Solutions for Act I Scene 4 - Shapeshifter

The first step is obviously to solve the jigsaw. It soon becomes apparent the resultant image is contained in a rectangle, and the final configuration should be:

We are left with a rectangle sliced into small triangles and quadrilaterals. This new shape is reminiscent of a tangram, and is in fact a form of one of the oldest puzzles in history, the Ostomachion. Whether or not we know this, since not much else can be done from this step (there's no obvious way to join the dots), it seems we want to cut up the rectangle into these new smaller polygons. It would be foolish to guess ostomachion as a final answer, not only because of the ambiguity of the name, but also because we haven't used all the information presented nor the fact that we have obtained new jigsaw style pieces.

Once again however there doesn't seem to be much more information to know what to do next with these new pieces. The only thing we have not used is the positioning of the dots, and we promisingly have exactly one dot on each Ostomachion piece. It's not clear how to use these dots however, until reviewing the puzzle in its original form again.

It might be evident that the original pieces of the jigsaw are arranged somewhat haphazardly in their original presentation. The intuitive leap to make here is that we want to put the new Ostomachion pieces back where they came from by matching up their dots to the dots in the original presentation of the jigsaw pieces. This idea is supported by the title, which suggests we want to "shapeshift" the jigsaw pieces into the Ostomachion pieces, and then shift the new shapes back to their starting positions.

Doing this gives the following shape:

The image should be recognisable as a unicorn, another fantastical beast to join the shapeshifter.

The answer is: unicorn