The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent.
My question is:
What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure?
I define a polygonal tile as a figure comprised of:
A finite set $P$ of at least three points.
The (straight) lines $X\pi(X)$ for each point $X\in P$ and some cyclic permutation $\pi$ of $P$ such that distinct lines do not intersect except at end points.
The region contained by the lines.
The hexagon and triangle are assumed to include the vertices, edges and interior.
The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with any other copy, but may share boundary points with one or more.
A congruent copy of a tile may be a "flipped" version (e.g. the pair of cyan tiles in the above digram are actually mirror images - this is allowed).