When I posted about Yukari’s Cube (A Puzzle with No Name – Yukari’s Cube), my good friend Tyler provided some information on research that he performed on the L tromino pieces used to pack a 3x3x3 cubic space. A polyomino is a shape comprised of cubes (or squares in 2D) attached by their sides and a tromino is simply a polyomino comprised of 3 cubes. There are 2 possible trominos: one with all the cubes in a row (the I piece) and one creating a corner (the L piece).
Tyler’s analysis was prompted by Roland Koch’s Trilogic puzzle designed in 2003. Trilogic is comprised of 9 L trominos that make a cube with a checkered pattern on each side of the puzzle. Roland’s website indicates that this puzzle is easy to moderate in difficulty. Tyler indicated that the pieces reminded him of licorice.
Since the Trilogic cubes were made from layers of contrasting wood, there are multiple ways that the L trominos could be constructed to make different patterns. Tyler discovered that due to the checkered pattern of Trilogic, there were only 3 possible constructions of the L trominos that would be valid for constructing the puzzle.
After reading Tyler’s comment, I was immediately inspired to devise the world’s most awesome 9-piece L tromino cube packing puzzle. My objective was to find a pattern that would expand the number of tromino constructions that could be used. Instead of starting with the design process, in my excitement, I started making the licorice cubes that would be glued together to make the trominos.
It turned out that the dimensions that I used for the cube layers look very similar to those used for Trilogic. This was completely unintentional. I was originally going to make all the layers the same width, but since I was working with 3/4" stock in a non-metric country, each layer would have been 3/20”, which oddly enough, my 5-piece gauge block set can’t accomplish. I ended up using 1/8”, 3/16”, 1/8”, 3/16”, 1/8” layers, which my gauge block set could handle.
Each piece was individually made by gluing the 5 square layers together, sanding the cubes, beveling the edges, and then gluing the cubes together to form the L trominos. Since the sanding was done somewhat aggressively by hand, the blocks only approximate cubic shapes but won’t hold up to close scrutiny. This is not the ideal way to make these cubes and is only sufficient to quickly make a prototype for testing. If you were going to make a large quantity of these, you would probably want to make a nice sheet of 5-layer plywood and then cut it down into cubes. Unfortunately, I don’t have the equipment for that so I end up using 3/4" x 3/4" red oak and poplar sticks from the local home improvement centers and then chopping them up using a miter saw.
After completing the required 27 cubes needed for the 3x3x3 packing puzzle, they sat around for months waiting for that brilliant design to burst forth from the void. Try as I might, I couldn’t come up with an interesting pattern that was different from the Trilogic checkerboard pattern that would yield an interesting puzzle. Interesting being defined as a pleasing pattern for the assembled cube and a unique solution. Oh, and it would be nice if all the pieces were different.
After a short diversion to work on another puzzle design, the +-x pattern came to mind and I set about testing this pattern and designing the L tromino pieces to make it. The resulting 9 puzzle pieces consist of 3 sets of duplicate pieces and 3 unique pieces. The completed cube has a +, -, or x on 3 sides of the cube with the same symbols on the opposite sides. The puzzle has 2 solutions.
Now comes the most difficult part of this tale to tell. I have not been able to solve it. In fact, nobody in my family has been able to solve it. Unfortunately, everyone in my home hates the designer of this puzzle including me. Well, actually my grandson likes to make a cube of the 9 pieces (ignoring the objective pattern) and thinks that we are all crybabies.
|Wrong Top Front Corner!|
Although we are having a problem finding a solution, I know that Tyler will come up with a way to analyze the pieces and derive the solution without trial and error.
Thanks for the vote of confidence, Ken. After some thought, however, I have concluded that I really need to have pieces in hand, to determine the case analysis which best reduces the trial-and-error. For example, I thought I could identify the piece with the "hidden" cubie, but that is no easy task. The remainder of the solution might be reasonable after that point. This is a tough puzzle. Good for your grandson to ignore the objective pattern. Cheers. -Tyler.ReplyDelete
Yes, sometimes my grandson has an clearer understanding of the way of things.Delete