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China Tangle & Octangle It's possible to apply number values to the cells equalling the number of beams. The board then has high numbers in the compact inner sections and low numbers scattered along the protruding parts, but ... ... for the Strands placement protocol, that's the wrong way around. To get it right, the numbers should be reversed, meaning that every cell will get a number equal to (7 - n), where 'n' is the number of adjacent cells. So the 6-beams cell gets a '1' and the 1-beam cells get a '6', as in the picture below: |
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The Strands number based placement protocol can be used on this board as is. The rules state that a player on his turn may place stones on up to 'X' vacant cells marked 'X', so it's allowed to place less stones. This is particularly suited if the first move is replaced by a swap offer, as is the case in China Tangle. Note that cells with many neighbours and thus a low number of placements are on average much closer together than cells with fewer neighbours and thus a high number of placements. This is inherent in the board's stucture, which is always the same. This is why I consider these boards a natural habitat for Nick's protocol. However, the shape is always different. You could start new games for a decade and never get an identical board. It's easy to verify this interactively. One step beyond: pattern mode Top left in the applet you can switch between number mode and pattern mode. The latter is the other way to look at it, and for what it's worth: the better one. The game is inherently 'hot' in both versions, meaning that players would want to place so long as it is useful. |
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Another way to look at it Having the program create boards for the above version of the game bypasses the China Labyrinth's defining property: its patterns. It doesn't matter for the one 6-beams cell and it doesn't matter for the 5-beams cells and the 1-beam cells, but cells with 2, 3 or 4 beams come in different patterns. Since beam patterns and neighbour patterns are inherently identical, the beams have been omitted.
Definition The set of a cell consists of the cell itself plus every subsequent 600 rotation of it, and all mirror images insofar these are not already included in the set of rotations. Only non-symmetric forked 3-beams cells have mirror images that are not part of their set of rotations. So this set consists of twelve cells instead of six. |
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These patterns predictably led to two variants: A number-based one like Strands:
And a pattern-based one unlike Strands:
The applet features both options and both lead to gameplay of the tactical kind. As already noted with Strands: strategy is no rocket science. Of course, number-based play gives on average more options at any given turn. The swap rule is sufficient to balance both variants to an equal degree. Yet I must designate pattern-based play as the main game because it follows the concept of the China Labyrinth more closely. |
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| The applet's hover function The applet has a hover function that is absolutely essential for playing the games. So long as the player to move hasn't placed any stone yet, it highlights in light-green the set of placement options of the cell above which the cursor hovers. Once one stone has been placed, the other vacant cells of the corresponding set are highlighted permanently with a green dot as shown below. The hover function then remains active, now highlighting cells that may be considered as either alternative placement options, or as possible replies of the opponent.  In the image above you see a snapshot of a pattern-based game that, till that point, went fairly even. The marked white stones are the ones that White placed on his last turn. Black has still the dotted vacant cells as options to place two more stones after the two he already did place (marked with a green circle). The cursor hovers over a symmetric 3-beams cell and the complete set - in this case consisting of two cells - is therewith highlighted. |
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A natural extension: China Octangle With the idea being implemented, an extension of the game based on the Octopuszle was a natural next step. It features squares instead of hexagons, with eight 'beams' that come in two kinds: four orthogonal ones in blue and four diagonal ones in red. Rotations are now in steps of 450 instead of 600. A 450 rotation changes all orthogonal beams to diagonal ones and vice versa. The main problem I saw was making a program that would generate the 255 squares layout of the big group of a two-groups transcendental solution, like this:  I wouldn't have a clue how to go about it and I was reluctant to ask Ed because the task seemed too daunting. So I only mentioned the idea without much hope that it would be realised. But as the image already gives away, Ed did it! I'm eternally grateful for that because the China idea was about to take an important turn, the one that led to China Squares. Meanwhile ... China Octangle isn't all that different from China Tangle. The beams come in two kinds but the beam-neighbour correlation is the same. As in China Tangle, the beams necessary to solve the puzzle can be omitted from the completed result without any loss of information.  The properties of a 255-squares transcendental board of the Octopuszle are the same as those mentioned in China Tangle:
Groups The goal of the game is to end with the largest group, cascading down in case of equality. In China Octangle a group is a maximal set of connected stones, whereby both orthogonal and diagonal neighbours count as connections. My take on China Tangle & Octangle China Octangle is an upscaled version of China Tangle, albeit not in the traditional sense. The board is four times as big and, contrary to China Tangle, opposing groups may intersect so its tactical resolution is higher. |
is assigned the digital number 10111001, which is the numerical value 185. If a stone is placed on it, then the notation gives only a square's numerical value, not its position because its place within the board varies.
