I am reading a book about mathematical games. One game is the 15 puzzle https://en.wikipedia.org/wiki/15_puzzle The answer whether it is solvable or not, was solved by using sign of permutations.
My question: Does anyone of you know who was the first human who discovered the principle of sign of a permutation (number of inversions)? And how did the person discover it? Was it because of this game?
Short answer: probably Cauchy.
Long answer: a bit of Googling "parity of permutations" eventually turns up this proof of the parity theorem for permutations by John Kiltinen, which has a pedagogical sidebar saying how different textbooks prove this theorem. In that sidebar Kiltinen says that Gallian, in his algebra textbook, says that the theorem is due to Cauchy. Comtet also gives a proof due to Cauchy in 1815. If you can read French, here's the original Cauchy paper. - see pp. 104-105 - Cauchy doesn't cite anybody here, as far as I can tell.