The phrase 'whichever places' is correct and usable in written English.
You can use it when you are referring to any place or places with no preference of one over another. For example, "I'm looking for a job in whichever places will accept me."
The artillerymen prepared their 18 howitzers for whichever place becomes the next mission, their cannons the bristles of a broom sweeping out the corners of a country.
It is one of the most popular places in town for lunch or dinner, and although you really can't go wrong whichever place you choose, Osteria Sottoriva stands out.
It's a theme I've discussed elsewhere, regarding this age of discontinuity and disruption, with its constant sense that, at a given moment, something happening somewhere else is more important than yet inescapably a part of whatever is happening here (whichever place "here" may be).
The point is, if there's a Starbucks and a Barstucks right next to each other, I'll buy my coffee from whichever place has free wireless access.
The hinge allows for 360-degree folding, and uses resistance to lock into place so whichever position you choose is sturdy.
Place the ball in whichever position you feel it is comfortable.
Assuming England finish as group winners, then beat whichever third-place team from one of the above groups in the round of 16, they could then face one of Portugal, Belgium, Italy (or even the Republic of Ireland).
You only have a limited time in whichever amazing place you've traveled to.
Age defined the time-dimension and the men exited from risk when they became fathers, died, or reached the end of follow-up, whichever took place first.
Lidl is an exception to the food shop stress, because everything is in the same place whichever store you visit.
