There is a puzzle on fault-free rectangles tiled by dominoes. It is rather known (it is described in Martin Gardner’s “Mathematical puzzles and diversions”, see here) and rather old (it is known at least from 1944). A few days ago Jyrki Lahtonen was riffing on an old contest training question he jousted with 40 years ago, see this question at Mathematics.SE. It was on a three-dimensional counterpart of fault-free rectangles, asking about the smallest side $N$ of a fault-free cube tiled by $2\times 2\times 1$ bricks. Jyrki Lahtonen proposed an elegant proof that $N\ge 22$ and now he offers a bounty +500 for a fault-free tiling of a smallest cube or for a proof that there is no such cube. Carl Schildkraut already improved the bound to $N\ge 24$.
On the other hand, I expect that a required construction exists for some cubes and it should be shown by a concrete example. But I think that a corresponding tiling is rather irregular, so hard to describe and its construction is a quest rather for a puzzle solver than for a mathematician. So I crosspost it here.
Since the considered tilings are too complicated to be dealt by hand, I wrote and shared an assisting program. It has a simple and intuitive interface, which looks a bit like “Tetris”, see a program screenshot. An the main working field are shown two consecutive layers of the cube of a selected size, parallel to one of three coordinate planes, which can be selected too. The bricks can be added or removed in a few clicks, see the program help for details. To facilitate diversity, each new brick obtains a personal random color. The red dots indicate the unblocked lines, perpendicular to the respective coordinate planes. Constructed partial tilings can be saved and loaded.
Downloads: an executable file for Windows, a zip-archive of Delphi 5 source files.
Good luck!
