Senket — Scoring Resolved

I have an answer for scoring. It doesn’t involve changing the text of the rules, just careful application of them. It may result in some odd scores for some very artificial board layouts, but since they don’t seem likely to ever turn up in actual gameplay, I’m comfortable with this as a solution. Here’s an example board that poses the question, how do I score this?

 

How to score this?

How to score this?

Here’s the sentence in the rules that describes how to identify territories: Territory is any region of the board your fences surround that does not contain any territory of your opponent.” So consider how to handle the above. There are two unequivocal territories: the one point blue in the upper right, and the one point red in the lower left. It only makes sense to work out from those, and it’s only fair to do it simultaneously. So at the same time, the one point blue in the upper right precludes the red three-post fence in the upper right from claiming territory, and the one point red in the lower left precludes the long blue fence from claiming territory. So this board is scored correctly as it appears: one point each for red and blue.

In the pathological case:

 

Not as hard as it looks.

Not as hard as it looks.

The one point red in the upper left and the one point blue in the lower right are the starting points. They invalidate the blue fence that starts at 1-9 and the red fence that starts at 7-1. That means that the red fence that starts at 1-8 and the blue fence starting at 5-1 are both valid and make territory. But that in turn means that the blue fence starting at 1-7 and the red fence starting at 3-1 are both invalid. And that means that the red fence starting at 1-6 and the blue fence starting at 1-1 are valid and make territory. And that means that the blue fence starting at 1-5 and the red fence starting at 1-2 are invalid, and finally the red fence starting at 1-4 and the blue fence starting at 1-3 are valid. So each player claims the same amount of territory and prisoners.

Here’s another example:

 

Who wins?

Who wins?

EDIT: The rule is subtle, and I didn’t apply it correctly here. This is wrong:

The critical point to understand is the lower left corner. Without that, Blue is toast. But Blue’s one point territory there invalidates Red’s fence from 1-3 to 5-1. The same thing happens in the upper right, leading to a score that looks like this:

Red's mistakes in the upper right and lower left are fatal.

Red's mistakes in the upper right and lower left are fatal.

Blue’s largest territory is worth (26 + 3)^2 = 841 points, while Red’s largest is just 25^2 = 625 points. Each of them have a 1 point territory, but Blue has an additional territory worth 9^2 = 81 points. Blue wins handily. If Red had managed to make territory in the lower left, he would have won easily.

 

Corrected Version

The lower left is critical, as I said, but the upper right is tricky as well. It is bounded by two unequivocal territories: the main Red territory in the center, and the one point Blue in the upper right. Working from each of those in, the Red invalidates the Blue fence from 8-11 to 11-5 at the same time as the Blue invalidates the Red fence from 9-11 to 11-7. The result looks like this:

 

Blue wins, but he doesn't get 81 points in the upper right the way I thought he did.

Blue wins, but he doesn't get 81 points in the upper right the way I thought he did.

 

 

Blue still wins, but he earns only 1 point in the upper right instead of 81.

I should add that these territories were generated automatically. I have that coded at this point. I now need to get the territories marked for area instead of post scoring, and generate scores automatically.

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4 thoughts on “Senket — Scoring Resolved

  1. Lars

    I have not read all your previous posts about Senket scoring, but …

    It seems to me that your “wrong” initial stab at the final example is correct, and your “corrected” one is misled. (I’m ready to stand corrected, because I know you’ve thought longer about this than I have.) Here’s why I think so.

    Your “corrected” analysis relies on a process of “invalidating fences”. It seems to me that this process is not justified by the rules. The rule for territory, as you know intimately, says “Territory is any region of the board your fences surround that does not contain any territory of your opponent.” Note in particular that it does not say “Territory is any region of the board surrounded by a fence of yours that does not surround any territory of your opponent.”

    The rule as written says that *regions* get “invalidated” (if they contain the other player’s territory), but why should *fences* then get “invalidated”? Every fence “surrounds” two regions, right? So if one of a fence’s regions gets invalidated, why should the fence (and thereby its other region) be invalidated?

