China Squares
China Squares uses the same ever-changing boards as China Octangle, consisting of 255 squares that come in 29 sets. Among them are two 3-beams sets, two 4-beams sets and two 5-beams sets that have squares of which the mirror images are not included in their set of 450 rotations. They each consist of 16 instead of 8 squares and thus offer a wider range of placement options. The game is basically the same as China Grove.

  • Squares have two kinds of neighbours (beams): orthogonal ones (blue) and diagonal ones (red). If a beam pattern is rotated 450 then every orthogonal beam becomes a diagonal one and vice versa.
  • Groups are defined by orthogonal and/or diagonal connections. A group is maximal, meaning that part of a group is not a group. The smallest group is a single stone.

The number of turns and the swap rule
The number of turns of a completed game of China Squares can be established the same way as it was in China Grove.

In the picture you see single or double representatives all 29 sets, with the number of squares of each set and the number of turns it takes to get it on the board. The total number of turns is 68. That is even, meaning that, contrary to China Grove, both players get the same number of turns: thirty-four. The second player thus has a double disadvantage and the need for the swap rule is obvious.

If a first placement were anything like it would be in a swapless game, it would surely be accepted. So a swap offer should be less than tempting, to make sure that an opponent who takes it to avoid having the last turn, will not have the additional advantage of a good first placement.

Personally I use to open with a single placement on the 8-beams square, or placements on the two all-symmetric 4-beams squares or on the four straight 2-beams squares. The last option gives the most placements, but straight 2-beams squares are usually far apart and can often be isolated quite easily.

How to play the game
For the occasion I'll put heuristics - rules of thumb that are right more often than not - in

  • unordered lists marked by red square bullets.

As in its ancestor Strands, a game begins by contemplating the board, but in China Squares there are more issues to consider, like the always present separated areas of higher square-density that inherently allow less placements. Such areas are sometimes connected by only a few bridges and surrounded by lower density areas that allow more placements. High density areas usually aren't occupied early on, because in that stage the number of placements should be maximised. A good strategy is to more or less surround them by peripheral placements in a Go-like matter, to ensure that the opponent can't invade them by extension, but only by creating a new group inside them. So a good heuristic is:

  • Maximise the number of placements in the initial stages of the game, but don't push it too far.

For instance, occupying the set of 1-beam squares obviously maximises placements. But these placements are also maximally isolated and unless there's a friendly stone next to them, on the penultimate squares (coloured dark green in the picture below), each placement costs P-1 points.

If a player occupies such a penultimate square, then an opponent's stone on the adjacent end-square is permanently isolated and thus penalised.

In this randomly picked board, one stone has been placed on an end-square, thus highlighting the other end-squares with a green dot.

In the picture there are four penultimate 3-beams squares and they do not all belong to the same set. By hovering the cursor above any of them, players can consider the complete set.

The fact that 3-beams squares determine the placement of six stones, or as many as vacant squares of the set are available, may make them attractive in the early stages of a game. If there are penultimate squares among the considered placements, these are a bonus that eventually will cost the opponent penalty points.

  • If considered placements happen to include penultimate squares, take it as a bonus.

Below you see an example from one of my games against Ed van Zon. Ed swapped the first turn, a single stone on the 8-beams square, so White will have the last turn and that will most likely mean having to take the end-squares.

Black already occupies three penultimate squares that will eventually cost White nine points, so having at least this one penultimate square softens the prospect, even if it's just part of a 6-stone placement with a different tactical goal.

In this case the main goal was to prevent Black from connecting in the lower left. The bottom white placement is the crucial one. The two squares that are now needed to connect the black groups belong to different sets, so they cannot be occupied in the same turn.

Cutting and connecting
In the picture below you see an example from one of my games against Aleh Tapalnitski, the author of "Meet Dameo". Black did indeed connect by taking the 8-beams square, thus cutting the white groups at that particular point. However, White has three other squares available to connect them and eventually did so. But cutting and connecting at the same time is obviously a good heuristic.

