Line-of-sight games
Rules
In Meridians, a hexagonal board with tessellated triangles like the one below is used, and the intersections of the grid are used to place the stones, just like in Go. Of the six sides of the board, one pair facing each other is one intersection shorter than the other four sides, so there is no single central intersection point on the board.
The standard size of the board is 6/7 intersections per side, with 5/6 intersections for beginners and 7/8 intersections for experts being the official sizes. Theoretically, it is possible to play with smaller or larger sizes.
In Meridians, a hexagonal board with tessellated triangles like the one below is used, and the intersections of the grid are used to place the stones, just like in Go. Of the six sides of the board, one pair facing each other is one intersection shorter than the other four sides, so there is no single central intersection point on the board.
The standard size of the board is 6/7 intersections per side, with 5/6 intersections for beginners and 7/8 intersections for experts being the official sizes. Theoretically, it is possible to play with smaller or larger sizes.
Initially, there is nothing on the board, and the two players take turns adding stones of their own colour, one at a time, to the empty intersections on the board. On each player's first move, they can place stones anywhere they want. From the second move onward, however, you place new stones according to the so-called "line-of-sight" mechanism.
When two stones are on the same line and there is no other stone between them, they are considered to "see" each other (even when adjacent). This is the “line-of-sight." After the second move, you place your stones on a point that can be “seen” from at least one of your stones. However, for a stone to remain on the board, it must be kept it "alive."
Let us define "connected group of stones of the same colour," or just "group," to have the same meaning as in numerous territorial or connection games. By convention, a single stone is a group of one.
When a stone is seen by a stone belonging to another friendly group, that line of sight is called a path. For a stone to live, it must have at least one path. However, the path is shared by the group: if any one of the stones belonging to a group has a path, it can keep the whole group alive.
When two stones are on the same line and there is no other stone between them, they are considered to "see" each other (even when adjacent). This is the “line-of-sight." After the second move, you place your stones on a point that can be “seen” from at least one of your stones. However, for a stone to remain on the board, it must be kept it "alive."
Let us define "connected group of stones of the same colour," or just "group," to have the same meaning as in numerous territorial or connection games. By convention, a single stone is a group of one.
When a stone is seen by a stone belonging to another friendly group, that line of sight is called a path. For a stone to live, it must have at least one path. However, the path is shared by the group: if any one of the stones belonging to a group has a path, it can keep the whole group alive.
Stones and groups that do not have any paths are called dead groups, and at the end of a player's turn, all dead groups of that player's colour are removed from the board. Thus, if your dead group is created by a path being blocked on your opponent's turn, you can save it by giving it a new path on your turn. Alternatively, you can abandon the dead group and place a new stone in order to gain an advantage elsewhere.
The objective of the game of Meridians is to eliminate all of your opponent's stones from the board. In most play, however, the winner is revealed by counting the number of points each player can still place stones before all of either player's stones are actually eliminated. Unlike in Go, there is no passing in Meridians, but you can resign if it becomes clear that you are losing.
Meridians has not been commercialized so far but given that it will be played with real components, it is possible that dead groups will tend to be overlooked. For this reason, an optional rule has been proposed that every time a dead group is formed, a token of the third colour should be placed to mark the group.
Meridians has not been commercialized so far but given that it will be played with real components, it is possible that dead groups will tend to be overlooked. For this reason, an optional rule has been proposed that every time a dead group is formed, a token of the third colour should be placed to mark the group.
Annotated game
This is an annotation of a Meridians game played turn-based on Mindsports from July 24 to August 5, 2021. It was the third game played between me, Kanare Kato, the designer of the game, and Kerry Handscomb, who had gotten the gist of it in the previous two games and beat me for the first time in this one. Kerry played light colours (first move), and I played dark colours (second move).
Meridians is a very young game, still only known to a few people, but I am convinced that it is one of those games where simple rules succeed in inviting depth. It would be my pleasure if you could get to find the basic tactics and clues for the strategies of the game.
This is an annotation of a Meridians game played turn-based on Mindsports from July 24 to August 5, 2021. It was the third game played between me, Kanare Kato, the designer of the game, and Kerry Handscomb, who had gotten the gist of it in the previous two games and beat me for the first time in this one. Kerry played light colours (first move), and I played dark colours (second move).
