A Game of Palindromes

On a really hot summer day, Merlin and Arthur are playing a game that their father gifted them a few days ago. The game consists of $N$ strings and each string is made up only of numbers from 0 to 9. Now Merlin and Arthur are going to take turns playing the game. In each turn the player chooses a string which is not a palindrome (let's call it $S$) and then chooses a number from 0 to 9 (say, $X$)