It's turing complete, so technically you can solve any problem that you can in any other language, you just need to model your problem down to a T. You know how some people complain about assembly for not supplying enough utility methods compared to modern tools? Screw those people. You know how Java has a convoluted array of methods of reading from files (like java.io.File, java.nio.File, InputStream, ...)? Screw Java, all of those minor obstacles are nothing compared to Brainfuck. You have to spend hours just parsing the input, and then spend days implementing data structures, and then tackle the problem in the most classic way possible: By programming a tape machine. You have to master all the structures you've done and tackle it cleanly otherwise you'd have glue all over you and a script that's only possible to debug if you've signed a contract in blood with satan.
Truly, finally a red / legend badge that is actually hard: "Solve 15 puzzles in brainfuck". I yearn for this.
If that's not possible, ><> is good enough too.