Meridians is a very young game, still only known to a few people, but I am convinced that it is one of those games where simple rules succeed in inviting depth. It would be my pleasure if you could get to find the basic tactics and clues for the strategies of the game.
In our first moves, we placed stones in a way that encloses the central points. In early games of Meridians, it is important to secure points through the "line of sight" of friendly stones, where you can place your own stones in the future. Therefore, it is unlikely that the first move will be placed on the edge of the board.
On the other hand, placing the first stone in one of the above central points may not be a bad move, but there is a concern that it will be surrounded by enemy stones in the future. So, placing the stones two or three steps away from the edge of the board, as in this case, seems to be a safer move.
In the following, we will tentatively call the straight line on the board on which the line of sight of your stone runs, where you can place your stone to create a path in the future, a "potential path."
On the other hand, placing the first stone in one of the above central points may not be a bad move, but there is a concern that it will be surrounded by enemy stones in the future. So, placing the stones two or three steps away from the edge of the board, as in this case, seems to be a safer move.
In the following, we will tentatively call the straight line on the board on which the line of sight of your stone runs, where you can place your stone to create a path in the future, a "potential path."
Kerry then places his second stone 3, keeping distance against the central points. This is a somewhat provocative move, because it is a position where the opponent could cut in with the path, but I did not. Even if I had cut in at this stage, he could have easily saved two stones with A or B (B in particular would have been a good move with many potential paths). On the other hand, my stone on C could have added potential paths only in two directions (North and South). For this reason, I chose 4 rather than C for the purpose of consolidating my ground near the centre.
Kerry places his stone 5, which is the same distance from the centre and edge as the previous stone. This shows that his forces are thinly and widely spread around the centre points. However, his stones on each side have only one path each and look a bit vulnerable.... Here, I placed a stone at 6 and cut in with his path from A to 5. This position not only allows me to cut into the opponent's path, but also to create a triangle shape of minimum size, as shown in the diagram.
The triangle is the most basic formation in the early game of Meridians. Each stone has two paths to the other two, making it difficult for the opponent to capture them, so it is the keystone of the early game positioning. However, it may be not advisable to try to build only triangles in the early game. Triangles created by placing a stone on the same line as two existing stones will discourage more potential paths.
Here I placed the stone 6 because I thought it would be a good move, minus the fact that it would not increase the number of my potential paths, since it would create both a triangle and a cut at the same time.
Triangles, cuts, and placements in free space that are neither of these—Meridians' early game is a combination of all three, with the goal of creating advantageous positions. The following description should shed some light on what an advantageous position looks like.
The triangle is the most basic formation in the early game of Meridians. Each stone has two paths to the other two, making it difficult for the opponent to capture them, so it is the keystone of the early game positioning. However, it may be not advisable to try to build only triangles in the early game. Triangles created by placing a stone on the same line as two existing stones will discourage more potential paths.
Here I placed the stone 6 because I thought it would be a good move, minus the fact that it would not increase the number of my potential paths, since it would create both a triangle and a cut at the same time.
Triangles, cuts, and placements in free space that are neither of these—Meridians' early game is a combination of all three, with the goal of creating advantageous positions. The following description should shed some light on what an advantageous position looks like.
Kerry still avoided the centre points and placed a stone at 7. This does not make a triangle, but it is a stable move that gives a path to both A and B. And now my little triangle is about to be encircled. My move 8 was, if I may make an excuse, an experiment to try a move I have not played often, but it was the worst move of the game. Because of its proximity to the corner, there are no potential paths in three directions, north, northwest and northeast. In addition, it is too far away from the small triangle and the path can be cut easily. C would have been a much better move because it could have interrupted my opponent's potential path from 7. The rest of the game seemed to be solely about whether or not I would be able to recover from this bad move.
Kerry blocked the only path my A had by placing a stone at 9 and completed a stable shape with paths to his three stones. Note that he has also completed now three triangles of Light colour stones on the board (albeit interrupted by my stones). This is a very strong shape that also allows him to place another triangle vertex outside Dark's presence.
