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?
Show solution
Solution
Answer: No, it is not possible.
Each allowed operation is a 3-cycle on consecutive positions. A 3-cycle is an even permutation, and composing even permutations always produces an even permutation. Therefore, any position reachable from $(1,2,3,4,5)$ must differ from it by an even permutation.
However, the permutation $(2,1,3,4,5)$ consists of a single exchange of two elements: it is an odd permutation. Since odd cannot be obtained as a composition of evens, this rearrangement—and many others—is unattainable.