As they were moving through the jungle, they met an old drunkard who insisted everyone should to swap their hats. The troop contains 64 people. In order to save time, each pair within the troop swaps hat with each other only once (i.e. no swapping is repeated). After a round of madness, Tintin notices that it is no longer possible to return all hats to their respective owners through more swappings. To sensibly resolve this maddening confusion, he decides to bring in even more hat-wearing guests in his troop, to allow for even more hat swappings. How many extra guests are needed to return all hats (including the extra ones) to their rightful owners? Note : Assume a person has exactly one mushroom.

Explanation

Two extra people are needed to return correct hats using the following method­  Let the extra people have names E1 and E2. 
Step 1: Arrange the 64 people into lines such that ­ In each line, every person is having the hat of the person directly behind. Except the last person whose hat is with the first person  in that line. 
Note : If two people have each others’ hats, they will make a separate line of only two  people.   
Step 2:: For a single line, first E1 swaps hats with the last person. Then E2 swaps hats with each person in middle one at a time from front to back. Finally E1 will swap hats with the first person.Then E1 & E2 swap each others’ hats. This way everyone in this line along with  E1 & E2 will have their own hats. 
  Step 3: Repeat Step 2 for each line.