I would prefer to interrupt the opponent’s line of B, 9, and C, but if I want to save my stone at A, I have to give it a path first. I thought 10 was the best move, since D was not in an opponent’s sightline, so it would not be interrupted immediately, plus it would also give E a path. However, given the poor location of A, it might have been better to abandon A.
I would prefer to interrupt the opponent’s line of B, 9, and C, but if I want to save my stone at A, I have to give it a path first. I thought 10 was the best move, since D was not in an opponent’s sightline, so it would not be interrupted immediately, plus it would also give E a path. However, given the poor location of A, it might have been better to abandon A.
Kerry's next move, 11, is clearly an attempt to corner my A, but it is also a very good move to create a new triangle with B and C. On the other hand, I went to the trouble of filling in my path between A and D on 12, because I feared that my opponent would create an "eye" shape by capturing my stone A.
In Meridians, an "eye" shape is a situation in which one or more stones are surrounded by another group of stones of one's own to prevent the opponent from invading, as shown in the diagram below. This formation, once established, is a rock and will remain on the board until there is no other place to place a stone and the player is forced to fill the eye himself. In addition, the stones connected to the eye shape will also never disappear, making it powerful weapon for breaking into the opponent's paths. Therefore, creating such an eye is a sub-goal in the middle game.
In Meridians, an "eye" shape is a situation in which one or more stones are surrounded by another group of stones of one's own to prevent the opponent from invading, as shown in the diagram below. This formation, once established, is a rock and will remain on the board until there is no other place to place a stone and the player is forced to fill the eye himself. In addition, the stones connected to the eye shape will also never disappear, making it powerful weapon for breaking into the opponent's paths. Therefore, creating such an eye is a sub-goal in the middle game.
However, the player who creates the eye first does not always win. You can minimize the influence of an opponent's eye by surrounding it and cutting off one's connection to the rest of the group, or you can also have a chance to win by creating a more advantageous eye than your opponent. Even so, it is important that you avoid letting your opponent make an eye early in the game.
Back to our game, if I don’t place the stone 12 and Kerry later places it there, the only way to save my stone A is to place a stone towards the southwest and make another path. However, the southwest path can be easily blocked at E, so A will eventually be captured. As a result, the arrangement of the north will be perfect for Kerry to create the "eye."
However, the creation of the eye would have been prevented if I later place a stone on F even if I had left A empty, so it can be said that the move at 12 was also the result of my impatience with the fact that an opponent’s eye was about to be created.
Back to our game, if I don’t place the stone 12 and Kerry later places it there, the only way to save my stone A is to place a stone towards the southwest and make another path. However, the southwest path can be easily blocked at E, so A will eventually be captured. As a result, the arrangement of the north will be perfect for Kerry to create the "eye."
However, the creation of the eye would have been prevented if I later place a stone on F even if I had left A empty, so it can be said that the move at 12 was also the result of my impatience with the fact that an opponent’s eye was about to be created.
Sure enough, Kerry began to cut off my north-south path. If I cannot cut in at A, it will be difficult to maintain my north-south path, but I can't touch A because I have to save the north group first to keep my opponent from making the eye.
Now let us look at what it means to keep a path between groups in this game: in Meridians, a group needs a path, or a line of sight from another friendly group in order to survive, but if a group is trapped in a position where its path is taken away by an opponent's cut, and it cannot make a new path anywhere, it can often prolong its life by connecting to another existing group. Therefore, it is basically important that each group should be able to keep its path and merge to other groups to make a larger group in case of emergency. Ideally, it will be more advantageous if you can merge and cut the path of your opponent at the same time.
Hereafter we will call a set of groups that are "connected" by paths a path-group. As far as my experiences goes, unless the opponent succeeds in creating an "eye," the player who succeeds in largely splitting up the opponent's path-groups while keeping all friendly groups in a single path-group likely wins. This is because the divided player will lose ground by having all the paths taken from a smaller path-group before the opponent's larger path-group runs out of all paths, thus allowing the opponent to create the "eye." Therefore, it is generally a good idea to try to keep your groups as one path-group as much as possible.
