• If the difference is $k$ on some day, what will the difference be on the next day?
• Examine if the difference increases, decreases or stays the same for some values of $k$.

Suppose for some day, we have $n-m=k$. Then,

$n-m = k $

$2n-2m = 2k $

$2n-2-2m-2 = 2k-2-2 = 2k-4 $

$ (2n-2)-(2m+2) = 2k-4$

That means the difference will be $2k-4$ on the next day. From this, it's easy to see that when $k<4$, the difference will keep decreasing each day and eventually Payel's number will become larger. On the other hand, when $k\geq4$, the difference does not decrease. Therefore, our answer is $x=4$.