Can a regular triangle be divided into

a) 2007,

b) 2008 smaller regular triangles?
Solution. a) A regular triangle can be divided into 9 smaller regular triangles using the points that trisect the sides. The regular triangle closest to one of the vertices of the original triangle is always further divided into more regular triangles using its midlines. Thus, the number of triangles always increases by 3. These numbers form the following sequence: $9, 12, 15, 18, 21, \ldots$

The 2007th term of this sequence is 2007 because 2007 - 9 is divisible by 3. Therefore, a regular triangle can be divided into 2007 smaller regular triangles.

b) A regular triangle can be divided into 4 smaller regular triangles using its midlines. The regular triangle closest to one of the vertices of the original triangle is always further divided into more regular triangles using its midlines. Thus, the number of triangles always increases by 3. These numbers form the following sequence: 4, 7, 10, 13, 16, . .

The 2008th term of this sequence is 2008 because 2008 - 4 is divisible by 3. Therefore, a regular triangle can also be divided into 2008 smaller regular triangles.