I need to solve this problem but I do not know how.
Sherlock Holmes and Dr. Watson have on the fridge magnets with a series of eight letters. It is an eight different letters. Every morning Sherlock swapped letters first to the seventh, eighth with the second, fourth with fifth. Every evening, Dr. Watson reposition letters that last three letters are on the beginning of the series. Thus, letters of 6th, 7th and 8 will become the 1st, 2nd and 3rd letter. Sherlock Holmes noticed that Mrs. Brown comes clean whenever the refrigerator repeats the same word. How often comes Mrs. Brown?
Hint:
You have some permutations, and the combination is yet another permutation.
Try to write it as mutually exclusive cycles.
The order of this permutation (least common multiple of the cycle lengths) is the number of days..