Example

Suppose someone rearranges the numbered stickers in the first grid so that the result looks like the second grid. Find out how many such rearrangements are necessary to return the stickers to their original locations.


1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
6 11 3 4
5 7 1 2
16 10 8 12
13 14 15 9


Comparing the new sticker locations with the old, we find:
$$\begin{array}{cccccccccccccccc}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ 6 & 11 & 3 & 4 & 5 & 7 & 1 & 2 & 16 & 10 & 8 & 12 & 13 & 14 & 15 & 9 \end{array}$$
Now we can find the cycles:
$$\begin{align} &(1, 6, 7), (2, 11, 8), (9,16), (3), (4), (5),\\ &(10), (12), (13), (14), \textrm{ and } (15)\end{align}$$
Note, the first three cycle lengths are 3, 3, and 2, respectively, while the rest are 1. So the stickers will return to their original positions collectively after 6 similar rearrangements (the least common multiple of the cycle lengths seen: 3, 2, and 1).