Let $f : [0, 1] \rightarrow \mathbb{R}$ be a monotonically increasing function such that
$$f\left(\frac{x}{3}\right) = \frac{f(x)}{2}$$
$$f(1 0 x) = 2018 - f(x).$$
If $f(1) = 2018$, find $f\left(\dfrac{12}{13}\right)$.

1. Given the function \( f : [0, 1] \rightarrow \mathbb{R} \) is monotonically increasing and satisfies the conditions:
   \[
   f\left(\frac{x}{3}\right) = \frac{f(x)}{2}
   \]
   \[
   f(10x) = 2018 - f(x)
   \]
   and \( f(1) = 2018 \).

2. First, let's analyze the first condition \( f\left(\frac{x}{3}\right) = \frac{f(x)}{2} \):
   - For \( x = 1 \):
     \[
     f\left(\frac{1}{3}\right) = \frac{f(1)}{2} = \frac{2018}{2} = 1009
     \]
   - For \( x = \frac{1}{3} \):
     \[
     f\left(\frac{1}{9}\right) = \frac{f\left(\frac{1}{3}\right)}{2} = \frac{1009}{2} = 504.5
     \]

3. Next, let's analyze the second condition \( f(10x) = 2018 - f(x) \):
   - For \( x = 0.1 \):
     \[
     f(1) = 2018 - f(0.1)
     \]
     Since \( f(1) = 2018 \), we get:
     \[
     2018 = 2018 - f(0.1) \implies f(0.1) = 0
     \]

4. Now, let's find \( f\left(\frac{12}{13}\right) \):
   - We need to find a pattern or a specific value that fits the given conditions. Notice that \( f(10x) = 2018 - f(x) \) implies a symmetry around 1009. For \( x = \frac{12}{13} \):
     \[
     f\left(\frac{12}{13}\right) = 2018 - f\left(\frac{1}{13}\right)
     \]
   - To find \( f\left(\frac{1}{13}\right) \), we use the first condition:
     \[
     f\left(\frac{1}{13}\right) = \frac{f\left(\frac{3}{13}\right)}{2}
     \]
     and so on, but this becomes complex without a clear pattern.

5. Given the constraints and the symmetry, we can infer that \( f\left(\frac{12}{13}\right) \) should be close to 1009, but since \( f \) is monotonically increasing and \( f(1) = 2018 \), we can deduce:
   \[
   f\left(\frac{12}{13}\right) = 2018 - f\left(\frac{1}{13}\right)
   \]
   Since \( f\left(\frac{1}{13}\right) \) is very small, \( f\left(\frac{12}{13}\right) \) is very close to 2018.

The final answer is \( \boxed{2018} \).