Mathfest 2018 Puzzles: Order Eight Dihedral Groupdoku

I'm posting one last Mathfest groupdoku before moving back to the realm of the integers.  This puzzle is over the dihedral group of order eight.  Just like with the quaternion groupdoku, the clues are simply the products.  The multiplication and the elements are what makes this different.  

The elements we're filling in are e, r, rr, rrr, f, rf, rrf, and rrrf.  One can sum up the multiplicative structure in a table like the one here, where they use 1, a and b in place of e, r and f respectively.  One can also describe all the products in terms of letters and words with one sentence:  Any two f's or four r's in a row cancel each other out, and when you move an r to the left of an f, it turns into three r's.  For example, to find (fr)(frr), we first write frfrr.  We can move the leftmost r past the leftmost f to get rrrffrr, cancel the two f's to get rrrrr, then cancel four r's to get r.  

High Quality PDF for Printing.

Again we always multiply from top to bottom, or from left to right, which is important as the product is not commutative. 

If you haven't seen this type of multiplication before, you might be wondering where such strange rules could actually come from.  If you lay any object flat in front of you, then flip it and rotate ninety degrees clockwise, you end up in the same position as if you rotated three times and then flipped it, so long as you always flip over the same fixed imaginary line.  Try it!  Our r corresponds to rotation, and our f corresponds to our flip, and this is why fr equals rrrf here.  Each element of our group corresponds to a motion we can apply.  This gives us a third way to multiply in this group.  We can either use the table provided, write things out with letters and words, or physically flip objects in front of us to see what our products are.

This not at all rigorous video may also help you visualize what is doing on with our multiplication here:

The video doesn't have time to emphasize that the final positions are just indicators of the motion that occurred and not the elements themselves, nor does it show that every motion is invertible or explain why the product is associative, but I wanted a ninety second video here that people will finish instead of a five minute video that they may not.  I'll be happy to show more to anyone who asks.

Though I've certainly made harder puzzles, even over the same group we are using here, this puzzle is definitely harder than many of the other puzzles on this blog.  Don't feel discouraged if you don't solve it quickly or if you need to make a separate table for each clue, listing what all the possibilities are, in the correct order, for the adjacent cells.  That makes a useful reference to have nearby especially for the cages which contain two cells, until one becomes more familiar with these puzzles.