Snow White has, in her huge basket, $2021$ apples, and she knows that exactly one of them has a deadly poison, capable of killing a human being hours after ingesting just a measly piece. Contrary to what the fairy tales say, Snow White is more malevolent than the Evil Queen, and doesn't care about the lives of the seven dwarfs. Therefore, she decided to use them to find out which apple is poisonous.

To this end, at the beginning of each day, Snow White prepares some apple salads (each salad is a mixture of pieces of some apples chosen by her), and forces some of the dwarfs (possibly all) to eat a salad each. At the end of the day, she notes who died and who survived, and the next day she again prepares some apple salads, forcing some of the surviving dwarves (possibly all) to eat a salad each. And she continues to do this, day after day, until she discovers the poisoned apple or until all the dwarves die.

(a) Prove that there is a strategy in which Snow White manages to discover the poisoned apple after a few days.

(b) What is the minimum number of days Snow White needs to find the poisoned apple, no matter how lucky she is?
1. **Part (a): Proving the Strategy**

   To prove that there is a strategy for Snow White to discover the poisoned apple, we can use a method inspired by binary search. Here is the detailed strategy:

   - Suppose there are \( n \) dwarfs available at the beginning of a day.
   - Snow White divides the remaining apples into \( n+1 \) piles.
   - Each dwarf is given an apple salad made from one of the \( n \) piles, leaving one pile ignored.
   - At the end of the day, Snow White observes the outcome:
     - If a dwarf dies, the poisoned apple is in the pile that the dead dwarf ate from. Snow White discards all other piles.
     - If no dwarf dies, the poisoned apple is in the ignored pile. Snow White discards all other piles.
   - This process is repeated with the remaining apples and dwarfs until the poisoned apple is identified.

   This strategy ensures that the number of apples to be tested is reduced by a factor of \( n+1 \) each day, where \( n \) is the number of dwarfs available.

2. **Part (b): Minimum Number of Days**

   To determine the minimum number of days required to find the poisoned apple, we need to consider the worst-case scenario. The worst-case scenario occurs when the dwarfs die one by one, and Snow White has to test the maximum number of apples.

   - Let \( k \) be the number of days required to find the poisoned apple.
   - Each day, the number of apples is reduced by a factor of \( n+1 \), where \( n \) is the number of dwarfs available.
   - In the worst-case scenario, the number of apples is reduced by a factor of 2 each day (since at least one dwarf dies each day).

   Therefore, the minimum number of days \( k \) can be found by solving the inequality:
   \[
   2^k \geq 2021
   \]
   Taking the logarithm base 2 on both sides:
   \[
   k \geq \log_2(2021)
   \]
   Calculating the value:
   \[
   k \geq \lceil \log_2(2021) \rceil
   \]
   Using a calculator:
   \[
   \log_2(2021) \approx 10.97
   \]
   Therefore:
   \[
   k \geq \lceil 10.97 \rceil = 11
   \]

   Hence, the minimum number of days required to find the poisoned apple in the worst-case scenario is 11 days.

The final answer is \(\boxed{11}\)