The Maze Cubeby Christoph Lohe 1. General ideas for a Maze Cube 2. How to design the corner cubies 3. How to design the edge cubies 5. Properties of the corner cubies |
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1. General ideas for a Maze Cube
The Maze Cube bases on a standard 3x3x3 cube. A twistable cube is not a precondition, but it can be done using a Rubik's Cube or Assembly Cube. Against them, the 54 mini-surfaces of the Maze Cube carry just a single color, but each mini-surface shows a small piece of a line on it, which connects to two neighboring minisurfaces. The lines on each mini-surface shall simply be either a straight line, or a right-angle:
All the 54 lines on the mini-surfaces shall build a single closed loop drawn over the 3x3x3 cube. In order to make that puzzle a real challenge, all the eight corner cubies shall be different to each other, and they must not look identical after rotation of a corner. Similar, the twelf edge cubies shall be different and must not look identical after a flip.
2. How to design the corner cubies
Let us consider all possibilities to design corner cubies for the Maze Cube. How can we arrange straight lines and right-angles on the three minisurfaces of a corner cuby, meeting the condition to allow a closed loop on the Maze Cube?
(a) three straight lines
It is not possible to arrange three straight lines on the mini-surfaces of a corner cuby so that you meet the demand to allow a closed loop on the 3x3x3 cube.
(b) two straight lines and one right-angle
The two straight lines can either be connected to each other, or be parallel. In any case, there remains just a single chance to arrange the right-angle:
Please note that the two corner cubies are complementary to each other as the first has the lines exactly where the second has lines missing, and vice versa. If you overlap the two images of the cubies then you would see a cross on each mini-surface. You will see later that all the cubies of the Maze Cube build such complementary pairs, and these pairs are shown close to each other in the following sections.
(c) one straight line and two right-angles
The two right-angles can either be connected, or are separated from each other. In any case you find two chances to arrange the straight line, building pairs of arrangements which look mirrored to each other:
(d) three right-angles
The three right-angles can either all be connected, or just two are connected, or all are separated from each other:
The last pair is special in some sense. The corner cuby before the last builds a closed loop in itself, and so does not meet the demand to allow a single closed loop on the 3x3x3 cube. It's complement with three separated right-angles is the only corner cuby with six open ends to neighboring mini-sufaces. All the other eight corner cubies offer just two or four open ends. As we only need eight corner cubies, we just neglect the last pair shown above.
Other arrangements of straight lines and right-angles which can meet our demands are impossible. Try it!
3. How to design the edge cubies
Let us consider the edge cubies now. The two mini-surfaces can get:
(a) two straight lines
The straight lines can either be connected or parallel to each other:
(b) one straight line and one right-angle
Straight line and right-angle can either be connected or separated. In any case you will find two arrangements, building mirrored pairs to each other:
(c) two right-angles
The right-angles can either be connected or separated. In any case you find three arrangements, two of the pairs look mirrored to each other:
There are no other possibilities to arrange the straight lines and right-angles in a way which will allow a closed loop on the 3x3x3 cube. In total we get twelf different edge cubies. What a lucky hit, we just need twelf edge cubies for the 3x3x3 cube!
4. How the cubies match
Can these eight corner cubies and twelf edge cubies really be arranged to build a closed loop on the 3x3x3 cube? The edge cubies connect towards the corner cubies with a pair of mini-surfaces. It is interesting to have a more detailed look to this interface between edge cubies and corner cubies.
Both mini-surfaces of the edge cuby can show "open ends" towards one particular corner cuby, or there may just be a single open end, or none. In case there is a single end towards the corner, this may be on the left side or on the right side - seen from the edge cuby towards the corner cuby. The interfaces between edges and corners can so be classified into four types:
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0 |
no open end towards the corner cuby |
You can apply the same classification for describing the corner cubies - this time seen from the corner cubies in the direction towards the edge cubies. Therefore, the following interface types must match:
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With this classification, it becomes clear that there must be the same number of type "0" interfaces on the edge and corner cubies in order to allow building a closed loop over the Maze Cube's surface. The same is true for type "2" interfaces. The number of type "R" and type "L" interfaces on the corner's side must match to the same number of type "L" and type "R" interfaces on the edges side. Let us check the eight corner cubies and twelf edge cubies regarding these demands.
5. Properties of the corner cubies
If we note the type of interfaces on the corner cubies clockwise, then these cubies are:
002 |
0RR |
0LL |
0LR |
220 |
2LL |
2RR |
2RL |
In this arrangement, complementary pairs of corner cubies are shown in the columns. It is no surprise that the interface notations appear also complementary: type "2" and type "0" build a complentary pair, and so do type "L" and type "R". The first row show cubies with a single line on it, the second row shows cubies with two lines.
What is the distribution of the interface types on the eight corner cubies? This is a surprise, we get the same number of six for all interface types! As a first result, it looks that the design of a 2x2x2 Maze Cube just using the corner cubies is possible. Indeed, there are several solutions!
