An 8 Circuit Labyrinth on Rubik’s Cube

After the 3 and the 5 circuit labyrinths on Rubik’s Cube, only the 7 circuit is missing. Unfortunately, the usual 3x3x3 cube is too small for that. And the next category (Rubik’s Revenge) with 4x4x4 is a little too big. In order to fill all areas, you need an 8 circuit labyrinth. The well-known 7 circuit (classical) labyrinth is best suited to this, supplemented by a kind of “lap of honor”, an additional loop. It can be added at the beginning or at the end.
Here I present the labyrinth with the additional loop at the end.

First the unfolded cube with the names of the different sides:

The unfolded, empty Rubik’s Cube

And here Ariadne’s thread:

Aridne’s thread on the unfolded Rubik’s Cube

The path sequence is: 3-2-1-4-7-6-5-8-9. This is the sequence for the classical 7 circuit labyrinth with an additional loop, hence “9” is the center (and not “8”). The entrance to the labyrinth takes place on the 3rd circuit. It is on the front side of the cube. And the center is next to this space on the same side. This shows that all other lines are only detours. Just like in the “right” labyrinth. The beginning and the end of the line (Ariadne’s thread) are right next to each other.
It should also be noted that the path on the cube is very distorted compared to the familiar look of the labyrinth, and not as balanced as we are used to. But the essential is there, the intricate and complex, not crossing lines. It is also nice that all small cubes are occupied, there are no empty spaces.

Again, a template that can be used to build a model. Here you can see, print or download it as a PDF file.

I tried it myself and made a more or less successful cube:

Aridne’s thread on Rubik’s Cube

In the large picture above, you can see the top, left and front sides from above. The three smaller pictures are each rotated by 90 degrees.

Here in a tilted representation:

Aridne’s thread on Rubik’s Cube

In the large picture above, you can see the front, left and lower side from above. The three smaller pictures are rotated by 90 degrees again.

I cannot judge how difficult the twisted cube would be to solve. And whether it is enough to simply say: connect the beginning and the end with each other without crossing the line?

Related Posts

• The 3 Circuit Labyrinth on Rubik’s Cube
• The 5 Circuit Labyrinth on Rubik’s Cube

Further Links

• Wikipedia: Rubik’s Revenge
• Wikipedia: Speedcubing

The Labyrinths with 2 Real Double-barriers, 3 Arms, and 5 Circuits

There exist 8 five-arm labyrinths using solely real double-barriers. In four of these, the sequence of the sector patterns in sectors II – IV is 3 8 3. In the other four, this sequence is 8 3 8 (see: related posts 1, below). It takes only a small jump from these five-arm to the respective three-arm labyrinths. As a matter of fact, of the three central sectors, simply two subsequent sectors have to be removed. Then, there remains only one central sector between the two sectors next to the main axis.

Figure 1 shows how for the first four patterns the sequence is reduced from five to three sector patterns. The sector patterns of sectors III and IV are removed (hatched area). Sector pattern no. 3 in sector II is directly connected with sector pattern no. 7 or no. 8 of the former sector V (red line). This becomes now the new pattern of the third sector.

Figure 1. Reduction of the Sequence of Sector Patterns from 3 8 3 to Sector Pattern no. 3

The same approach is also used in fig. 2 in order to reduce the second sequence from 8 3 8 to a central sector with the sector pattern no. 8.

Figure 2. Reduction of the Sequence of Sector Patterns from 8 3 8 to Sector Pattern no. 8

In both cases, this doesn’t change the remaining sequences of the patterns. The patterns of the sectors III and IV are simply skipped.


Figure 3 shows the four patterns and labyrinths, that have been obtained by reducing the first sequence. These all have sector pattern no. 3 in the central sector.

Figure 3. The four Three-arm Labyrinths with Sector Pattern no. 3 in Sector II

Figure 4 shows the four patterns and labyrinths, that have been obtained by reducing the second sequence. These all have sector pattern no. 8 in the central sector.

Figure 4. The four Three-arm Labyrinths with Sector Pattern no. 8 in Sector II

Just as in the five-arm labyrinths, there exist also 8 different labyrinths with exclusively double barriers and three arms. They are named following the rule known from previous posts (related post 2). Thus, the name is made-up of a capital letter followed by two horizontal marks.

Related Posts:

1. The Labyrinths with 4 Real Double-barriers, 5 Arms, and 5 Circuits
2. Classifying the Labyrinths with 3 Double Barriers, 4 Arms, and 5 Circuits

The 5 Circuit Labyrinth on Rubik’s Cube

I started with the 3 circuit labyrinth on Rubik’s Cube (see Related Posts below). However, quite a few areas have remained empty. Therefore I have now dealt with a 5 circuit labyrinth.

