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 Loops & groups in transcendental solutions of the 16-puzzle Contrary to compact solutions like the ones in BackSlide, transcendental solutions of the 16-puzzle don't allow contact between blank sides. If the lines are removed from the squares, no information is lost. Each square is still uniquely determined by the pattern its neighbours form around it. This transcendental solutions generator illustrates the point. It will very occasionally spawn the one possible solution with four groups and four loops.   In both types of solution the number of separate groups, with the blank counting as one group, will always be equal to the number of loops in the figure. The 16-puzzle is a first order puzzle in which diagonal adjacencies don't play a role. The second order variant is the Octopuszle that consists of 16x16 suares and has a similar groups/loops relationship, though not the same one.

Imagine the lines or 'beams' in a connexion as walls where you can walk along. If you keep the wall on your right, you walk clockwise around the outside, and anti-clockwise around the inside.

If you walk around a structure while keeping your right hand against the wall, back to your starting point, either on the outside or on the inside, you will always make a net turn of 360º regardless of the corners or angles you encounter.

There are convex (positive) and concave (negative) angles and straight lines on any tile except the blank. When walking on the outside, the first ones take you closer to the full 360º turn, the second ones take you away.

It should be noted that the deviation from walking straight ahead is to be seen as the degrees value of an angle (clockwise is positive, anti-clockwise negative). So the straight lines count as 0º and, e.g., the tiles in the top row of the image below each have a 60º convex bend and a -60º concave one. Remember your right hand touches the wall when passing a bend.

The 16-puzzle has sixteen -90º bends, 4 on the piece with four lines, 8 on the four pieces with three lines and 4 more as single bends. Sixteen bends can make a maximum of four loops: the one 'four' in the centre, the four 'threes' adjacent to the sides and the four bends in the corners. That's one group. There are seven pieces left. The horizontal straight and two horizontal ends make a second group and the vertical straight and two vertical ends make a third. The blank makes up the fourth group.

In any puzzle the number of groups x (360º) plus the number of loops x (-360º) equals the sum of all counted convex and concave angles in all tiles, since any counted angle is part of an outside or an inside walk, and no uncounted angle is part of a walk by definition or implication (the 16-puzzle only has counted angles, the China Labyrinth and Octopuszle also have uncounted ones). That sum is a fixed number. For the 16-puzzle it means:

#groups x (360º) + #loops x (-360º) = 0
Ergo: #groups - #loops = 0, in other words the number of groups always equals the number of loops.

Loops & groups in transcendental solutions of the China Labyrinth
In the China Labyrinth the reasoning is the same but now with regard to -60º bends (the concave ones). There are 48 of them and they can theoretically be used to make 8 loops.

 Type # of -60º bends 6 6 6 6 12 6 6

Again the number of groups x (360º) plus the number of loops x (-360º) equals the sum of all counted convex and concave angles in all tiles. In counting the total, we disregard the -120º angles in a tile (the inside sharp angles), because they are never encountered in the walk. Their beams imply certain beams in adjacent tiles, which inevitably shields these angles from a walk.
For the China Labyrinth the result is the same as for the 16-puzzle, namely:

#groups x (360º) + #loops x (-360º) = 0

 That means that in the China Labyrinth the number of groups will also always be equal to the number of loops. However, the maximum of 8 groups seems out of reach because to our knowledge no-one ever managed to get beyond 7. Like the one on the left. Notice the six 60º bends on the outside.

Loops & groups in transcendental solutions of the Octopuszle
In counting the total for the Octopuszle, we disregard the -135º angles in a tile, as well as the orthogonal -90º angles, because they are never encountered in the walk. Their beams imply certain beams in adjacent tiles, which inevitably shields these angles from a walk. See the top-right group in the picture below for some examples of the orthogonal -90º angle (as well as some -135º ones).

 Taking the above into account, there's 16 x 360º of convex angles in the set and 48 x -360º of concave ones so the theoretical maximum number of groups is 16. And, here the number of groups x (360º) plus the number of loops x (-360º) equals (16 x 360º + 48 x -360º) or #groups - #loops = -32 That means that in the Octopuszle the number of loops equals the number of groups + 32. The solution depicted here illustrates the inevitable: it has 6 groups and 38 loops.