Question 20
A room is an equilateral triangle side $100$ meters. It is subdivided into $100$ rooms, all equilateral triangles with side $10$ meters. Each interior wall between two rooms has a door. If you start inside one of the rooms and can only pass through each door once, show that you cannot visit more than $91$ rooms.
Suppose now the large triangle has side $k$ and is divided into $k^2$ small triangles by lines parallel to its sides. A chain is a sequence of triangles, such that a triangle can only be included once and consecutive triangles have a common side. What is the largest possible number of triangles in a chain?
Suppose now the large triangle has side $k$ and is divided into $k^2$ small triangles by lines parallel to its sides. A chain is a sequence of triangles, such that a triangle can only be included once and consecutive triangles have a common side. What is the largest possible number of triangles in a chain?