Whichever means any one of them that you choose. The choice is open and the result is the same.
It works as a pronoun and a determiner, so take whichever you like and whichever route you take are both correct. It differs from whatever by implying a limited set, since whichever assumes you are choosing from options already in view. There is a common fixed use in whichever comes first and whichever way you look at it, both of which close off an argument rather than opening one.
1pronounwhicheverAny one from a set, chosen freely.
“Take whichever you prefer.”
From whiche ever, whiche evere, which ever, which evere; equivalent to which + -ever.
Etymology from Wiktionary, CC BY-SA