Senket — A Complex Problem

Has Red seen something Blue missed?

Has Red seen something Blue missed?

The Question
Blue is looking to take almost 40 squares of territory in the corner — note the lone sentinal guarding the corner itself. That’s worth 1600 points even if he doesn’t add to it outside.
Red invades with 1, and Blue responds with 2, thinking that Red is handing him a prisoner. Who’s right, Red or Blue?

Answer

Blue can't stop Red now.

Blue has made a mistake and is about to pay.

First, a simple demonstration that Blue cannot capture Red:

Red 3 threatens to make territory at 4, so Blue plays there.

Red 5 does the trick: Red can now make territory at A, B, or C. Blue can’t even stop two of those options, let alone all three; Red will make territory on the next move if he chooses to.

One way it might play out.

One way it might play out.

Here is one way the local conflict might play out. Blue accepts his losses and plays at 6, closing Red in and giving him a strong position on the left side.

Red plays at 7, taking 2 squares and also threatening to take 2 more at 9.

Blue 8 takes 9 squares.

Red 9 takes 2 squares.

The original play at B remains up for grabs: Blue can play there to deny Red 2 squares or just seal off the left side as he chooses.

Red ends up with a territory worth 16 or 36 points, and reduces Blue from 1600 points to about 600, so this isn’t a good outcome for Blue. But Red can do better…

A Better Answer

Red plays smarter, Blue suffers more.

Red plays smarter, Blue suffers more.

Red plays 3, threatening to make territory immediately at 4. Blue plays 4, blocking and taking a square for himself.

Red responds with 5, threatening to make territory at A, which Blue blocks with 6, taking 5 more squares.

Red collects his due with 7, after which he can make territory either at 9 or B. Blue can only block one of them.

Instead of blocking directly, Blue plays 8, sealing off the damage. Red 9 takes 4 squares, ending Blue’s hopes of a large territory.

The end result is 16 points for Red (or 49 if he plays at B), and over 300 for Blue after he seals off the left completely.

So Red did some damage — he reduced Blue by 1300 points — but he can do better.

A Deep, Even Better Answer

The key points to really hurting Blue.

The key points to really hurting Blue.

The key points to really hurting Blue are A and B. Using either of those, Blue can seal off the upper left, salvaging roughly 12 squares.

Red wants to take those points away. He has no legitimate way to do so — Blue needs just one move to stop him, and he needs two to break through. So he needs to trick Blue. Blue has to think that Red will lose the fight completely, leaving nothing but prisoners.

If Blue falls for it, Red cuts him off at the knees.

If Blue falls for it, Red cuts him off at the knees.

To pull this off, Red starts with 3, threatening to make territory at 4, which Blue obligingly blocks. Red pushes at 5 (the first of the two critical points). Blue blocks at 6, dreaming of prisoners. Red at A isn’t a threat because Blue can make territory in the corner, invalidating Red’s territory.

Then Red 7 threatens to take territory at B and C, and Blue starts to get nervous. Blue 8 blocks B, and A and C are both still answered by Blue making territory in the corner, but Red seems to have a purpose…

Red 9 does the damage. Red can now make 3 squares of territory at A or 2 squares at C. Blue can’t stop both, and taking a point in the corner won’t help. To make matters worse, 9 threatens to break out at the top. Resigned to his suffering, Blue 10 seals off the damage.

Blue is wishing he had points instead of regrets.

Blue is wishing he had points instead of regrets.

Red 11 prevents Blue 12 from taking 9 squares. Blue 12 salvages 4 squares and prevents Red from taking 4 squares, so it is worth 8 overall. Red 13  takes away 6 squares.

Blue has no answer but 14, threatening to take 6 squares, and Red 15 prevents 4 of those and threatens to take 4 of his own.

Blue 16 (finally) takes away one of Red’s options for making territory, so Red has to play at 17, or he will end up as prisoners the way Blue originally hoped. Blue has only one option left as well, so Blue 18 takes three squares. Note that Red 17 at 18 would prevent Blue from making territory at the bottom, but then Blue would make 2 squares connecting up to the left from 16, and Red would have no way to make territory himself, again leaving him as prisoners.

So what was once about 1600 points for Blue is now (4^2 = 16) + (3^2 = 9), a total of 25 points for Blue, 4 for Red, so Red has stolen almost all of Blue’s points.

The question is, how likely is Blue to fall for this? The critical moment was Red 5. If Blue simply closes Red in, allowing Red to make territory at 6, the outcome is much better for Blue.

A Tricky Solution

This doesn't look good for Red -- which is what he wants Blue to think.

This doesn’t look good for Red — just what he wants Blue to think.

The previous solution isn’t too likely: Blue would likely see where Red was headed and cut him off after 5, or perhaps after 3. Here’s a variation that’s a little more subtle, and hence more likely to fool Blue.

Red starts with 3, and Blue blocks with 4, making territory in the corner immediately. Red 5 is blocked by Blue 6, and Red 7 is blocked by Blue 8. Finally, Red 9 threatens to make territory at A, but Blue 10 shuts that down.

Blue is more likely to let Red through with this sequence because it doesn’t look like he’s going to be able to make territory. The critical plays are again Red 5 and Red 7. When Red plays 5, Blue would do better to close Red in. But it certainly looks like Blue will be able to prevent Red from making territory inside. The trick is that there is no trick: Red won’t be able to make territory — but he’ll survive anyway.

Blue realizes he has a problem.

Blue realizes he has a problem.

Red 11 shows Blue the problem: he has no way to cut in between the territory Red will make in the upper left and the lone fence below, and if he can’t get between them, the fence is safe.

Blue 12 is a good effort. It threatens to cut off Red and then loop around to connect to Blue’s fence outside somewhere down to the right, so Red 13 makes territory immediately. That’s not enough, of course — he still has to make sure Blue doesn’t get to the wall betwen that territory and the ailing fence below. Blue 14 is the kicker. Now Blue can connect at A or B to cut Red off. B immediately makes territory and takes Red prisoner. A connects to 12, and from there Red will have a hard time preventing Blue from joining his two fences.

Unfortunately for Blue, Red 15 blocks both connections at once. Note that 15 at C doesn’t block A, and will fail.

Blue licks his wounds.

Blue licks his wounds.

Finishing up, Blue 16 picks up 7 squares, and Red 17 prevents 5 squares.

The result is 7^2 = 49 + 1 = 50 points for Blue, and 4 points for Red, not quite as good as the previous solution, but this sequence has the advantage that Blue is much more likely to fall for it rather than just locking up Red early on. It is also better because Red made it to the outside, giving him a better position above.

Conclusion

In “How Deep is Senket?” I talked about the overall game complexity of Senket. This problem speaks to the local depth of Senket. The sequences shown all involve looking forward 10-15 moves or more. This problem also shows that it is possible for a fence to be safe even if it surrounds no territory. Blue cannot surround Red’s fence at the bottom without also surrounding Red’s 2 point territory on the upper left.

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