Home > Riddles > The triple rotation
Moving three pieces seems like a wide freedom, but it is not so much. The surprise is that a very specific operation leaves intact a property of the complete order.
You start with the line
\[ 1,\ 2,\ 3,\ 4,\ 5. \] The only operation allowed is to choose three consecutive cards and rotate them cyclically: \[ abc\to bca \quad\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{or}\quad abc\to cab. \] Is it possible to obtain any permutation of the five cards in this way? \]
Answer: No, you can't. Each allowed trade is a 3-cycle on consecutive positions. A 3-cycle is an even permutation. Composing only even permutations always produces an even permutation. Going from $(1,2,3,4,5)$ to $(2,1,3,4,5)$ is a single exchange of two elements, that is, an odd permutation. Odd cannot be obtained as a composition of evens.
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