A box contains answer $4032$ scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of the scripts and keep the other outside. If there is one script with even score and the other with odd score, we put back the script with the odd score and keep the other script outside. After following this procedure a number of times, there are 3 scripts left among
which there is at least one script each with odd and even scores. Find, with proof, the number of scripts with odd scores among the three left.
1. **Initial Setup:**
   - The total number of scripts is \(4032\).
   - Exactly half of these scripts have an odd number of marks, so initially, there are \(2016\) scripts with odd marks and \(2016\) scripts with even marks.

2. **Procedure Analysis:**
   - We choose 2 scripts randomly and analyze the possible outcomes:
     - **Both scripts have odd scores:**
       - We add one mark to one of them, making it even.
       - We put the script with the new even score back in the box and keep the other script (with an odd score) outside.
       - This decreases the number of odd scripts by \(1\).
     - **Both scripts have even scores:**
       - We put one script back in the box and keep the other script outside.
       - The number of odd scripts remains unchanged.
     - **One script has an odd score and the other has an even score:**
       - We put the script with the odd score back in the box and keep the script with the even score outside.
       - The number of odd scripts remains unchanged.

3. **Effect on the Number of Odd Scripts:**
   - From the above analysis, the number of odd scripts can only decrease by \(1\) when both chosen scripts have odd scores.
   - Otherwise, the number of odd scripts remains unchanged.

4. **Parity of the Number of Odd Scripts:**
   - Initially, the number of odd scripts is \(2016\), which is even.
   - Each action either decreases the number of odd scripts by \(1\) (making it odd) or leaves it unchanged.
   - Therefore, the parity (even or odd nature) of the number of odd scripts can change only when we decrease the number by \(1\).

5. **Final Condition:**
   - We are given that after following the procedure a number of times, there are \(3\) scripts left, with at least one script having an odd score and at least one script having an even score.
   - Since the number of odd scripts must be even (as it started even and can only change by \(2\) or remain the same), and there is at least one odd script, the only possible even number of odd scripts among the \(3\) left is \(2\).

Conclusion:
The number of scripts with odd scores among the three left is \(2\).

The final answer is \(\boxed{2}\).