  • Cutting and connecting are often good, doing it at the same time is almost always good.

Cutting opponent's groups without connecting can be done in two ways: by extending one of a player's own groups, or by an isolated placement.

  • Cutting by extension is usually good.
  • Cutting by an isolated placement may be conditionally good. A connection brings the opponent P+1 points while a cut costs the player who makes it P-1 points.

Invading or extending into contested territory
Territory is contested if both players can place there by extension. There will also emerge territories in the course of a game that are partly or wholly surrounded by one player in a way that prevents the opponent from doing so. The picture below, taken from one of my games against Laurentiu Cristofor, shows how White surrounded several such areas in the centre, on the right and bottom-right and in the lower-left. It's a penalty-8 game, hence the large groups.

If the opponent wants a foothold in any of these areas, he'll have to invade with one or occasionally more stones and accept the penalty. The territory in the centre has a relatively high square-density, so the number of stones that can be placed there is low. Invasions in such dense areas usually don't come early in the game.

With two of the three stones of his last placement (the marked stones) Black secured future connections in the top-left area by creating two options for each pair of groups to do so.

  • If you can connect groups via different squares that do not belong to the same set, these intended connections are relatively secure.

Black left the possibility for the big cental white group to connect to the 11-stones group top-right or to the adjacent single white stone. However, the two squares involved do not belong to the same set, so Black could have prevented only one connection. For the same reason, White cannot make both connections in the same turn, so Black still has the the option to cut one of them if the other is taken.

Orthogonal and diagonal extensions
If we count both orthogonal and diagonal adjacencies, then an orthogonal straight line of length L on a regular square grid has 2L+6 adjacencies, while diagonal straight line of length L has 2L+2(L-1)+6 adjacencies. So ...

  • On average diagonal extensions give more room to extend than orthogonal ones, and they're harder to block.

Placements of seemingly equal value
It sometimes happens that the number of placement options exceeds the number of stones that a player is entitled to place. In such cases a stone, usually the last one, may have more than one option of seemingly equal value. In such cases ...

  • Consider how attractive each option may be for the opponent.
  • Use the hover function to consider the adjacent squares of each option and how it might serve the opponent.

Permanency and its effect on lookahead
China Squares is a perfect information game, but humans, uness blessed with superhuman powers of perception, can only see what's there with the help of the applet. In the realm of abstract strategy games, this is an exceptional feature. It reduces the factual extent of the players' lookahead to two ply: choosing their own placements among several mixed bag options and viewing possible replies of the opponent to each of them. But China Squares is also a pure placement game where every placement is permanent. That makes the actual extent of the lookahead much deeper.

  • Establishing more than one way to connect two groups may not be a 100% guarantee that it will happen, but the option will remain for many moves and it may take exceptional circumstances to prevent it.
  • Occupying a bridge that isolates an area means that the opponent cannot invade it by extension. If at all, it must be done with one or more isolated stones and accepting the penalties.
  • Placing a stone in isolated territory may allow extending there and prevent the opponent from claiming it altogether. If such a group grows to the size of the penalty, it will itself be neutral, but even if it doesn't, it will prevent the opponent from occupying the same squares.
  • Occupying penultimate squares means either being in a position to eventually take the end-squares without getting a penalty, or it will force the opponent to take them and being penalised for it.

These are all examples of moves that can be predicted to carry their weight for many turns to come, if not indefinitely. So the actual lookahead goes much further than the extent of the factual one.

The good, the bad and the ugly: mixed bag placements
Mixed bag placement options differ from the kind of multi placement options that are featured in the game's ancestors Strands and Symple. In Strands you're free to choose up to X placements on cells marked X. In Symple you're obliged to either place a single or grow all groups, but you're free to choose where to grow those groups.

In China Squares every single first placement of a turn comes with a number of additional mandatory placements on squares of the same set.

