Easter Island Dominoes
Flush with the successful design of the "Perkinson Guest Bathroom Tile" puzzle, the obvious next step was to consider dominoes instead of trominoes. This time, I allowed the pieces to be flipped over, and I also counted pieces whose two tiles had their tilted edges at right angles to each other. This leads to a complete set with 13 members, some of which are slightly strange looking, and one of which (with the two tilted edges joined together) is a quite boring perfect rectangle.
Leaving out the boring rectangle, we get twelve roughly one-by-two-unit dominoes; the obvious tray shape is a six-by-four-unit rectangle. Can the (nearly) complete set fill that tray? Sadly, my packing program said no, and this time it also didn't work to change the top and bottom edges into the angular "sine wave" pattern from the bathroom-tile puzzle. I had my local laser cutter, Joe Pelonio, make me up a set of the pieces anyway, so that I could play with them and try to get a feeling about why the rectangular tiling wouldn't work. The first thing I noticed when I got the pieces in my hands was that one of them looked a lot like the profile of one of the famous "moai" heads from Easter Island; thus the eventual name of the puzzle.
It's not well known (especially to archaeologists), but many, many sets of these 12 pieces have been discovered in excavations on Easter Island. Never, though, have they come across a copy of that elusive 13th piece, the perfect rectangle. From this, we can infer that the ancient Easter Island culture, now long lost to us, did not approve of straight lines and perfect rectangles. Being a culturally sensitive fellow, I've created a tray that has one tilted tile edge exposed on each edge of the tray, thereby avoiding violating the islanders' taboos.
Your first challenge in solving this puzzle is simply to lay all twelve pieces flat in the tray; there are 250 ways to do that, and it's not very difficult if you just have a bit of patience. You'll find, though, that almost every such packing has at least one blemish (as least from the point of view of ancient Easter Island culture): there will either be (a) a straight-line crack all the way across or down the puzzle, or (b) a subset of the pieces that form a perfect rectangle or square, or (c) both!
There are just 83 ways to pack the pieces without a straight-line crack, and only six ways to do so without forming a perfect square or rectangle. Your real challenge is to find one of merely three solutions that have neither "blemish". That'll take you a little bit longer to achieve, I think.