3. (20 points) An equilateral triangle with unit sides is divided by three lines, parallel to its sides, into 4 equal triangles, and the middle triangle is discarded. Each of the remaining three triangles is in turn divided by three lines, parallel to its sides, into 4 equal parts, and its middle triangle is also discarded, after which the same operation is performed on each of the remaining triangles, and so on. Suppose the operation is repeated 12 times. How many equilateral triangles with side $1 / 2^{12}$ remain?
Answer: $3^{12}$.

Solution: In this case, after each operation, the number of triangles triples (since one of the four triangles obtained from the division is discarded), and the side lengths of the resulting triangles are half the side lengths of the original triangle. Therefore, after twelve operations, there will be $3^{12}$ equilateral triangles with side lengths of $1 / 2^{12}$.