Seven L-ements
Other Variations and why this
one was chosen
Warning: The following exposes you to the solution hints!!!
Rick Eason
First a Note on Piece Dimensions and Cube Size
The initial objectives in creating this puzzle were to use seven “L-shaped” pieces based on two different base dimensions to create a solid cube where all pieces are different and there is (hopefully) a single solution, ignoring rotations and reflections.
The base dimensions chosen for this puzzle were 1 and 2.
The cube size is necessarily 4x4x4. 5x5x5 is too large – no piece spans more than four cells, so no piece can fill more than one corner cell, and there are only seven pieces, so there is no way to fill all eight corner cells. 3X3x3 is too small, as the seven smallest pieces, all different, have a combined volume of 36 cells which is too large for the 27 cells available.
The Selection
Using the above objectives, 72 different sets of pieces can make a “Seven L-ements” type puzzle having only a single solution. The file here gives a description of the possible pieces, while the file here gives a list of all 72 sets having a single solution. Note that the piece designations used on this page are different from those given on the other “Seven L-ements” pages.
A main objective was to create a puzzle that was not so easy to solve if one blindly tries to assemble it, but which became much easier if a bit of simple logic is used. I focused on the fact that the 4x4x4 cube has 8 corner cells and 8 (hidden) center cells and there are only seven puzzle pieces with a limited number of ways of filling these cells. Most puzzle pieces span at most three cells and can therefore fill only one corner cube each. Seven such pieces can not possibly fill the entire 4x4x4 cube, but by choosing a set of pieces that includes at least one piece that spans four cells in one or more directions, a solution becomes possible.
In order for this solution logic to help as much as possible, it was decided to use a set of pieces which included only a single piece which spanned four cells in one direction, thereby forcing the condition that the six “shorter” pieces each claim one corner of the cube, while the single “long” piece claims two. This immediately eliminates sets using pieces “j” and “J” (described in the file of pieces above), as they each span four cells in two directions and can therefore potentially fill three corner cells. It also eliminates any set of pieces containing two or more from the set (f, g, i, F, G, I). The file here gives the 28 sets of pieces which remain.
Note that if each piece must fill some corner cell, then only the two-unit-thick pieces can contribute to the eight center cells. A count was made for each of the 28 sets of pieces of the maximum number of potential center cells, given that each piece must occupy a corner cell. A count was also made of the number of options in picking pieces from the set to fill the 8 center cells, not counting orientations – i.e., some pieces can be used in more that one way to contribute to the center cells. This file gives these counts for each of the 28 sets.
As above, in order for the solution logic to help as much as possible, it was decided to only consider sets of pieces having at most two options for filling the center 8 cells. It was also decided to eliminate sets which included piece “G” as this piece is a bit larger than the others, creating an “unbalanced” set of pieces. It was also felt that sets which included “G” might also be easier to solve by just stumbling on the solution. This reduced the number of sets under consideration to these 10 sets.
Each of these 10 sets was tried and the one that seemed best was picked. Several of them seemed to be too easy to stumble upon the answer. Also, sets using pieces “F” or “I” were not as highly valued for the same reasons as “G” mentioned in the previous paragraph.
© Rick Eason, 2005