Again we deal with the simple, alternating, transit mazes, defined by the New York Professor of Mathematics Tony Phillips. In his calculations he ascertains a number of 1014 theoretically possible variants of interesting 11 circuit labyrinths (12-level mazes).
He also describes a simplified method for calculating these variants, which John E. Koehler developed in 1968 to solve a related problem of stamp-folding.
The following pictures should explain this method. To this I first use the already known path sequence for the 11 circuit labyrinth which can be generated from the basic pattern, namely: 5-2-3-4-1-6-11-8-9-10-7-12.
The path sequence must begin with an odd number and then the row must be composed of even and odd numbers alternately. The center is named with “12”, as it is the outside.
I draw a circle and divide it into 12 parts, as for a dial. Now I have to connect all points with lines, but same-colored lines must not cross.
I start with blue in 12 and go to 5, 2, 3, 4 (Fig. 1). Then from 4 to 1, thereby I change the colour (Fig. 2). I continue with 6, 11, 8, 9, 10 (Fig. 3). I again change the colour and complete the lines from 10 to 7 and 12 (Fig. 4).
But you can do it differently. For example, draw all the lines first in one color and then the intersecting ones in the other. Here again, the same-colored lines should not cross each other. But more than once, as long as they are different (see 4 – 7).
But since we are looking for new labyrinths, we now go the opposite way: We draw a network of 12 lines, which connects all 12 points according to the above specifications and derive from this the path sequence.
Here is an example:
I write the first path sequence in line 2 (here in blue), starting at 12 and reading the lower digit, here 5. This is the beginning of the path. Then I follow the polygon until I land at 12 again and get: 5-2-3-4-1-6-11-10-9-8-7-12. That’s the original.
Now I go backwards and write the path sequence in line 3. So from 12 to 7, etc. That gives: 7-8-9-10-11-6-1-4-3-2-5-12. This is the complementary to the original.
I receive the lines 1 and 4 by arithmetic. I add the corresponding numbers of each row to “12”. In line 4, I get the dual to the original. In line 1, I get the complementary to the dual.
I verify this by comparing the numerical columns thus obtained with the others in “reverse”. This applies to the lines 1 and 4, as well as 2 and 3.
This is reminiscent of what has been described before when dealing with the dual and complementary labyrinths (see Related Posts below).
But there are alternatives. I turn the dial around, write the numbers for the 12 dots to the left, counterclockwise.
This is how it looks like:
The left side shows the dial as before. I start at 5, count to 12 and get the original. Then I start at 7 and count again to 12 and get the complementary to the original.
Now the right dial. I also start at 5 and count to 12 and so get the dual to the original. Then again from 7 to 12 and I get the complementary to the dual.
What should the blue written path sequences mean? They point out that the entry into the labyrinth can be placed on the same axis as the entry into the center. Here on circuit 5 and 7. Walter Pullen calls this that a labyrinth layout is mergeable. This allows you to construct a small recessed spot in the labyrinth, which some name the heart space. Especially in the concentric style, this can be implemented well.
From these two newly obtained path sequences, I now construct two new 11 circuit labyrinths in concentric style:
They have a different pattern of movement than the upt to now known labyrinths. In addition, we see 6 turning points for the circuits.
This is the dual to the previous labyrinth. Again, there is another “feeling”.
Who makes the beginning and builds such a labyrinth?
The other two paths sequences also result in new labyrinths, which I don’t show here. These belong to the remaining 1000 variants that are theoretically possible for 11 circuit labyrinths.