I'm try to design a game in which the board is made up of a 3x3 grid of square tiles. Each tile is a 5x5 grid of spaces. Each tile has 4 exit spaces each located on 1 of the middle 3 spaces along each edge of the tile. All exits must be connected along a path of orthogonally adjacent spaces. The paths can only be 1 space wide and must all lead to a single intersecting space in any of the middle 9 spaces of a tile. Spaces in this path that are not exits cannot be on an edge space. How many possible tiles can exist within these parameters?
It is not necessary that paths connect from tile to tile. I'm curious to know the math behind this in case I need to adjust tile sizes.