A tricky probability question. I have been asked the following question, and unfortunately I have no idea how to proceed. Here is the question:

Suppose we have 99 empty papers and we wrote numbers from 1 to 99(using each number) on one side of the papers randomly. Then we mixed all the papers randomly and started to write numbers from 1 to 99 on the other sides (empty sides) of each paper randomly. What is the probability of having the same number on both sides of at least one paper?

Thanks!
Hint: Derangements.

Hint: Complement.
the answer is very simple . say in one piece of paper it is '5' chance that we end up  with wrong number on backside  out of 99 possible number is 98/99. if we write 5 ,we will be right. so chance of writing wrong number on backside in all 99 paper  is (98/99)^99 . so probability of at least one correct  = 1-(98/99)^99= 63.34%