The China Labyrinth is over forty years old and all that time one of the loose ends in my mind has been to make a game that uses solutions of the puzzle as a board. The dominating idea was to use it as a 'castle' with yet to determine pieces moving along its corridors in search of treasure or safety. Basically a movement game. It may and probably will not matter within this idea whether the solution is transcendental or 'compact'. Because there is a dungeon, an inherently inaccessible room in the castle, 'warp wands' were envisioned, that allow you to move instantly to a predetermined room, regardless of where it may be situated. Under circumstances this predetermined room may be the dungeon. The idea never went anywhere.

Then in May 2022 Nick Bentley invented Strands, a game played on a board with cells that have numbers, 1 to 6 or thereabouts, and a protocol that allows you on your turn to place as many stones on like-numbered cells as the number indicated on the first cell where a stone is placed. Different distributions of cells are possible. In the inventor's words:
The board's center is the most powerful place to put pieces, but you can place more pieces per turn near the edge. You must navigate this tradeoff to win. An AI was used to determine how many pieces you can place on each region of the board, so all regions are balanced.

At least according to the algorithm. The goal was to create the largest group with the count cascading down in case of equality.

This rang a bell. Random setups with values 1 to 6 emphatically suggested the use of the China Labyrinth as a board. The solution to my 40 years old loose end suddenly emerged like a ship out of the fog. The Labyrinth literally seemed tailored to the goal and it even harboured a refinement because it features either number-based or pattern-based follow-up placements. I felt elated!

Since the solutions tend to have low numbered cells wide apart and high numbered cells closer together, the placement options have been inverted: you may not place the number of stones equal to the number of neighbours of the first cell where a stone is placed, but instead you may place "seven minus the number of the first cell".

One step beyond
The above resulted in China Tangle and Ed van Zon made the applet. In January 2023 a twin game emerged that I named China Grove, after a Doobie Brothers song. That finalised the implementation of the idea, or so I thought.

But it didn't. Somewhere in February I thought "what if boards based on transcendental solutions of the Octopuszle were used, instead of on the China Labyrinth?". You can't unthink those kind of ideas so I shared them with Ed van Zon. I was very surprised and glad about the speed with which he managed to modify the Labyrinth generating program to one that does the same with the 256 squares of the Octopuszle, namely generating 255 of them, connected to make up one board. Like this:

transcendental solution Here you see the big group of a transcendental solution in two groups, with the lines needed to create it, at least by humans, still visible. The single blank square has been omitted.

It may be that till Ed had completed the program, no-one had seen a transcendental solution of the Octopuszle for almost a quarter of a century. My archive of photographs of hand-made solutions went up in the air together with the set I used to make them, and all my other belongings for that matter, in the explosion of SE Fireworks in May 2000.

Being able now to generate a solution, if not in seconds then at least in minutes, is extremely pleasant. Moreover, Ed has expanded the theory by proving the loops/groups relationship in both the China Labyrinth and the Octopuszle. So I can tell you that the number of open loops in a transcendental solution of the Octopuszle will always be 32 + the number of groups.

The minimum number of groups of a solution is 2 and the theoretical maximum is 16, with 34 and 48 loops respectively. It may be that the theoretical maximum cannot be reached. In the China Labyrinth transcendental solutions have a theoretical maximum of 8 groups, but no solution beyond 7 groups has till now been found.

But all this doesn't concern the game since it is inherently played on a 255-squares connected lay-out.

Enschede, March 2023

christian freeling