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One notable difference is my decision to feature only pattern-based play in Octangle.The reason is that in number-based play the number of placements to choose from would be very high. There are, for instance, seventy squares with four beams, divided over eight sets. The picture on the left shows one or two representatives of each set. Playing number-based the number of placement options might thus be up to seventy. Pattern-based placement is restricted to the set that the first placed stone belongs to. |
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Another difference is the percentage of squares of which a mirror image is not part of the set of their 450 rotations. Such sets are double the size of sets without that feature. So they, on average, allow a wider choice of placement options. In China Tangle this concerns only the set of forked threes, consisting of 12 cells. That's less than 20% of the total number. In Octangle there are two 3-beams sets, two 4-beams sets and two 5-beams sets with that particular feature. That's six sets of sixteen squares each, 96 squares in all, more than 37% of the total number. So Octangle has a larger number of 'increased choice' options, not only because of its larger size, but also relative to its own size. |
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A major difference A Strands board is compact, so barring the board edge, the only thing that can prevent you from getting somewhere, anywhere, is the opponent. An Octangle board isn't compact. It usually has different high-density areas that are often connected by one-square or two-squares bridges. Areas can thus be isolated by one or two stones. Several examples can be seen in the picture. Obviously a group cannot grow beyond the size of an isolated area it occupies. Given the goal of the game that is an important consideration and it plays no role in Strands. It suggests that occupying bridges should conditionally precede growing groups behind them. It also suggests there's more to the game than Strands' strategy. |
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Labyrinth boards like the ones used for the different China games in this article, do not have a centre in the usual sense. Instead they have regions of higher square-density that tend to be nearer to the centre and regions of lower density scattered around the periphery, as shown in the picture. In high density areas the allowed number of placements is low, in accordance with the Strands protocol, but there are exceptions. There are usually some rogue squares with very few neighbours situated in or around the centre, and they belong of course to sets with a high number of placements. However, the other squares of their sets are mostly scattered around the periphery, so placements will as a rule still be scattered accordingly. |
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Summary Playing China Tangle in number-mode feels closest to its ancestor. In pattern-mode, placement choices are more restricted and players usually get a mixed bag of options, some of which may be crucial, while others may be anything from possibly useful to totally irrelevant. That's not wholly unlike Strands. China Octangle is four times as big as China Tangle and even though it doesn't allow playing in number mode, it offers a wider choice of options. There are two main reasons why it offers more intricate and intriguing tactics:
Both Tangle and Octangle are 'hot', meaning that players would want to place stones so long as it is useful. Placement options are never disadvantageous, but they may be useless in some situations where the actual fight for the bigger group takes place elsewhere. True, if the fight for the biggest group is even, then the next biggest groups play an important role, but cascading down the balance is usually broken rather quickly. Games remain hot only so long as they're balanced. In all three games, broken balance is easily spotted by comparing the largest groups. If the difference exceeds a certain point, then it may become all too clear which side will win. Further placements are then inconsequential and the game has gone from hot to lukewarm to stone-cold dead. For any abstract strategy game to remain interesting, it must remain balanced and any advantage should not be too easily spotted. The strategies of China Tangle and Octangle are no rocket science. Given the largest group goal they're decent games, but they're not aspiring to be great games. For that, we need to adapt the placement protocol to serve a far more interesting goal, the one of China Grove and China Squares. |
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