“There are two primary types of strategies; those involving one’s memory and those involving algorithmic methodology.” -Al Zobrist
Memory Based Strategy
First, consider memory. The large number of codes in the code book seems daunting, but a clever method may be available. Some rare geniuses may be able to pictorially memorize a huge number of positions that they have seen, but a list based memorization may produce good results for the average player. Consider the set of “1’s” in the CLASSIC section of the code book. There are 918 of these. Now consider the subset of codes starting with the letter “A” (219 of these). Since A is the first letter, let us solve these with the A in its canonical position. There are only four shifts for the position of the A piece and we want to make sublists for these. The order of the sublists will be:
- The base position closest to (0,0,0) in the right-handed coordinate system, where each zero represents the x, y, and z coordinates respectively
- A 1 unit shift in the z-coordinate (0,0,1)
- A 1 unit shift in the y-coordinate, with the z-coordinate back to zero (0,1,0)
- A 1 unit shift in both the y and z coordinates (0,1,1). The reader can understand this as “alphabetic/lexicographic” ordering on the coordinate system
So, if an “A” code is given to the player, and he/she can remember the four lists, then he/she will be able to place the A piece exactly and only have to solve a five piece puzzle. To make memorizing the lists easier, only memorize the three shortest of the lists. In fact, list one is way longer and list four is very short. You would then have to figure that the code is on list one by knowing for sure that it is not on one of the three lists that you have memorized. Then, repeat this memorization for the other letters from B to M. The lists get shorter as M is approached since very few codes start with the later letters. This is because the codes themselves are sorted into alphabetical order.
Algorithm Based Strategy
Now consider algorithmic strategies. Here is one very important strategy that everyone should be aware of for the 3x3x3 cubes. The cube has eight corners and the codes usually give you six pieces to solve. This means that two of the pieces must touch two corners. Keeping this in mind cuts down on searching, and a good strategy may be to place the pieces that can touch two corners first in the varying ways that they can go. Note the four positions of piece A in the discussion above. Only the first position of piece A touches two corners. The fourth position of piece A touches none of the corners. For A to be in fourth position, the remaining five pieces must cover the eight corners (impossible?).
Another such strategy may involve alternate coloring (black/white) of the subcubes of the pieces. The standard 3x3x3 cube enforces a 13/14 split of the alternating colors, which can come into play with certain pieces such as the M piece, which is four cells of one color and one cell of the other color. This is an imbalance of three cells that will have to be made up for by other pieces, and thus places a restriction on where those pieces can be placed since many can only make up for an imbalance of either zero or one cell. A similar method would be to count the number of faces that a piece can put on the outside of the cube for the various shifts that the piece can take. All of the pieces have a set of numbers, for example, piece A can have (11,10,8,5). For a solution, the number realized for each piece add to the total cube area, which is 54 for the 3x3x3 cube. A lot of restrictions will pop out from this strategy, such as:
- The average for a 6 piece puzzle is 9 faces per piece
- The number of odd integers must be even
A more general approach that seems very important, and especially important for solving the harder cases, could be stated as “drive through the combinations without excessive repeating”. It is too easy to repeat promising sequences over and over again when the solution needs a very different start.
A player will notice, after solving hundreds of different puzzles, that the last two pieces can be seen as possible or impossible without even placing the pieces. This is a perceptual skill that will develop naturally with experience. Gaining this skill means that only the first four pieces have to be placed to see if a solution is obtained. It should be possible for players with great geometric perception to extend this skill to three pieces. Since the first piece can be placed with the list keeping method described above, and the last three pieces may be a gimme to the experienced and perceptive player, solving each puzzle may simply be determining the positions of the second and third pieces.
What speedup can be expected for the skilled player? The codes in the CLASSIC section of the code book take the beginner about three hours each to solve on average. We would expect to see players at the tournament level solving these puzzles in about ten minutes or better after a few years practice, a speedup factor of at least eighteen.