
4. 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.


Solution: There are three types of processes. In the first type, the scripts with odd scores decreases by 2. In the second and third types, there is no change in the number of scripts with odd scores. Hence at each step, the number of scripts with odd score decreases by 0 or 2 . Since there are 2016 scripts with odd scores, the number of scripts with odd scores at the end is either 0 or 2 . Since it is given that there is at least one script with odd scores, two of the three must have odd scores.
