Misère picking stones game with labels Alice and Bob are playing a picking stones game with a pile of n stones labeled from '1' to 'n'. The rule is simple: Alice and Bob alternately take off 1 arbitrary stone or 2 consecutive stones(e.g. take off 2 stones labeled by '3' and '4', but '3' and '5' are not considered consecutive even if '4' has been taken). Whoever takes the last stone loses. Alice takes first. Proof that Alice always has a winning strategy for all n other than 1,4,9,12,20. I personally think this must have been understood already, so references are also very welcome.

This game is commonly known in combinatorial game theory as Kayles. And, yeah, it is well-explored. This seems to be a short paper on the process, although I haven't read it myself.