Back to our game, my groups are just about to be shut out to two path-groups, so I wanted either to cut at A or extend the north group to the still open northwest direction to keep in touch with the south group. With this in mind, I played 14, but at a glance it seemed hopeless to save my north group, as the enemy stone at B and the line of enemy stones southwest of C have already set up many sight-lines around my north group.
Now let us look at what it means to keep a path between groups in this game: in Meridians, a group needs a path, or a line of sight from another friendly group in order to survive, but if a group is trapped in a position where its path is taken away by an opponent's cut, and it cannot make a new path anywhere, it can often prolong its life by connecting to another existing group. Therefore, it is basically important that each group should be able to keep its path and merge to other groups to make a larger group in case of emergency. Ideally, it will be more advantageous if you can merge and cut the path of your opponent at the same time.
Hereafter we will call a set of groups that are "connected" by paths a path-group. As far as my experiences goes, unless the opponent succeeds in creating an "eye," the player who succeeds in largely splitting up the opponent's path-groups while keeping all friendly groups in a single path-group likely wins. This is because the divided player will lose ground by having all the paths taken from a smaller path-group before the opponent's larger path-group runs out of all paths, thus allowing the opponent to create the "eye." Therefore, it is generally a good idea to try to keep your groups as one path-group as much as possible.
Back to our game, my groups are just about to be shut out to two path-groups, so I wanted either to cut at A or extend the north group to the still open northwest direction to keep in touch with the south group. With this in mind, I played 14, but at a glance it seemed hopeless to save my north group, as the enemy stone at B and the line of enemy stones southwest of C have already set up many sight-lines around my north group.
In this sequence Kerry seems to have made a mistake. 15 seems the move to capture my north group consisting of A, B, and C, but this point is vulnerable with only one path (to D). Perhaps he thought it might be advantageous to have as many groups as possible to have potential paths. However, he could have safely captured the group by placing his stone at E instead of 15, so that even if I gave my north group a path to the southeast, it could be certainly blocked at F and captured.
In fact, I was able to prolong the life of the north group by placing 16 to block 15's only path. Kerry's only options to save 15 are to place a new stone in the south direction to create a path for it, or to place a stone E or G to merge with H. If he decides not to save it, 15 will be captured and empty on my next turn, so I can save the north group by extending it to the southwest.
Kerry chose to extend the life of 15 with 17. I immediately placed 18 to block his 15-17 path as well as enclose 15 and H.
In fact, I was able to prolong the life of the north group by placing 16 to block 15's only path. Kerry's only options to save 15 are to place a new stone in the south direction to create a path for it, or to place a stone E or G to merge with H. If he decides not to save it, 15 will be captured and empty on my next turn, so I can save the north group by extending it to the southwest.
Kerry chose to extend the life of 15 with 17. I immediately placed 18 to block his 15-17 path as well as enclose 15 and H.
Here Kerry chose to put his stone 19 to capture A, abandoning both B and C, which will be captured and removed at the beginning of my next turn! This may look like a bold move, but it is an accurate one. If he had placed a stone anywhere other than 19 to save either B or C, I would have taken D first, and later placed a stone on E or F to capture the enemy stones in the north and eventually complete the eye there. Losing two stones early in the game is painful, but it is probably more disadvantageous to let the opponent make the eye.
This choice of Kerry's also forced me to choose to abandon the stone on A and extend my group to 20 in order to prolong its life. As I mentioned, if the G, H, I group is captured, my opponent will be able to create an eye shape there, so preventing the capture of this group is a priority for me.
I was no longer able to surround Kerry's stones in the north, but as a result I was able to extend my north group to the southwest, leaving my line-of-sight influence in the northwest area. It seems that I was able to make up for some of my mistakes earlier in the game by my opponent's mistakes.
This choice of Kerry's also forced me to choose to abandon the stone on A and extend my group to 20 in order to prolong its life. As I mentioned, if the G, H, I group is captured, my opponent will be able to create an eye shape there, so preventing the capture of this group is a priority for me.
