There are three grasshoppers in a row. They jump over each other: at every second one of the grasshoppers jumps over one of her neighbors (they still stay on the same straight line). Can they be back in their original positions after 1999 jumps? (Their distances and the gap sizes do not matter; the legal moves are: ABC → ACB, ABC → BAC.)
2026-03-26 17:30:50.1774546250
Invariance - Three grasshoppers jumping
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The answer is no, this is impossible.
The mathematical abstraction here is to see this as a sequence of $1999$ transpositions on the set with $3$ elements. As already hinted at in the comments, a transposition is always an odd permutation and the composition of an odd number of odd permutations is odd. However, the identity permutation is even, therefore it is impossible to compose $1999$ transpositions to yield the identity permutation.