Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?
1. **Define the Problem and Setup:**
   Peter and Basil have ten quadratic trinomials, and Basil calls out consecutive natural numbers starting from some natural number. Peter then evaluates one of the ten polynomials at each called number, and the results form an arithmetic sequence.

2. **Formulate the Arithmetic Sequence:**
   Let the sequence of numbers called by Basil be \( n, n+1, n+2, \ldots, n+k-1 \). The results of evaluating the polynomials at these numbers form an arithmetic sequence. An arithmetic sequence has the form \( a, a+d, a+2d, \ldots, a+(k-1)d \), where \( a \) is the first term and \( d \) is the common difference.

3. **Polynomial Evaluation:**
   Let the ten quadratic polynomials be \( P_1(x), P_2(x), \ldots, P_{10}(x) \). Each polynomial is of the form \( P_i(x) = ax^2 + bx + c \).

4. **Conditions for Arithmetic Sequence:**
   For the sequence to be arithmetic, the difference between consecutive terms must be constant. This implies that for any three consecutive terms \( P_i(n), P_j(n+1), P_k(n+2) \), the following must hold:
   \[
   P_j(n+1) - P_i(n) = P_k(n+2) - P_j(n+1)
   \]
   Simplifying, we get:
   \[
   P_j(n+1) - P_i(n) = P_k(n+2) - P_j(n+1) \implies P_k(n+2) - 2P_j(n+1) + P_i(n) = 0
   \]

5. **Constructing Specific Polynomials:**
   To achieve an arithmetic sequence, we can construct specific polynomials. For instance, choose \( P_i(x) \) such that:
   \[
   P_i(2i-1) = 2i-1 \quad \text{and} \quad P_i(2i) = 2i
   \]
   This ensures that the polynomials evaluated at these points form part of an arithmetic sequence.

6. **Maximum Number of Called Numbers:**
   To determine the maximum number of numbers Basil could call, consider the Pigeonhole Principle (PHP). If Basil calls out 21 numbers, then by PHP, at least one polynomial must be used at least three times. However, a quadratic polynomial can have at most two roots for any linear equation \( ax + b \). Thus, having three roots would contradict the polynomial's degree.

7. **Conclusion:**
   Therefore, the maximum number of numbers Basil could call out without violating the conditions is 20.

The final answer is \( \boxed{20} \)