Question 29
A regular hexagon has side $1000$. Each side is divided into $1000$ equal parts. Let $S$ be the set of the vertices and all the subdividing points. All possible lines parallel to the sides and with endpoints in $S$ are drawn, so that the hexagon is divided into equilateral triangles with side $1$.
Let $X$ be the set of all vertices of these triangles. We now paint any three unpainted members of $X$ which form an equilateral triangle (of any size). We then repeat until every member of $X$ except one is painted. Show that the unpainted vertex is not a vertex of the original hexagon.
Let $X$ be the set of all vertices of these triangles. We now paint any three unpainted members of $X$ which form an equilateral triangle (of any size). We then repeat until every member of $X$ except one is painted. Show that the unpainted vertex is not a vertex of the original hexagon.