There are other interesting properties of the eight corner cubies: there is the same number of corners with a single line, and corners with two lines, building four complementary pairs. If you count the number of right-angles and the number of straight lines on the 24 mini-surfaces then their ratio is 16:8, or 2:1.
So far, everything looks fairly symmetric. But it isn't! For example, if you count the number of corners with a type "R" interface then the result is four. The same is true for corners with the "L" interface. However, the number of corners with a type "0" or type "2" interface is five! The cubies with an "R" or "L" interface are more rare somehow.
6. Properties of the edge cubies
We can classify edge cubies in the same way, but what about the interfaces between an edge cuby and a center? Let us note a "+" if there is an open line end at the center side of an edge cuby, otherwise let us note a "-". If we note the interfaces clockwise again then we get the following twelf edge cubies:
0+0+ |
0+R- |
0-L+ |
R-R- |
L-L- |
0-2- |
2-2- |
2-L+ |
2+R- |
L+L+ |
R+R+ |
2+0+ |
Again, complementary pairs of corner cubies are shown in the columns, the first row shows edges with a single line on it, and the second row shows edges with two lines.
What is the distribution of the interface types on the twelf edge cubies? Again, we get the same number of six for all interface types which connect to the corners. This is really magic!
Both, the number of type "+" and type "-" interfaces between edges and corners is twelf. This means that the twelf edge cubies will fit to the six centers! Independant on whether a center has a straight line or a right-angle on it, the center will show two open ends. So, each center offers two "+" and two "-" interfaces as it connects to four edge cubies.
Similar to the corner cubies, there are other interesting properties for the twelf edge cubies: there is the same number of cubies with a single line, and cubies with two lines, building six complementary pairs. If you count the number of right-angles and the number of straight lines on the 24 mini-surfaces then their ratio is 16:8, or 2:1. It is the same result which we have already seen for the corner cubies! We also see the same property in the cubies with an "R" or "L" type interface being more rare. Last not least, six of the twelf edge cubies look the same after they are a flipped, while the other six look different then.
7. The Maze connection
Let us summarize:
- the eight corner cubies show the same number of six for all interface types "0", "L", "R", and "2"
- the twelf edge cubies also show the same number of six for all interface types "0", "L", "R", and "2"
- the twelf edge cubies show the same number of twelf for the interface types "+" and "-"
- the six center cubies show the same number of twelf for the interface types "+" and "-"
All this is a precondition - not a proof - that the 26 cubies can match and build a single closed loop over the 3x3x3 Maze Cube. For example, the number of "L" type interfaces of the corner cubies must fit to the same number of "R" type interfaces on the edge cubies, and vice versa. This does not necessarily mean that the number must be six, but the design of the edge and corner cubies automatically leads to this very symmetric distribution of interface types. We have seen that you can hardly find other designs of the single cubies, so this symmetry is one mystery of the Maze Cube.
How to design the center cubies? The distribution of right-angles and straight lines for both, edge and corner cubies' mini-surfaces was 2:1. Shall we keep that ratio for the six centers? There is no demand to do it but let us do it, just to keep some additional symmetry. Four centers get the right-angle, and the other two centers get a straight line.
8. There is a solution
Now, the work starts. Can you take the cubies in your hands, and assemble them to a Maze Cube which shows a single closed loop line on it? It is not easy, but it is great fun trying it. For everybody who quickly wants to look for a solution, here is the original Maze Cube design:
9. More challenges
You can take the approach of the Maze Cube to develop other challenges. As an easier starting point, you can try to assemble a 2x2x2 Maze Cube, just using the corner cubies. There are a number of solutions to this problem, one is shown here:
You can also try a 4x4x4 Maze Cube, using the twelf different edge cubies for the "left side edges", and the same set for the "right side edges". When flipped, left side edges are mechanically forced to become right side edges, and vice versa. Therefore, we do not get 24 different Maze edge cubies for the 4x4x4 cube but only 18. Do you see why?
Also your 3x3x3 Maze Cube rise some more questions. Are there different solutions for the problem than the initial state? Are there solutions for different designs of the centers, for example, if the two straight center lines lay on the same axis?
Consider the single closed loop on your Maze Cube divides the total surface into two areas. Are these areas equal in size?
What about solutions where all the lines on the Maze cubies fit to each other, but do not build a single loop? They can build multiple loops instead:
In this example, some islands build up, it is the area surrounded by a smaller loop. Is there a solution with a maximum number of loops or islands? Are all the loops separated, or is one loop surrounding another one? You can colorize the area between the multiple loops so that incoherent areas get different colors. How many colors would be necessary in the example above? What may be the maximum number of colors you get for any solution?
Can you assemble your Maze Cube so that no (!) open end on any cuby fits to another cuby? Is there a solution of the Maze Cube which shows a ratio 6:3 (it is 2:1) between right-angles and straight lines on all the the six sides? Or is there a solution which shows the ratio 8:4 on all the 2x2x2 sub-cubes surrounding each corner?
As you see, there are still a lot of questions, and most are not answered so far. Have fun in solving these puzzles or in creating even new ones!
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updated: 09.01.2004 |
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