  • The total number to place may range from one to eight, depending on the number of beams of the square of the first placement and the number of vacant squares of the set that it belongs to that are still available.
  • The number of squares to choose from may range from one to sixteen depending on the size of the set and the number of its available vacant squares.

A player whose turn it is, looks for placements that have a high priority in the position at hand. Should I connect, cut, extend into contested territory, or invade? Very much the same considerations one might have in any territorial battle. There will be a number of focus points for a first placement. China Squares differs from other games in that they can only be considered by actually making such placements, one after the other, thus having the applet highlight the vacant squares of the corresponding sets by a green dot.

Here's a tip: if you search a position to see the number of placements that a square will allow, even before hovering, it may be easier to look at the number of absent neighbours around a square.

  • The total number to place - 9 minus the number neighbours - equals 1 plus the number of absent neighbours.

I find it a more convenient way to scan a position with eyes only because in my experience it takes less time than locating the squares of a set by hovering and then counting them. But that may just be one of my idiosyncracies.

Let's take the above game against Laurentiu Cristofor again, a few turns onwards. White's last placements are highlighted. The bottom one, connecting two groups, is the most important one, especially since it is a penalty-8 game. With it came two placements of lesser impact, both adding one point to White's score while one of them prevents a black stone from growing.

The alternative, connecting the bottom two groups at the square that now holds Black's first placement on the following turn (the stone marked with a green circle), looks better to me: it comes with the squares marked with a green dot and thus would have connected two pairs of groups, while extending in contested territory at the top and invading in the area on the left that is dominated by White. Here a white stone would have had the option to connect to the one single white or the group of three, and possibly both.

As it is the square was taken by Black who consequently took the three dotted squares that came with it, now threatening to cut the white single bottom-left from the two stones above it. This prompted White to yet connect them. The cursor hovers above the connection square and highlights it and the squares that come with it in green: four additional placement options, each of them adding one point. Of course White can only occupy two of them because it concerns 6-beams squares that limit the number of placements to three. And this is precisely what happened in the actual game.

To conclude this example, White now threatens to connect the group with the two marked stones to the single one beneath it so Black of course considered the option to cut, and did so. There were only two squares of the set still vacant, on a knights move distance of one another. In the picure on the left they're marked.

Since there was no swap in the game, Black will also have the last turn and in a penalty-8 game that hurts. I fear for Black, even more so because I'm Black, but you can see for yourself how it panned out.

On the whole White did create more 'breathing space' throughout the game, vacant territory surrounded in a Go-like matter. Black has a few patches too, on the left and top-right, but it will not be enough. It brings us to recognising two important heuristics:

  • Extending into contested territory is better than extending in one's own territory.
  • Surrounding territory in a Go-like matter is very important because it gives breathing space, once contested territory has run out.

Allow me to yet again emphasise the property that gave rise to Strands: in areas of greater square-density the number of placements is low. That's one of at least two reasons to avoid occupying them in the beginning. The other is that they're inherently surrounded by squares that allow more placements per turn. Maybe not the large numbers one gets to place at the periphery, but enough to avoid falling behind too much on an opponent who leans towards placing more stones at the price of less connectivity.

A strategy aimed at more connectivity allows on average less stones to be placed. The picture above shows how Black resigned the game in a position after thirty-one full turns, three full turns before the end. White got 111 stones on the board in 17 groups, Black has 122 stones in 21 groups. White's Go-like enclosure of certain areas clearly paid off, and he did it with less stones than Black needed to lose. It shows that White's strategy has been more efficient.

Mixed bag options: a fundamental difference?
I've called mixed bag options a fundamental difference because barring its little ancestor China Grove, I know of no other game that has unintended placements coming with intended ones. With the good may come the bad and the ugly. If anyone at the BGG's Abstract Games Forum knows of other abstract strategy games with this feature, I'd very much appreciate to see a post about the game(s) in question.

A final question
I wonder what the members of the BGG Abstract Games Forum community's take is, on the the effect of raising or lowering the penalty value.

It has been fun writing this little essay :)

Enschede, December 2023

Christian Freeling