Question 188
Let $ABCDEF$ be a regular hexagon of side length $2$. Through the vertices and the sides' midpoints we construct parallels to the sides, which divide the hexagon into $24$ congruent equilateral triangles, whose vertices are called *nodes*. A *sheet* is any (non-degenerate) equilateral triangle whose vertices are nodes. A *trio* of a node $X$ is the figure formed by three adjacent sheets such that their intersection is only $X$ and they are not congruent in pairs.
a) Find the maximal possible area of a trio.
b) Show that there exists a node whose trios can cover the whole hexagon, and a node whose trios cannot cover the whole hexagon.
c) Determine the total number of trios associated to the hexagon.
Dana Heuberger
a) Find the maximal possible area of a trio.
b) Show that there exists a node whose trios can cover the whole hexagon, and a node whose trios cannot cover the whole hexagon.
c) Determine the total number of trios associated to the hexagon.
Dana Heuberger