4. The diagram shows a shape made up of 36 identical small equilateral triangles. What is the smallest number of small triangles identical to these that could be added to the shape to turn it into a hexagon?
A 10
B 12
C 15
D 18
E 24
Solution
D

To turn the figure given in the question into a hexagon by adding the smallest number of triangles, two triangles should be added to create each vertex of the hexagon (shaded dark grey) and one triangle added to create each edge (shaded light grey) as shown on the right for one edge. Since a hexagon has six vertices and six edges, the smallest number of triangles required is $6 \times(2+1)=18$.