The smallest of the assembled cubes, measuring 3x3x3 units, can be built in a single color; for the larger cubes, stones from more than one color are needed.
In regard to the monochrome 3x3x3 cube task, one can ask several questions:
(i) What are the different classes of cubes that are possible?
(ii) Can an example for each class be constructed?
(iii) Are there different cubes in each class with the same stones but connected in different ways?
Illustrations and details of the analysis are provided below; here are some answers:
a) Of the 11 stones, two cannot be used for assembling a 3x3x3 cube.
b) Of the remaining nine stones, two main series of cubes can be built by leaving either one or the other of two specific stones out. That stone defines the class of cubes. The one not left out is used in all cubes of that series.
c) From the remaining seven stones there are then seven 3x3x3 cubes that could, in principle, be built where each is different by one of the seven stones being left over.
d) I have found at least one example for each cube in one of the two main series of cubes.
e) In the other series, I have found only six examples with the last cube so far having resisted successful assembly. It is my conjecture that that cube cannot be built.
f) For many of the other examples in both classes, there are several different assemblies possible.
Details of the analysis:
The stones consist of all possible arrangements of 2, 3, and 4 unit cubes such that they cannot be transformed into each other by rotations, see the picture; the single unit cube itself is not part of the stone set. Using alphabetic letter analogies to describe their shapes, the possible stones in a set are:
2I -- forming the letter I;
3I, 3C -- forming a larger letter I, and the letter C (or a corner in plane);
4I, 4L, 4T, 4R, 4Q -- these are planar stones where the letter designations I, L, and T are obvious, R resembles the lower-case r while Q stands for 'quad' as it forms a 2x2 square;
4C, 4S, 4D -- these stones extend into the 3rd dimension, C forming a corner, and D and S signifying the right-handed (dexter) and left-handed (sinister) versions of the two remaining 4-unit stone configurations (which are mirror images of each other).
An image with projections of the stones into the three spatial planes is also shown (to be added) to relate the stone shapes to the solution drawings below.
The total unit count of all the stones is 40. To form a 3x3x3 cube one needs 27 units. The 4I stone cannot be used since it is too large. That leaves 36 units and, therefore, stones with a total unit count of 9 will have to remain unused when a 3x3x3 cube is assembled. The only way to achieve this with stones consisting of 2, 3, or 4 units is to leave out one of each, that is, always stone 2I, and then either the 3I or the 3C stone plus one of the remaining 7 4-unit stones. This suggests that two series of 3x3x3 cubes can possibly be assembled; they are designated as the 3I and the 3C Main Series. Within each series, one of the 7 stones also left over identifies the specific 3x3x3 cube.
The two main series with different assemblies of monochrome 3x3x3 Rumis cubes are:
3I-C, -D, -L, -Q, -R, -S, -T
3C-C, -D, -L, -Q, -R, -S, -T
So far, at least one example was found for each of those cubes except for the 3C-T cube. The analysis of several attempts at the 3C-T cube indicates that it is not possible to assemble this 3x3x3 monochrome cube; that is currently my conjecture.
Any thoughts or suggestions?
Udo Pernisz
