Home > Riddles > The triple rotation
You start with:
$$ 1,\ 2,\ 3,\ 4,\ 5. $$
The only operation allowed is to choose three consecutive cards and rotate them cyclically:
$$ abc\to bca\quad\text{o}\quad abc\to cab. $$
Can you reach:
$$ 2,\ 1,\ 3,\ 4,\ 5? $$
Didactic closure of the level
The difference between “normal” and “misère” seems small, but it changes the closure of the strategy.
Back to problem
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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