Life in Blooms: 3rd eyes, minimal life, and one-sided seki

game-2-85-bgg

I’m trying to understand a game I invented called Blooms, and I’m sharing my thoughts here in the hope readers will offer criticism as a bulwark against my looming idiocy. Feedback welcome.

Below are observations about how a concept called life works in Blooms, compared to how it works in Go, its closest relative. It’ll help to know Blooms’ (short) rules and the basics of Go to understand what follows. But you can squeak by with these facts:

In both Blooms and Go you claim territory on a board by surrounding groups of empty spaces with your stones:

territory
An example of territory in Blooms (left) and Go (right). On the left, one player owns both the red and the yellow stones, so she owns the 3 points of territory surrounded by those colors. Likewise on the right, one player owns the white stones and owns the 4 intersection points surrounded by white (in Go you count the intersections, not the spaces).

Your opponent can ruin your plans by capturing your stones. You capture an enemy group when there are no empty spaces adjacent to it:

capture
The yellow bloom and the white group will be captured because there are no empty spaces next to them.

So it’s important to prevent your opponent from capturing your groups. You do that by making groups your opponent can’t capture. Such a group is said to be alive. That’s what the rest of this post is about: how to make living groups in Blooms, and how it compares to making living groups in Go. (Note: in Blooms, groups are called “blooms” – I’ll refer to them as such from now on for clarity).

Now let’s discuss life:

3-eyed blooms are the equivalent of 2-eyed groups in Go

One way to make a living group in Go is to make sure it surrounds at least two non-adjacent empty spaces, called eyes:

 

two-clear-eyes-3

In Go you’re not allowed to fill an eye unless it causes an immediate capture. But since you can only place one stone each turn, you can’t fill either eye of a two-eyed group. Thus a group with two eyes is alive.

However in Blooms, you can place two stones per turn, instead of one. So you can capture a bloom with two eyes by filling them both on the same turn. Therefore, a bloom needs three eyes instead of two to be safe, for example like so:

3-safe-eyes

The minimum number of turns needed to make enough eyes to live is smaller in Blooms than in Go

Here are the minimum number of turns required to make a 2 eyed group on different parts of the board in Go:

Corner of the board – 6 turns:

go-corner-group

Side of the board – 8 turns:

go-min-side

Interior of the board – 10 turns:

go-min-interior

Now, for comparison, below are the minimum number of turns required to make the equivalent 3-eyed blooms in Blooms. To understand these pictures, you should know that in Blooms, each player owns two stone colors. If you place two stones on your turn, they must be different colors. In all the following pictures, the two colors shown belong to the same player.

Corner of the board: 5 turns, with 1 stone left over to place elsewhere:

3-eye-min-corner

Side of the board: 6 turns, with 2 stones left over:

3-eye-min-side

You may wonder why, in each of the pictures above, I say the blooms are alive when clearly the grey stones are capturable. The reason is, if the opponent tried to capture those grey stones, the Grey-Black player can replace them with black stones without losing initiative elsewhere, and thus ensure the black bloom lives. I won’t show this sequence so you’ll have to trust me or prove it to yourself.

Interior of the board: 7 turns, with no stones left over:

3-eye-min-center

In summary:

chart

Two Observations:

  1. You need fewer turns to make enough eyes to live in Blooms than in Go. You might thus conclude it’s easier to make living groups in Blooms than in Go. However, other factors complicate things. For example: it also takes fewer turns to capture a bloom than a group in Go. Another factor: to make the 3-eyed blooms above, you have to place your own stones of different colors next to each other, which means your 2 colors partly surround each other. Self-surrounding makes it easier for your opponent to capture them before they’re alive. In fact, if anything, securing life is harder in Blooms than in Go.
  2. The differences in number of turns needed to secure life on different parts of the board are smaller in Blooms than in Go, which will affect the arc of the game. In Go, players tend to start the game by placing stones where it’s easiest to make life: the most common openings are near the corners, the next most common are near the sides, and the least common are in interior. Here’s a heat map of opening moves distilled from a popular online Go server illustrating this:

fuseki_heatmap

So players tend to start near the perimeter and progress inward.

