There are 42 different one-arm alternating labyrinths with 7 circuits. Among these there is one pair of complementary interesting labyrinths. Now how does it look in pairs of complementary unintersting labyrinths? This question has already been indirectly answered in my last post (see related posts below): There is none! This sounds surprising. Therefore I address it further here. The 42 labyrinths form 21 complementary pairs. One of it is composed of 2 interesting labyrinths. We also know there are 22 interesting labyrinths. So the other 20 pairs are made up of an interesting and an uninteresting labyrinth each. Therefore no possibility remains for a pair with two complementary uninteresting labyrinths. What is the reason for that?
As we have seen, only in alternating labyrinths with an odd number of circuits it is possible to derive a complementary (see related posts). In such labyrinths the pathway always enters on an odd-numbered ciruit and also reaches the center from an odd-numbered circuit. Further, in one-arm labyrinths the pathway cannot enter the labyrinth on the same circuit from which it reaches the center.
In uninteresting labyrinths the pathway always must enter the labyrinth on the outermost circuit or reach the center from the innermost circuit. The complementary is derived by mirroring. By this, the outermost is transformed to the innermost circuit and vice versa. If in an original labyrinth the pathway enters on the first circuit, it is an uninteresting labyrinth. In the complementary the path will enter on the innermost circuit. Thus the complementary is not an uninteresting labyrinth, unless the path would reach the center from the innermost circuit. This, however is not possible, as it already enters the labyrinth on this circuit. The original is an unintersting, but the complementary an interesting labyrinth. The other alternative would be that the path in the original labyrinth reached the center from the innermost circuit. But then in the complementary it would reach the center from the outermost circuit what is not an unintersting labyrinth. Therefore the complementary could only be an unintersting labyrinth, if the path would enter it on the outermost circuit. This, however is impossible, as the path reaches the center from this circuit.
These results are only valid for one-arm labyrinths with up to 7 circuits. In labyrinths with mulitiple arms, the pathway may reach the center from the same circuit on which it enters the labyrinth. Thus, for example it could enter the original labyrinth on the first circuit and also reach the center from the first circuit. This would consitute an uninteresting labyrinth. In the complementary, the pathway would then enter the labyrinth on the innermost circuit and also reach the center from the innermost circuit, what again would qualify for an uninteresting labyrinth. In one-arm labyriths with more thean 7 circuits the definition of what constitutes an uninteresting labyrinth can be extended. In these cases trivial circuits can be added not only at the outside or inside of smaller interesting labyrinths (what generates uninteresting labyrints) but also on central circuits between other interesting elements at the inside and outside of the labyrinth, what also may generate uninteresting labyrinths.