Visualize go this way, instantly make it 30 times easier to count

…and the stones are little cubes.


this image is so jarring. Thank you for cursing me with it

personally I’ve just never considered that this might be helpful to anyone… are you visualizing each point as a square and thus visualizing the dual of the graph that is the go board? Not to be rude, but I don’t have a strong visual imagination (always thought “close your eyes and imagine…” was just a way to get kids to shut up), and a bit curious about what ppl who exercise it regularly do


My initial feeling was that this is a ridiculous suggestion. But on second thought, it is somewhat arbitrary to use intersections or its dual of squares. It is only a visualisation choice.

What I like about this visualisation:
Sides and corners cover more visual area. So it is more evident that sides and corners are big territorially. Maybe bigger than you feel when using intersections rather than squares, because half of the side intersections are visually “outside” the edge of the board.

What I don’t like about this visualisation:
Somehow I find it harder to spot ataris with these squares. I feel that outgoing lines are a more clear visualisation of liberties and occupation of those liberties. With squares I only have a less accurate, more general impression of space.


Haha! Just looks so wrong! Nice work though :wink:


How ancient players view and record a game

The stones on top don’t matter, they are just markers. The locations, the intersections, are the concept behind, not to “occupy a square”, but to control intersections of roads. What matters isn’t the area encircled, but how well establish these connections are. How to better visualize the area wasn’t the top priority in our ancestors’ minds. And from my digging and hypothesis, I don’t think the original goal of the precursor games that evolved to Go is about encirclement at all, There are solid evidence and traditions to know that before the time Go was spread to Japan, the goal was about how many stones to “live” on the board. The bigger encircled area is just convenient for not having to play until the board is filled.


I addressed this question several years ago in this discussion: Tafl game Alea Evangelii hints at 11th century go influence in Europe? - #4 by Conrad_Melville
Specifically: “I have been unable to find any theory about the origin of intersections versus squares in games, but I offer my own speculation, which I think is plausible and even fairly obvious. I suggest that badly drawn lines are less disturbing aesthetically than badly drawn squares, and therefore are easier to make and preferable to use for the common people in ancient times. Aristocrats, on the other hand, could afford to have craftsmen make boards with neat squares. The antiquity of Go is consistent with this, and I suspect that it originated among the common people (contrary to legend).”


I agree that it would make more sense Go has a more mundane origin, have some roots related to simple commoner’s games. I looked up a lot of ancient sources and have some theorized evolving path for Go discussed here

Although I don’t feel it is the “difficulty” or aesthetic reasons of drawing squares prompt placing stones on intersections. Ancient unearthed Go boards are huge compared to modern boards. They can be up to nearly 70 cm on each side, and at least around 60cm with 17x17 or fewer “roads”. This matches ancient sources refering Go as 三尺之局 - a game of 3 ancient Chinese feet. However, the stones unearthed are about the same size as modern ones. To me, it is pretty obvious ancient players don’t view “inside the grid” as the place of contention, but the lines and intersections are.

Also, in my hypothesis, the precursor of Go, might not be a game played on a grid-like pattern board. But like Merels game where they are about the movement and blocking/capturing of stones. Where the lines indicate where the legal movement directions can be. This is also very telling if you saw a Chinese Chess board, also played on the intersections. With a weird “center of base”, I feel they are also a clue as to relics for earlier existing games why pieces are treated like markers, and moving on roads instead of occupying territory.


also - counting by two in Japanese rules means that a captured stone is one two-count, just like two empty spaces.

it works a lot better for me. for some reason my only remaining problem is if there are an odd number of points in a territory and I have to carry it. I don’t know why that’s so damn hard

I do really like the idea of keeping the difference rather than the totals - but I just can’t seem to make myself do it in practice


another thing I haven’t practiced in a while, but was pushed when I did AYD under Inseong Hwang is to recognize all the square shapes 1,4,9,16,25 (hopefully you don’t need 36 – or hopefully you do), as well as common 5,6,10,12, etc shapes to actually speed up the counting.

One thing I did multiple times IRL (particularly towards endgame) is to count Chinese style cuz you only need to count one side, so you can’t forget a second number and just look at how much you need to secure to have a victory.


Ah! This also makes sense. I do count the stones this way as well, but forgot.

