Moving three consecutive tiles seems like a wide freedom. But not all rearrangements are achievable, and the reason is more subtle than it seems.
The triple rotation
Riddle statement
You start with the row
\[
1,\ 2,\ 3,\ 4,\ 5.
\]
The only operation allowed is to choose three consecutive tiles and rotate them cyclically:
\[
abc \to bca \quad\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{or}\quad abc \to cab.
\]
Is it possible to obtain any permutation of the five tiles in this way?