Pyramid go 🔺

Go Variants
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One natural way to generalize scoring in go is to assign different point values to different intersections. This can be done in lots of different ways, but I think a simple pattern is preferable, so that it’s easy to keep track of point values without some sort of special overlay.

My suggestion is to start with a value of 1 along the first line, and then increase by 1 point per line all the way into tengen. On a 9x9 board the point values would be distributed like so:
image

My working title for this variant is “Pyramid go”, since the point values could also be visualized like a square pyramid:
image

Area scoring is used (or should I say volume scoring? :wink:): In the position below black has 16Ă—3 + 8Ă—4 + 5 = 85 points, while white has 32Ă—1 + 24Ă—2 = 80 points, so black is winning (assuming no komi).

image
image728Ă—728 142 KB

If we were to change one of the 3-3 points from black to white, this would reverse the result (the new score would be black 82 to white 83).

Although I’ve used a 9x9 board to demonstrate the idea, I think bigger boards would be even more interesting. I expect a 19x19 game to start with fighting in the center, with tengen quite possibly being the optimal first move.

Pyramid go could be played like a normal game on OGS, except you will have to do the scoring manually or with some separate tool (maybe I’ll even try to figure out how to parse SGF files if I’m feeling adventurous :stuck_out_tongue:).

Anybody interested in trying it out? :slight_smile: I’m open to any board size, time settings, etc.

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Edited

The issue I see is that black is winning in the image after playing 16 stones, while white is still losing after playing 20 (maybe that’s not a real issue maybe just a bad illustration :stuck_out_tongue: - it’s a good scoring illustration though)

Tengen was already a good starting point on 9x9 :slight_smile: so would the komi needed be large?

What about reversing the pyramid scores giving 5 points to the first line and 1 point to tengen?

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Or wait maybe you just add like ~4 (2?, 1?) points to the usual 7.5 komi and it’s ok?

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Indeed, hence why I think it will do better on bigger boards! Although my example position is not realistic, I think it’s safe to say that the first move is extremely valuable on 9x9. Making komi bigger can make the game fair, but it will still be a bit boring (too similar to regular 9x9 game). On 19x19, it will not be possible for one side to claim all of the center, and it will look very different from a normal game.

I certainly wouldn’t mind trying other distributions later though! Two other playable ideas:

  • Star points worth 10, other points worth 1.
  • 1st to 4th line worth 0 points, rest of board worth 1.
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:slight_smile:

But back to the game, it would be very interesting on like 19x19.

So tengen is 19 points. I guess enclosing the corner, normally ~ 15 points in two moves is probably going to be worth less than an empty triangle in the middle? :smiley:

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Staircase Go: Intersections in the i-th row are worth i points :laughing:

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10 points! Also, a corner enclosure is worth more than usual too (in absolute terms). But Tengen will still be super valuable, because it exerts influence over all the most valuable points on the board.

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Neat idea, I’d like to play sometime :slightly_smiling_face:

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Doh, silly me :stuck_out_tongue:

What actually happens with mirror go in this variant? Would black win by 1 point without komi or?

I might watch the first game if yous give it a try :slight_smile:

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Mirror go can be broken in all the normal ways, although the considerations on how to to do it in the best way (without giving the opponent a good chance to stop mirroring after making you play some bad move) would be a bit different from usual.

If black opens on tengen and mirrors for the rest of the game, and white doesn’t break it, black will win by 10 points (the value of tengen) assuming no komi.

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I can’t do numbers today.

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Me and @martin3141 are planning to try this variant out in a live game later today, probably around 2021-06-23T16:30:00Z!

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The game just started:

We used a pie rule instead of komi: Martin decided that G7 would be the first move, and I then decided to take black.

Edit: It was an exciting game! Things started in the center and moved outwards, as expected in this variant. Martin was ahead by alot in the middlegame, but towards the end we got into a close capturing race which I won by 1 liberty. The final score was B733 - W597.

Final position in custom scoring tool

(This hypothetical board shows that white is easily winning as long as the top group doesn’t die)

If anyone else wants to try this variant, the scoring tool should be pretty easy to use - just click (or click and drag) to place stones, hold down shift to place white stones. Territories are detected automatically, but you need to remove all dead stones from the board.

Other board sizes should be working as well, just add the size to the link like so:
https://pyramid.4tc.xyz/?size=9

This was very hastily thrown together so there might well be some bugs that I missed :slight_smile:

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I’ve been thinking about how this fun alternative scoring rule could be extended to any board (in other words a general graph). I have an idea in mind but would love to hear your thoughts about it. What do you think about this proposition?

Definition:

The distance of two vertices v, w of a connected graph is the length of a shortest v-w-path.

The excentricity of a vertex v is the maximum distance of v to another vertex.

The diameter of a graph is the maximum excentricity among its vertices.

Let G be a graph and v one of its vertices, let d(G) denote the diameter of G, ex(v) denote the excentricity of v.
The pyramid-go-value of v is 1 + d(G) - ex(v)

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Does this definition give the same values for square boards as your original proposal? I’m not sure if I’ve understood it, and maybe something simpler to understand would be better. Could you provide some examples to justify this particular choice?

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I thought this is the case but now that you ask I believe it may not be the case, let’s see …

The diameter of a square nxn board is the distance of two diagonal corner vertices, i.e. 2(n - 1)
Lets say n is odd, then the vertex of minimal excentricity is Tengen with ex(v) = n - 1

I think the point values would form this pattern:

pyramid
pyramid876Ă—877 227 KB

… so unfortunately its not compatible with pyramid go :pensive:

But it’s applicable to any graph.

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To be clear, this is not my original idea. See for example https://en.wikipedia.org/wiki/Graph_center

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This certainly seems like the most natural generalization of the “pyramid” idea! (even though it doesn’t quite agree on square boards, it’s obviously very similar)

And for a disconnected graph it seems most natural to do the same thing component-wise.

(not that I think the pyramid-scoring has any particular benefit over other interesting scoring distributions one could come up with)

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Like some scoring based on

? :smiley:

If you made the scoring for an intersection in some way opposite to how early it tends to get played in a game, one could get some sort of anti-go, at least in the usual learned sense :stuck_out_tongue:

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If instead of generalising to any graph, you generalise to any graph which has a boundary, then it becomes easier. The value of an intersection in pyramid-go is simply its distance to the boundary.

You can either draw the boundary around the board, or if your board is almost-a-grid, you can perhaps say:

  • Every intersection which has fewer than 4 liberties is considered to be at distance 1 from the boundary;
  • The distance of an intersection u to the boundary is one plus the minimum distance of u to a vertex at distance one from the boundary.
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