In Blooms, the differences in making life between corner, side, and interior aren’t as pronounced, so openings will likely be spread a little more evenly across the board (but still with a bias toward the perimeter). There’s a caveat though: right now, Blooms is mainly played on small boards where there isn’t much distinction between interior and perimeter. On those boards, we expect openings to cluster near the center (since it confers the advantages of both interior and perimeter), and in fact that’s what I’ve seen so far. For this reason, Blooms may only reach full potential on larger boards where openings can be fully varied. How large remains to be seen (likely smaller than a full size Go board).

One wrinkle: the board I’ve played Blooms on most is a hexagonal board with five spaces on each side (see the gif at the top of this post), and on that board, the minimal bloom ratios change slightly. Specifically, you can make a three-eyed bloom spanning two corners in 4 turns, with no stones left over:

3-eyes-min-hh5

I doubt this will have a major effect on the game, but I mention it for completeness.

A new kind of seki: one-sided seki

In Go there’s a situation called seki, where two enemy groups share adjacent empty spaces neither player can fill without dooming her group:

seki-go

Since neither player fills them, both groups live. Blooms has seki too, but it also has a kind of quasi-seki that doesn’t exist in Go (as far as I know):

one-sided-seki

In the above picture, one player is Red-Yellow and the other is Grey-Black. Here, the black bloom lives: there are three spaces to fill around the black bloom, which means it would take 2 turns for Red-Yellow to capture it. But Red-Yellow can’t fill them because she would create surrounded red or yellow blooms for a turn, which is illegal. So Red-Yellow’s red stones get in her way here.

Grey-Black could fill the spaces next to the red stones to capture them, but doesn’t want to, so it’s a seki-like situation for her. Note: Grey-Black has a chance to capture the red and yellow blooms down the road, so the black bloom is thus more of a threat to the red and yellow blooms than vice-versa!

Red-Yellow’s weakness in this situation illustrates a key difference between Blooms and Go: in Blooms, your own stones can screw you. It also leads us to a strategy aphorism:

Don’t put your colors where they’ll doom each other, and make your opponent put hers where they will.

One-sided seki happens in real games, and it affects the difficulty of making life in Blooms, by providing an extra way to live. In fact, the game shown in the gif at the top of this post ended with just such a situation. Here’s the final board state:

end-one-sided-seki

Look at the lone yellow stone on the second row from the bottom. Thanks to that stone, one-sided seki protects the black bloom. If that yellow stone were red, the Red-Yellow player could eventually capture the black bloom and win in dominating fashion. That’s why Grey-Black decided not to capture the yellow stone: if the yellow stone were removed, Red-Yellow could replace it with a red stone, thus dooming the black bloom. As it was, Red-Yellow won narrowly, 31-29.

One-sided seki exists in Blooms because self-capture is illegal, as in Go. But I wonder: should it be illegal in Blooms? I don’t understand the implications, except it would make the rules more complicated (to prevent cycles, at least).

Simple Ko: not a thing in Blooms

I’ll write about this in a future post, but I mention it here because it affects life: in Blooms, simple Ko doesn’t happen. Ko is a situation in Go where alternating captures would lead to a board position identical to what it was before the first capture. This could lead to an infinite cycle of alternating captures between the players, the simplest of which looks like this:

simple-ko-400

Go has a rule to prevent such cycles, and it protects groups in certain situations where they would otherwise die. Since Ko doesn’t happen in Blooms, Blooms has no such rule, and blooms have no such protection. This makes life harder to secure in Blooms than it otherwise would be.

There’s lots more to say but there’s no way anyone has read this far so I’ll stop now.

Try Blooms for yourself

You can play Blooms against a weakish AI, using Stephen Tavener’s AiAi system (download here). To run it, you’ll need Java and you may need to change your security preferences. You’ll get a bunch of zipped files in the download. The file to run is “ai ai.jar”. Once opened, load Blooms by going to: File –> Choose Game, then select Blooms.mgl. You can change the AI settings from the AI menu. Note: by default the AI is set to think for only 1 second, which leads it to play essentially randomly. You’ll need to increase the AI thinking time if you want it to do anything interesting.

Kickstarter?

There’s a chance I’ll make a boxed version of Blooms through Kickstarter. Sign up here to receive an alert if it happens. The more people sign up, the more likely I am to do it. I have an idea about how to make an unusually beautiful physical set.

2 thoughts on “Life in Blooms: 3rd eyes, minimal life, and one-sided seki

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