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Subject: Interesting endgame position rss

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Malcolm S
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Here's an interesting endgame position from a game z vs bennok on Little Golem that I happened to be browsing. It's Orange's play here. I believe that Orange has a winning play available here, although in the game, Orange went on to lose. But it's tricky. What do you think?


http://littlegolem.net/jsp/game/game.jsp?gid=1687449&nmove=1...
 
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Russ Williams
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play on first cell of third row (to wall off space for the main group) and last cell of 3rd row (to threaten sealing off big space on the right from blue's main group)?

this gives 3 moves to blue. if blue spends 3 moves to move along the east and connect to that blue singleton, then orange plays 3 on the northeast to kill that space for blue. if blue does anything else, orange's response includes cutting off that blue singleton? but lots of possible ways it could go, and i didn't spend a lot of time trying to rigorously analyze it.
 
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Malcolm S
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Interesting idea. My intuition is that blue's 3 move play on the east would be too high value for orange to survive the endgame.

It's not too hard to analyze how it would play out with that continuation, starting from orange's move:

Orange: C3 (wall off on left) I3 (partial wall-off on right); now orange's group is size 17
Blue: F8 G7 H6 (connect to singleton on right); now blue's group is size 20
Orange: H1 H2 I2 (the 3 in the northeast)
Blue: B4 B5 (threaten to wall off the only empty hex that isn't adjacent to some player's largest group)
Orange: A6 (deny that threat) A7; now orange's group is size 19

At this point, Blue has next move, Orange has 3 adjacent empty hexes, and Blue has 4 adjacent empty hexes; Blue has group size 20 and Orange has group size 19. So Blue appears solidly ahead, having initiative and a lead in group size and adjacent empty hexes. Remaining moves would be: Blue fills 2 of Orange's, Orange fills 2 of Blue's, Blue fills remaining 1 of Orange's and plays 1 next to his own group, Orange fills remaining 1 of Blue's. So final result is 21 to 19 in Blue's favor.
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Malcolm S
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What I had in mind here:
Spoiler (click to reveal)

Orange: I3 F8 (to complete the wall on the right)


Suppose Blue pushes on the left. This is the first response I thought of, although it's not Blue's optimal response.

Blue: C3 B4 (pushing along the left); largest group now 18
Orange: A5 B5 (blocks on the left) I2 (push up right); largest group now 18
Blue: H2 I1 (defend right); largest group now 20

Now Orange has a catchup move, group sizes are Blue 20 Orange 18, adjacent empty hexes are Blue 3 Orange 4. Orange begins by filling Blue's empty hexes. They then claim two each of Orange's empty hexes, leaving largest group sizes equal at 20, but second largests are Blue 5 Orange 9, so Orange easily wins by tiebreaker after they finish filling in the useless empty hexes on the right.

If Blue doesn't defend on the right and instead plays

Blue: A7 B7 (connects second-largest group in Orange's territory)
Orange: H1 H2 (complete wall in upper-right)
Blue: A6 A9 (fill remainder of Orange's empty hexes)
Orange: G4 E9 (fill remainder of Blue's empty hexes)

Then final largest group sizes are Blue 18 Orange 18, but second largests are Blue 7 Orange 11 even before filling in the useless hexes, so Orange again wins by tiebreaker. (The reason the largest group sizes are also tied in this scenario is that Blue lost 1 point by not defending upper right, but gained 1 point by not giving a catchup move the next turn.)


Alternatively, suppose that instead of Blue responding to Orange's initial move by pushing on the left (which gave a catchup move), instead Blue tries to force Orange to give a catchup move. This will turn out better for Blue, but it's still not quite good enough.

Blue: A6 (threatens to block A5 next turn) I1
Orange: C3 (blocks Blue's main group on left) B5 (prevents losing A5); largest group now 20
Blue: I2 (blocks and connects on upper right) A7 B7 (connects second-largest group, now size 6); largest group now 20
Orange: G4 E9 (fills 2 of Blue's 4 empty hexes and enlarges second-largest group to 8)
Blue: A5 B4 (fills 2 of Orange's 3 empty hexes and enlarges second-largest group to 8)
Orange: H1 H2 (fills Blue's remaining 2 empty hexes)
Blue: A9 (fills Orange's remaining empty hex and enlarges second-largest group to 9) I4 (start trying to limit Orange's second-largest)
Orange: G7 H6 (enlarge second-largest group to 10)
Blue: I5 (ends the game)

Largest group sizes are now both 20, but second-largests are Blue 9 Orange 10, so Orange takes a narrow victory by tie-breaker.
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Justin Blank

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Nice analysis, it looks rightish to me.

I'm pretty sure you can tell that F8 is better than A5 or C3 without sketching out the playout. A5 is just a point (and not even one in the actual playout...), whereas C3 and F8 actually change the potential future points you can get. I think you can say that both moves are between 2 and 3 points, and C3 is slightly bigger. However, since C3 gives Blue an entire point worth of compensation, it's not as good.
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Malcolm S
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How do you calculate the point estimates for C3 and F8?
 
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Nick Bentley
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You guys can't imagine how happy it makes me to see threads like this popping up.
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Justin Blank

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blueblimp wrote:
How do you calculate the point estimates for C3 and F8?


Badly I looked at it today, and realized that I'd been thinking something very silly about the right side. It actually might be less than 2 points--I'm not sure how to count when you have a double ended corridor like that with a stone in the middle.

I'm drawing a loose analogy to counting corridors in go, and I'm not sure how valid it is. Maybe this'll be the thing that pushes me to actually learn combinatorial game theory.

Edit: in fact, the idea of even giving the moves point values is probably a bit misleading, because of the tiebreaking mechanism.
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