    Thus where you say “the Red [central territory] invalidates the Blue fence from 8-11 to 11-5”, I don’t see how that’s justified by the rules. Sure, the Red territory invalidates the *leftward region* of the Blue fence from 8-11 to 11-5, but it does not affect the rightward region of the same Blue fence. That rightward region is still a “region of the board your [Blue’s] fences surround that does not contain any territory of your opponent.”

    So, while you propose this “answer for scoring” as a solution that “doesn’t involve changing the text of the rules, just careful application of them”, it seems to me that it *does* involve changing the rules, and does not arise from mere careful application of them.

    Again, my apologies if I’m glossing over issues you’ve already dealt with in other posts about Senket scoring. I’m going to go read those other posts out of curiosity (especially “Scoring Senket is Hard” parts 1 & 2). At this point I’m hopeful that you can resolve ambiguities about scoring Senket without introducing complicating factors, like having the scoring be dependent on an iterative algorithm that starts with outermost fences and invalidates opposite ones simultaneously. (An algorithm like that might be *useful* for calculating score, but if the game is meant to be simple, the scoring must not be dependent on a complex algorithm. IMHO.)

    Reply
  2. Lars

    OK, after reading scoring-senket-is-hard-pt-ii, I’m not as hopeful that the ambiguity can be resolved without changing the rules. The algorithm you suggest above sounds fair, and it may be the best resolution, but it does augment the rules for territory. As you pointed out in scoring-senket-is-hard-pt-ii, this simple 4-fence case threatens a contradiction: that the region containing point 9-9 is both Red and Blue territory. But of course it can’t be both.

    Your procedure above for invalidating fences is fair and unambiguous, as far as I can see; but if you decide this procedure is authoritative, I think you’ll have to add something to the rules to make it agree with the result of this procedure.

    Also, I think instead of saying “the one point blue in the upper right precludes the red three-post fence in the upper right from claiming territory”, it would be more accurate to say “the one point blue in the upper right precludes the red three-post fence in the upper right from claiming territory *to the upper right*”. It’s apparent from the example that this is what you mean, and in this example it makes no difference; but when the rule is elaborated and applied to your 3rd example, it becomes clear that it *does* make a difference. (This is along the same lines as what I was saying in a previous comment about invalidating *regions*, not fences.)

    How to adjust the rules to fit this procedure? Maybe something like the following:
    “Territory is any region of the board your fences surround that does not contain any territory of your opponent. Regions that could be claimed by both sides belong to no one’s territory.”

    I think this would be equivalent to the algorithm where you start from the obvious corner cases (regions that contain no opponents’ regions), and invalidate regions based on that. When you check whether a region is surrounded by blue and doesn’t contain any red territory, you’ll also have to check whether it’s surrounded by red and doesn’t contain any blue territory.

    Reply
    1. gcanyon Post author

      You are absolutely correct about terminology: I should always say that a territory precludes a fence from claiming territory on that side (whether it is inside, left or right, etc.).

      The recursive nature of the application of the rule doesn’t bother me so much since (with a little thought) it flows from the rule fairly easily. It’s like the principles of snapback and seki in Go: neither is explicit in the rules, but both flow directly from the rules.

      That said, the recursive application of invalidation does feel a bit like a hack. I’m tempted to simply go with something like what you said, “Territory is any region of the board your fences surround that does not contain any territory of your opponent. Regions that could be claimed by both sides belong to no one’s territory.”

      I hesitate because it changes how scoring is done, since (for example) in the pathological case it would mean that each side claims just the one unambiguous point of territory. That feels wrong to me because what happens in the middle should matter in a game like that.

      I’m also unhappy simply with the number of words it takes to describe it 😉 Maybe this is equivalent: “Territory is any region that can reach only one side of your posts that bound it, but which can reach both sides of all your opponent’s posts within it.” Although that seems confusing, and is still too long to make me happy.

      All of this is tempered by the fact that I’ve played about a hundred games of Senket and never come across a situation where recursion was needed or applied. That’s not to say that it won’t come up in more advanced games, but I think it’s unlikely. It’s very easy to make territory, and hard to prevent it. Senket is about making the biggest territory, not about surviving. Unlike Go where you can have large wars over life and death, in Senket every time I’ve tried to set up a puzzle like that, I fail. But again, that may be because of my limited perception. I’m the undisputed Senket world champion, but I still think I suck at it!

      Reply

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