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## The Sequence of Segments in Multiple-arm Labyrinths

In one-arm labyrinths, each circuit is represented by one number. Therefore it is possible to capture even quite large labyrinths appropriately with the level sequence. In labyrinths with multiple arms, the pathway may repeatedly encounter the same circuit. Various possibilities exist to take account of this in the level sequence. For this, according to the number of arms, the circuits have to be further partitioned to segments. Here I will show a method in which all segments are numbered through.

For this I use an example of a labyrinth that has repeatedly been presented on this blog. It has 3 arms and 3 circuits.

First, each circuit is partitioned to three segments. One segment corresponds with a unit of the pathway between two arms. Next, the segments have to be numbered through. This can be done in different ways. Here I number them from the outside to the inside and one circuit after each other.

Now we can track the course of the pathway through the various segments. This results in the sequence of segments encountered by the pathway. In labyrinths with multiple arms the level sequence thus extends to a sequence of segments.

The sequence of segments of this labyrinth is 7 4 1 2 5 8 9 6 3. The length of this sequence of numbers is a result of the number of circuits multiplied with the number of arms. Thus, for a labyrinth with 3 circuits and 3 arms, 9 numbers are required. Whereas in a one-arm labyrinth with 3 circuits only 3 numbers are needed.

However, besides the numbers no other information is needed. The sequence of segments itself determines where the pathway makes a turn or traverses an axis. In one-arm labyrinths this had to be indicated additionally by use of separators.

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