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# Another dissection puzzle

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?

Small print:

• 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 mutual 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 the copy of any other tile, 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).