I was no longer able to surround Kerry's stones in the north, but as a result I was able to extend my north group to the southwest, leaving my line-of-sight influence in the northwest area. It seems that I was able to make up for some of my mistakes earlier in the game by my opponent's mistakes.
Kerry tries to extend his power to the west but is unable to, because sight-lines of my stones are concentrated in the northwest, thanks to the preservation of the north path-group. On the contrary, by placing my stones to southwest, I was able to reunite my north and south path-groups into one path-group, dividing my opponent's path-group into northeast and southwest.
Note the positions of 22 and 24. In the future, 22 can be merged with A by B or C, and 24 can be merged with 22 by D or E. This kind of placement, where a stone can be connected to other stone by two points, is familiar to us in connection games such as Hex (yes, Meridians certainly has an aspect as a connection game!). What is interesting is, in this game, in order to place a stone in such a position, you need to have a line of sight from a different stone than the one being connected to.
At this point, the board is not so bad for me, with my path-group almost encircling my opponent's path group of the north. However, I do not have the advantage yet: my little triangle of F, G, H is in a position where my opponent can easily break into it, so it will be difficult for me to complete the encirclement of Kerry's north path-group. Also, even though in the northwest I have the advantage of sight-lines from my stones, Kerry's forces are stronger from southwest to east.
Note the positions of 22 and 24. In the future, 22 can be merged with A by B or C, and 24 can be merged with 22 by D or E. This kind of placement, where a stone can be connected to other stone by two points, is familiar to us in connection games such as Hex (yes, Meridians certainly has an aspect as a connection game!). What is interesting is, in this game, in order to place a stone in such a position, you need to have a line of sight from a different stone than the one being connected to.
At this point, the board is not so bad for me, with my path-group almost encircling my opponent's path group of the north. However, I do not have the advantage yet: my little triangle of F, G, H is in a position where my opponent can easily break into it, so it will be difficult for me to complete the encirclement of Kerry's north path-group. Also, even though in the northwest I have the advantage of sight-lines from my stones, Kerry's forces are stronger from southwest to east.
As I expected, Kerry has cut into the triangle. Now it seems hopeless to connect all stones of the triangle. My proper move here might have been A to keep my opponent's path-group divided, but I was worried about reducing my paths at this point: after placing my stone on A and being blocked at B, it might be difficult to win if I were attacked at the northwest. So, I ventured to place 28. This may seem like a reckless move since this position is in the middle of Kerry's four groups and can be easily captured..., but I did not place the stone out of desperation!
Note here that C is an empty intersection where both I and Kerry have a chance to merge our stones while preventing the other from doing so. And if he wants to capture my stone 28, then needs to place a stone on this C, which will inevitably make his two groups into one group. If this happens, then I can extend the life of 28 by placing a stone on D or E, so he will need to place a stone on F or G to capture it.... Eventually, four of Kerry's five groups, which have kept in touch with each other by paths, would merge into one group with only two paths left, H-27 and H-I!
It is tricky to capture groups that are in touch with each other by many paths, but if it is one big group, depending on the condition of the board, there is a chance to round them all up. In other words, 28 was a strategic move to make the board more favourable to me. The goal of the Meridians is not to capture more stones. Therefore, sacrificing your own stones to reduce the opponent's paths can sometimes be a good move for the strategy.
Middle Game to Endgame
So far, I have commented on the early game to the middle game, and I think I have pretty much achieved my goal of explaining the basic approach to the strategy and how to see the situation on the board in Meridians. So, I will finish this annotation with a brief description of how our game developed up to the endgame.
Note here that C is an empty intersection where both I and Kerry have a chance to merge our stones while preventing the other from doing so. And if he wants to capture my stone 28, then needs to place a stone on this C, which will inevitably make his two groups into one group. If this happens, then I can extend the life of 28 by placing a stone on D or E, so he will need to place a stone on F or G to capture it.... Eventually, four of Kerry's five groups, which have kept in touch with each other by paths, would merge into one group with only two paths left, H-27 and H-I!
It is tricky to capture groups that are in touch with each other by many paths, but if it is one big group, depending on the condition of the board, there is a chance to round them all up. In other words, 28 was a strategic move to make the board more favourable to me. The goal of the Meridians is not to capture more stones. Therefore, sacrificing your own stones to reduce the opponent's paths can sometimes be a good move for the strategy.
