A square has a side of 2 dm. A new square is formed by joining the midpoints of the sides. If this procedure is repeated 6 times, what is the perimeter of the smallest square?

Answer 1

After joining the midpoints of the sides of a square, a smaller square is formed inside it. The ratio of the side length of the smaller square to the side length of the original square is $1:2$. Thus, the side length of each square after the first square is half the side length of the previous square in the sequence.

Therefore, the sequence of side lengths of the squares is $2, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}$ dm. The perimeter of each square is four times its side length, so the sequence of perimeters is $8, 4, 2, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}$ dm.

The perimeter of the smallest square is $\boxed{\frac{1}{8}}$ dm.