Dear Anonymous A,
In general, if your office requires a Problem Solving Test for full-time hires, then it will also require the Test for internship candidates.
It should be mentioned that there are some McKinsey offices around the world that do not require the Problem Solving Test for full-time hires, and subsequently, may not require it for internship candidates. That said, my limited understanding on this matter is that those offices that do not require the Test are quite few and quite far between.
The solid advice is that you if you wish to intern at McKinsey, you must prepare to take this Test. Diligent and customised study is crucial to pass this most challenging of examinations.
Check out this link for a fun perspective on the Test (see below)
All the very best to you!