Middle Game to Endgame
So far, I have commented on the early game to the middle game, and I think I have pretty much achieved my goal of explaining the basic approach to the strategy and how to see the situation on the board in Meridians. So, I will finish this annotation with a brief description of how our game developed up to the endgame.
As I had planned, Kerry connected three of the four groups into a larger group that has only two paths (to A) at this point. However, out of fear that if I took 37, he would attack from the northwest, I let my opponent take 37 absently and my path-group was divided into three. Considering that I would have lost in the end, I might have had a better chance of winning if I had taken 37 before B....
Kerry threatened me by extending the larger group to the northwest, and I responded by connecting the northwest stone in a circle. This left my largest group with only one path as well, but hopefully it would create an eye to the northwest. However, my small path-group in the northeast is getting isolated.... If it is captured and Kerry creates an "eye" there leading to the large group, I won't stand a chance given the board situation. My remaining chance is to continue to "threatmate" Kerry's largest group by blocking their few paths and prevent them from attacking the northeast.
After connecting the largest group with the smaller group A, B, C, and D, Kerry increased the paths of the largest group by two with a single stone at 59. I, on the other hand, managed to extend the life of the small group in the northeast by extending it while attacking the enemy's largest group. At this point, Kerry's largest group has two paths (E, F), and it will take at least three moves for Kerry to surround my northeast path-group, so Kerry is still unable to attack my northeast group completely.... But with 63 closed, my largest group has now also only two paths (G, H)!
Kerry accurately changed the target to my largest group, extending his largest group and crushing the only path of my group with 67. I have no choice but to place a stone to the northwest to prolong the life of my largest group. If I make a path to the northeast, it will be crushed in one move, so the two paths I made by placing on 70 are effectively the last paths for my largest group. But Kerry managed to give his outstretched largest group two more paths by placing at 71. No matter how I moved, I would not be able to fill in the paths of my opponent's largest group before my largest group was captured, so there was no hope for me. After 71 moves, I gave up and Kerry won.
These are the annotations of the most interesting Meridians’ game I have ever experienced. As I mentioned at the beginning, Meridians is still a young game and there are no established theories yet. And since neither Kerry nor I are professionals, I'm sure we'll find more mistakes in our play as we continue to study the game, but I hope this annotation has given you a basic idea for tactics and strategies.
Meridians is now available for turn-based play at Mindsports.nl thanks to Christian Freeling and Ed van Zon, and also on Ludii thanks to Michael Amundsen (Ludii also has a weak AI). If you are interested in playing, please give it a try! ◾️
These are the annotations of the most interesting Meridians’ game I have ever experienced. As I mentioned at the beginning, Meridians is still a young game and there are no established theories yet. And since neither Kerry nor I are professionals, I'm sure we'll find more mistakes in our play as we continue to study the game, but I hope this annotation has given you a basic idea for tactics and strategies.
Meridians is now available for turn-based play at Mindsports.nl thanks to Christian Freeling and Ed van Zon, and also on Ludii thanks to Michael Amundsen (Ludii also has a weak AI). If you are interested in playing, please give it a try! ◾️
🔴⚫️🔴⚫️🔴⚫️🔴
Meridians is one of several line-of-sight games that emerged in 2020-2021, including Tumbleweed, featured here and in AG21, and Stigmergy and its square cousin Pletore—Stigmergy inspired by Tumbleweed, and in turn Pletore inspired by Stigmergy. The line-of-sight mechanism in Meridians is minimal, and needs no more counting that does identifying the number of eyes in a group in Go. In fact, all you need for Meridians is a set of Go stones and a correctly configured board. The second aspect of the genius of Meridians is the very shape of this Meridians board. Square boards with an even number of squares do not have a single, unique central square; on the other hand, all hex-hex boards have a unique central hexagon. Meridians overcomes this potential imbalance by using hex boards shaped to have a pair of points placed centrally. The strategy must be subtly different depending on closeness to the corners where two long sides meet and corners where a short and a long side meet. ~ Ed.