Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine the mass of any material (the mass can be any real number between $0$ and $20.2$ kg) with an error of at most $10$ g.
1. **Understanding the problem**: We need to determine the smallest number of weights, \( n \), such that Ana can measure any mass between \( 0 \) and \( 20.2 \) kg with an error of at most \( 10 \) g. Each weight must be at least \( 10 \) g.

2. **Initial approach**: Let's consider the weights \( a_1, a_2, \ldots, a_n \). Each weight \( a_i \) must be at least \( 10 \) g. We need to ensure that any mass between \( 0 \) and \( 20.2 \) kg can be measured with an error of at most \( 10 \) g.

3. **Checking \( n = 2019 \)**: 
   - If we pick \( a_1 = a_2 = \ldots = a_{2018} = 10 \frac{1}{10000} \) g and \( a_{2019} = 20200 - 2018 \cdot 10 \frac{1}{10000} = 19 \frac{7972}{10000} \) g, the smallest possible sum of masses of weights is \( 10 \frac{1}{10000} \) g.
   - This means the smallest measurable mass is \( 10 \frac{1}{10000} \) g, and we cannot measure masses smaller than this with the required precision. Hence, \( n = 2019 \) does not satisfy the requirements.

4. **Checking \( n = 2020 \)**:
   - If each weight is exactly \( 10 \) g, then the total mass of the weights is \( 2020 \cdot 10 = 20200 \) g or \( 20.2 \) kg.
   - This setup allows us to measure any mass in increments of \( 10 \) g, from \( 0 \) to \( 20.2 \) kg.
   - Since each weight is \( 10 \) g, we can measure any mass with an error of at most \( 10 \) g by determining which multiple of \( 10 \) g the mass is closest to.

5. **Conclusion**: The smallest possible value of \( n \) for which Ana can measure any mass between \( 0 \) and \( 20.2 \) kg with an error of at most \( 10 \) g is \( 2020 \).

The final answer is \( \boxed{ n = 2020 } \).