I am not a topologist, so please excuse me if the question is trivial.
Suppose I am given three nice, path-connected spaces. Then I can think of two ways to wedge these spaces: join all three at a common basepoint, or have one space in the "middle" with two basepoints (so I guess there are four ways in total). Do these constructions result in the same space up to homotopy? My guess would be to take a path between the two basepoints in the "middle" space and contract it to a point, but I understand that it is sometimes a subtle matter to tell if this sort of thing is a homotopy equivalence.