First of all again the scheme for Rubik’s Cube with the notation of the different sides of the unfolded cube.

The unfolded, empty Rubik’s Cube

From the 8 possible combinations for a 5 circuit labyrinth, I chose the variant with the path sequence: 0-3-4-5-2-1-6.

Ariadne’s thread on the unfolded Rubik’s Cube

The path begins on the “front” side on the bottom left corner stone, then leads upwards (circuit 3), jumps around the corner to “left” and runs over “back” to “right” around the cube. From here it goes up (circuit 4) and back again left around to the “front” side. Then follow the circuits 5, 2, 1 and finally the jump to the center (6).
The number of the circuits begins on the bottom (the “down” side) of the cube, not on the “front” side. The goal, the center of the labyrinth is in “6” on the “up” side. And only one stone remains empty on the underside (the “down” side).

There is also a template that can be used to build a model. Here you can see, print or download it as a PDF file.

I tried that myself and made a more or less successful model out of paper:

Ariadne’s thread on Rubik’s Cube

In the large picture above, you can see the top (“up”), left and front sides from above. The three smaller pictures are each rotated by 90 degrees.

Here in a tilted representation:

Ariadne’s thread on Rubik’s Cube

In the large picture above, you can see the front, left and lower (“down”) side from above. The three smaller pictures are rotated by 90 degrees again.

I cannot judge how difficult that would be to solve. Especially if you do not say beforehand what that should be and you should find the beginning and end of the thread yourself.

Related Posts

• The 3 Circuit Labyrinth on Rubik’s Cube
• The 3 Circuit Labyrinth Cube

Further Link

• Wikipedia: Rubik’s Cube

The Labyrinths with 4 Real Double-barriers, 5 Arms and 5 Circuits

Up to now I have examined only sector labyrinths with four arms. The real double-barrier, however, originates from the five-arm labyrinth type Gossembrot 51r. As we know, this is not a sector labyrinth and it has 7 circuits. Now I want to find out, how many sector labyrinths with five arms and exclusively real double-barriers there are. Such labyrinths must have five cirucits. Therefore, we can use the same six sector patterns we have already used for the four-arm sector labyrinths.

In these four-arm labyrinths, only two sector patterns could be placed in every quadrant, i.e. sector patterns no. 3 and no. 8. Four sector patterns could only be placed in the quadrants next to the main axis (related posts 2). Now this is not different either in labyrinths with five arms. However, we then have to fill not only four quadrants, but the five sectors I til V with sector patterns. Sectors I and V lie next to the main axis. In the three sectors II, III, and IV between them, only sector patterns no. 3 or no. 8. can be placed. These can be arranged in only two different sequences, 3 8 3 or 8 3 8.

Figure 1 shows how the first sequence can be completed with patterns for sectors I and V. In sector I the two sector patterns no. 5 or no. 8 can be connected to the sequence 3 8 3. In sector V, sector patterns no. 7 or no. 8 can be attached.

Figure 1. Combinations with the Sequence 3 8 3 in the Sectors II – IV

In fig. 2 we can see, how the second sequence can be completed to a full five-arm labyrinth. Here, in sector I the sector patterns no. 3 or no. 4, in sector V the sector patterns no. 2 or no. 3 can be attached to the sequence 8 3 8 between them.

Figure 2. Combinations with the Sequence 8 3 8 in the Sectors II – IV

Figure 3 shows the four patterns and labyrinths that can be generated with the first sequence (3 8 3 from fig. 1).

Figure 3. Patterns and Labyrinths with the Sequence 3 8 3 in the Sectors II – IV

Figure 4 shows the four patterns and the corresponding labyrinths that can be generated with the second sequence (8 3 8 from fig. 2).

Figure 4. Patterns and Labyrinths with the Sequence 8 3 8 in the Sectors II – IV

Just as in the four-arm labyrinths, there exist also 8 different five-arm labyrinths made-up exclusively of real double-barriers. Even though they have only five circuits, they strongly remind us to the labyrinth type Gossembrot 51 r. We can also name them following the same rule as for the four-arm labyrinths (related posts 1). The name, thus, is composed of an uppercase letter followed by a number of marks. However, since these labyrinths have four side-arms, also four marks have to be attached to each uppercase letter. These marks indicate how the sectors are connected to each other. We have here exclusively real double-barriers with direct connections. Therefore, each name is composed of an uppercase letter followed by four horizontal marks.

Related Posts:

1. Classifying the Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits
2. The Labyrinths with 3 Double-barriers, 4 Arms, and 5 Circuits