Aashna and Radhika see the integers $1$ to $211$ written on a blackboard. They alternate turns and in every step each of them wipes out any $11$ numbers until only $2$ numbers are left on the blackboard. If the difference of these $2$ numbers (by subtracting the smaller from the larger) is $\geq 111$, the first player wins, otherwise the second. If you were Aashna, would you chose to play 1st or 2nd and why?
By intuition only – I don’t see how I can prove it:
Since there are 19 turns ($19\times 11=209$ numbers + 2 on the blackboard), I would choose to play 1st. In my 1st move I would wipe out numbers 101 to 111 since these are the only ones that do not have a pair to meet the rule. Then for whichever numbers Radhika would remove, I would respond by removing the numbers that would have a difference of 111, for example for 92 I wipe out 203 etc. If Radhika chose to remove pairs with difference equal to 111, there would still be a single number, for which I would remove its pair and then I would also remove pairs with difference 111 or 112 and so on.
Does this method guarantee a win?