I don’t think I actually understand Chinese style counting. It is something I should learn! I also don’t 100% understand your comment about memorizing all of the common square shapes. Is this just doing things like length x width? I know a few maths people who count like that.

I’m not sure how much I recommend Chinese counting exactly, as it is counting as if chinese scoring (territory+stones). Basically because there are 361 points on the board and 7.5 komi in Chinese (and only seki makes it differ sometimes), you can know if you won (in Chinese Rules) if you have more than half of all the points on the board (361+7.5)/2 = 368.5/2 = 184.25, meaning that if you count that you have some number of points (say, 160) then you know how much you need to win (in this case 25 to make 185>184.25), and then look for a way to get that (maybe estimating how much space there is left in dame/small endgame/whatever)

for the squares themselves, just recognizing any square of territory and how much it’s worth, like you see a lot of 3x3 squares when counting and you wanna recognize instantly that it’s 9 cuz that speeds things along. 10, 12, and 5 shapes are also very common and can be an aid to counting if you recognize them (especially the 10s). The details of the technique aren’t really something I feel capable of teaching right now, although I think Inseong does a great job if you can catch one of his lectures.


There are whole books that talked about evaluation and estimation.

Counting is not just about counting, everyone has the time can add numbers up, but also count where and how much confidence is the estimation. Block estimation can be more useful where the situation judgment isn’t about the counting difference, but more about the overall situation and how much space left can be done to pull back, or play it safe. And there are counting needed in details like yose and ko situations, or a big swap of positions. Do enough rough estimation and build intuition about how each closure or extension, etc. can gain is also very important, how the edges and borders between groups before they are settled, and get an idea of how valuable different moves are.


The lines have a purpose. If you have problems counting territory with stone on intersections, I can imagine you will have bigger problems counting liberties, or reading cut/connect when stones are in squares.

A good math student will see this is a duality. Intersections and squares are totally equivalent, just a point of view. Thanks for bringing up this interesting topic. Everyone should train themselves by switching foreground and background when appreciating music or visual art.

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I’m not sure just recognising a word/concept would qualify someone as being a good math student.

The two 19x19 type boards probably aren’t dual in any strict sense (like in a graph theory sense) since the numbers of faces in a direction don’t match the number of lines in a direction. Similarly not being translationally invariant, as if the board were a torus, means the two 19x19 boards aren’t dual in a lattice sense either.

In any case, the fact it’s even a regular grid isn’t what’s important, it’s purely the adjacency. One could skew all the squares out of shape into various quadrilaterals maintaining the adjacency and it wouldn’t matter to the rules, but it would certainly impact how one reads and counts.

This is the point, presentation can matter, and to me, I could see that once I tried putting stones in the squares, it did seem to lead to easier estimation of for example corner territories. I’m not necessarily advocating that that’s how it should be but it’s just an observation that was made and I kind of agree with it at the moment at least.


One main property of duality is that it’s its own inverse: taking the dual of the dual should give you back your original.

I’m not seeing either in which way the squares-board or the intersections-board are dual. They’re equivalent representations of the same idea (thus a good mathematics student might find the way by which you count, intersections or squares, a rather uninteresting distinction), but not dual to each other. Here’s what the graph theoretic dual of a 19x19 board looks like:

It looks a bit nicer if we move that one point in the top right to infinity:


Is that, like, trolling to (e.g.) Ride of the Valkeries?


This has nothing to do with the specific concept of duality from optimization theory. Further, one does not have to use other formal mathematical notions of duality to understand this basic equivalence.

I think it is obvious to everyone (not just “good math students”) that these are just two different graphical designs for presenting the same information.


I’ve being thinking a lot, what would Go be like if not playing on a grid like “board”, on a hex honeycomb pattern board? On spider web like ring? there seems to be endless possibilities.

Playing on a hex grid would greatly change the game, it doesn’t seem to have ko, and ladder? not even seeming false eyes. Would it be more intuitive?

There are some scattered articles on Sensei’s library that document experiments with unusual gobans:

I’ve played a couple of games on the surface of a cube, which can be viewed as a form of “edgeless go” but without the supersymmetry of toroidal Go.

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Depending on whom you ask, that might be Romantic trolling…