To solve this, we calculate the total perimeter of the garden and determine how many fence pieces are needed based on the available fence sizes.

Step 1: Calculate the perimeter

The garden is rectangular with dimensions:

• Length = $6 \, \text{m}$
• Width = $3 \, \text{m}$

The perimeter of a rectangle is given by: $P = 2 \times (\text{length} + \text{width})$ Substituting the values: $P = 2 \times (6 + 3) = 2 \times 9 = 18 \, \text{m}$

Step 2: Find the smallest number of pieces

We need a total length of 18 meters, and we cannot cut or bend the fence pieces. The available fence sizes are $0.6 \, \text{m}$, $1.8 \, \text{m}$, and $2.1 \, \text{m}$. We aim to minimize the number of pieces.

Strategy:

• Use as many large pieces ($2.1 \, \text{m}$) as possible.
• Then use smaller pieces to make up the remainder.

Calculation:

1. Number of $2.1 \, \text{m}$ pieces: $\text{Maximum number of } 2.1 \, \text{m pieces} = \left\lfloor \frac{18}{2.1} \right\rfloor = 8$ Total length from 8 pieces: $8 \times 2.1 = 16.8 \, \text{m}$ Remaining length: $18 - 16.8 = 1.2 \, \text{m}$

2. Use $1.8 \, \text{m}$ or $0.6 \, \text{m}$ pieces for the remaining $1.2 \, \text{m}$:

   • $1.2 \, \text{m}$ can be exactly made up with: $2 \, \text{pieces of } 0.6 \, \text{m}$

Total number of pieces:

• $8 \, \text{pieces of } 2.1 \, \text{m}$
• $2 \, \text{pieces of } 0.6 \, \text{m}$

Total pieces: $8 + 2 = 10$

Final Answer:

Esme needs 10 pieces of fence to surround her garden.

Let me know if you have any questions or want further details!

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