1. **Problem statement:** We need to find the smallest number of fence pieces to surround a rectangular garden measuring 3 m by 6 m without cutting or bending any fence pieces. 2. **Calculate the perimeter:** The perimeter $P$ of a rectangle is given by the formula: $$P = 2 \times (\text{length} + \text{width})$$ Here, length = 6 m and width = 3 m, so: $$P = 2 \times (6 + 3) = 2 \times 9 = 18 \text{ meters}$$ 3. **Fence piece sizes:** The available fence piece lengths are 0.6 m, 1.8 m, and 2.1 m. 4. **Goal:** Use the fewest pieces possible to make exactly 18 m without cutting. 5. **Check divisibility:** We want to find integers $n$ such that $n \times \text{fence length} = 18$. - For 0.6 m pieces: $$n = \frac{18}{0.6} = 30$$ - For 1.8 m pieces: $$n = \frac{18}{1.8} = 10$$ - For 2.1 m pieces: $$n = \frac{18}{2.1} = \frac{18}{2.1} = \frac{180}{21} = \frac{60}{7} \approx 8.57$$ (not an integer) 6. **Conclusion:** Only 0.6 m and 1.8 m pieces fit exactly without cutting. Among these, 1.8 m pieces require fewer pieces (10 vs 30). **Final answer:** The smallest number of fence pieces needed is **10** pieces of 1.8 m length.