I recently asked a question on the puzzling site, where I placed three colored T-tetraminos on the plane and asked for a tiling of the plane with T-tetraminos that fulfilled the following three properties:
It contains my provided tiles in the same positions.
No two tiles of the same color share an edge.
With a finite number of exceptions, all tetraminos should be one of two colors.
The three tiles I put down were intended to make a solution challenging but possible and there are some solutions there that show it is possible (you are going to have to try it yourself to see if it is challenging). However I want to be a little more devious and choose a starting position (of a finite number of T-tetraminos) that permits at least one tiling following rules 1 and 2 but none that follow all three rules. However coming up with one has been very challenging.
The question here is does such a starting position exist?