House of mirrors

Here's the puzzle, and here's an interactive I wrote in PaperJS (Javascript). This interactive allows you to play around with the puzzle much more efficiently than by printing out grids, coloring them with markers, and using a pen to draw your lines of symmetry.


Let's find an upper bound on the number of lines possible.

Let \(r\) and \(c\) be the number of rows and columns respectively. Then the number of red diagonal lines from top left to bottom right, is \(r + c - 1.\) The number of black horizontal lines, is \(r + 1.\) The number of red horizontal lines which are coincident with a black line, is \(r + 1 - 4 = r - 3.\) The number of red horizontal lines that cut the middle of a square is \(r - 2.\) Therefore, the total number of horizontal lines is

$$r - 3 + r - 2 = 2r - 5$$

Thus, the total number of red lines possible, in a grid containing \(r\) rows and \(c\) columns, is

$$\begin{align} & 2(r + c - 1) + 2r - 5 + 2c - 5 \\ =\,& 2r + 2c - 2 + 2r + 2c - 10 \\ =\,& 4r + 4c - 12 \end{align}$$