem solving In how many ways can we construct a $6\times 6$ chart with only $1$ and $-1$ such that in every row and column, the product is always positive?
 [/INST] There are $\color{red}{2^{25}}$ ways of filling a $6\times 6$ grid with $0$ or $1$ in such a way that along every row or column there is an even number of $1$s. We may first fill a $5\times 5$ subgrid in any way we like, then complete those rows and columns in the only way they fit our constraints. The only square we still have to fill has to be filled with the parity of the number of $1s$ in the chosen $5\times 5$ subgrid:
$$\begin{pmatrix} 0 & 0 & 1 & 0 & 0 & ? \\
                  0 & 1 & 1 & 1 & 1 & ? \\
                  1 & 1 & 0 & 0 & 1 & ? \\
                  1 & 0 & 1 & 1 & 0 & ? \\
                  0 & 1 & 1 & 0 & 1 & ? \\
                  ? & ? & ? & ? & ? & ?  \end{pmatrix}\mapsto
\begin{pmatrix} 0 & 0 & 1 & 0 & 0 & \color{red}{1} \\
                  0 & 1 & 1 & 1 & 1 & \color{red}{0} \\
                  1 & 1 & 0 & 0 & 1 & \color{red}{1} \\
                  1 & 0 & 1 & 1 & 0 & \color{red}{1} \\
                  0 & 1 & 1 & 0 & 1 & \color{red}{1} \\
                  \color{red}{0} & \color{red}{1} & \color{red}{0} & \color{red}{0} & \color{red}{1} & \color{blue}{0}  \end{pmatrix}.$$
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