More than Three Polygons

  1. In a previous problem, Three Polygons, I asked: Three regular polygons, all with unit sides, share a common vertex. Each polygon has a different number of sides, and each polygon shares a side with the other two; there are no gaps or overlaps. Find the number of sides for each polygon.
  2. I now expand on that idea: If the restriction that each polygon has a different number of sides is removed, how many more solutions are there? There can be more than three polygons (up to six triangles). What are all the possible combinations?
  3. In a plane, one set of polygons may be found at one vertex, while different sets could be found at others. There could be all triangles (1 unique polygon), or a combination of triangles and hexagons (2 unique polygons), and others. What is the largest number of unique polygons which can tile a plane?

Source: Original.

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