Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small.
[img]https://cdn.artofproblemsolving.com/attachments/a/1/aedfbfb2cb17bf816aa7daeb0d35f46a79b6e9.jpg[/img]
Nair build triangular figures by following these rules:
$\bullet$ Figure $1$ is made up of $4$ small pieces, Figure $2$ is made up of $2$ large pieces and $8$ small, Figure $3$ by $6$ large and $12$ small, and so on.
$\bullet$ The central column must be made up exclusively of small parts.
$\bullet$ Outside the central column, only large pieces can be placed.
[img]https://cdn.artofproblemsolving.com/attachments/5/7/e7f6340de0e04d5b5979e72edd3f453f2ac8a5.jpg[/img]
Following the pattern, how many pieces will Nair use to build Figure $20$?
1. **Identify the pattern for small triangles ($S_n$):**
   - For Figure 1: $S_1 = 4$
   - For Figure 2: $S_2 = 8$
   - For Figure 3: $S_3 = 12$
   - Observing the pattern, we see that the number of small triangles increases by 4 for each subsequent figure. Thus, we can generalize this as:
     \[
     S_n = 4n
     \]

2. **Identify the pattern for large triangles ($L_n$):**
   - For Figure 1: $L_1 = 0$
   - For Figure 2: $L_2 = 2$
   - For Figure 3: $L_3 = 6$
   - For Figure 4: $L_4 = 12$
   - Observing the pattern, we see that the number of large triangles follows the sequence of $n^2 - n$. Thus, we can generalize this as:
     \[
     L_n = n^2 - n
     \]

3. **Calculate the number of small and large triangles for Figure 20:**
   - For small triangles:
     \[
     S_{20} = 4 \times 20 = 80
     \]
   - For large triangles:
     \[
     L_{20} = 20^2 - 20 = 400 - 20 = 380
     \]

4. **Sum the number of small and large triangles to find the total number of pieces:**
   \[
   S_{20} + L_{20} = 80 + 380 = 460
   \]

The final answer is $\boxed{460}$