There are many wonderful math circle activities related to tiling problems.
There is a collection of tiling problems that may be solved via coloring. This is one of the more difficult collection of tiling problems. It is due to Tom Davis. http://www.geometer.org/mathcircles/
Dave Auckly provided a list of tiling problems related to combinatorial group theory (the ME tiles file below.) It also has a mentor guide. This one should be paired with the Turning Laughter into Insight activity.
At the Joint Math Meetings in 2015, Dave Auckly led a demo circle, and Japheth Wood discussed
the lesson for circle leaders. The hand out and a compact version of the hand out are included in the attachments. The idea of remodeling - changing one grid into another is explored in this session.
Leszek Gawarecki prepared the Math Club Tetris Tiles hand out below. It is of intermediate difficulty.
There is a lesson plan created by Matthias Kawski for the Circle on the Road workshop.
This session focuses on planar problems: Use a set of flat tiles of rectangular, triangular and other shapes to tile the floor of a more or less odd shaped bathroom (without cutting any tiles, without any gaps or overlays).
Mathematically, sometimes impossibility can be established using some kind of parity argument parity. As the rooms get larger (or the tiles get smaller) it becomes natural to use solutions of smaller problems. This naturally suggests and leads to formal
His presentation is most suitable for younger students.
More discussion and templates may be found at
The square and triangular grids attached may be useful.