Difference between revisions of "Edge template VI1a"

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== Elimination of irrelevant Blue moves ==
 
== Elimination of irrelevant Blue moves ==
  
Red has a couple of direct threats to connect, using smaller templates. Blue must play in the carrier of these threats in order to counter them. To prevent Red from connecting Blue must play in the intersection of Red's threats carriers.
+
Red has a number of direct threats to connect, using smaller templates. Blue must play in the carrier of these threats in order to counter them. To prevent Red from connecting Blue must play in the intersection of Red's threats carriers.
  
=== [[edge template IV1a]] ===
+
=== [[Edge template IV1a]] ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 20: Line 20:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R i4 j2 E +:d7 +:e6 +:e7 +:f5 +:f6 +:f7 +:g5 +:g6 +:g7 +:h4 +:h5 +:h6 +:h7 +:i3 +:i5 +:i6 +:i7 +:j3 +:j5 +:j6 +:j7"
+
   contents="R i4 j2 S i4 d7 e6 e7 f5 f6 f7 g5 g6 g7 h4 h5 h6 h7 i3 i5 i6 i7 j3 j5 j6 j7"
 
/>
 
/>
  
Line 27: Line 27:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R i4 j2 E +:e7 +:f6 +:f7 +:g5 +:g6 +:g7 +:h5 +:h6 +:h7 +:i3 +:i5 +:i6 +:i7 +:j3 +:j4 +:j5 +:j6 +:j7 +:k5 +:k6 +:k7"
+
   contents="R i4 j2 S i4 e7 f6 f7 g5 g6 g7 h5 h6 h7 i3 i5 i6 i7 j3 j4 j5 j6 j7 k5 k6 k7"
 
/>
 
/>
  
=== [[edge template IV1b]] ===
+
=== [[Edge template IV1b]] ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 36: Line 36:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R i4 j2 E +:d7 +:e6 +:e7 +:f5 +:f6 +:f7 +:g5 +:g6 +:g7 +:h4 +:h5 +:h7 +:i3 +:i5 +:i6 +:i7 +:j3 +:j4 +:j5 +:j6 +:j7 +:k5 +:k6 +:k7"
+
   contents="R i4 j2 S i4 d7 e6 e7 f5 f6 f7 g5 g6 g7 h4 h5 h7 i3 i5 i6 i7 j3 j4 j5 j6 j7 k5 k6 k7"
 
/>
 
/>
  
  
=== Using the [[parallel ladder]] trick ===
+
=== Using [[Tom's move]] ===
  
6 moves can furthermore be discarded thanks to the [[Parallel ladder]] trick. Of course, symmetry will cut our work in half!
+
6 intrusions can furthermore be discarded thanks to [[Tom's move]], also known as the [[parallel ladder]] trick. Of course, symmetry will cut our work in half!
  
We can dispose of 3 moves on the left (and, using mirror symmetry, the corresponding 3 moves on the right), as follows:
+
If Blue moves in any of the cells marked "1" on the left (and, using mirror symmetry, in the corresponding 3 cells on the right), Red can respond as follows:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 50: Line 50:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R 3:h5 5:h6 1:i4 j2 B 4:g7 6:h7 2:i5 E +:e7 +:f6 +:g5"
+
   contents="R 4:h5 6:h6 2:i4 j2 B 5:g7 6:h7 3:i5 B 1:(e7 f6 g5)"
 
/>
 
/>
  
At this point, we can use [[Tom's move]] as follows:
+
At this point, Red can use [[Tom's move]] to connect:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 59: Line 59:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R h5 h6 i4 3:i6 j2 7:k3 1:k5 5:l4 B g7 h7 i5 4:i7 6:j5 2:j6 E +:e7 +:f6 +:g5"
+
   contents="R h5 h6 i4 4:i6 j2 8:k3 2:k5 6:l4 B g7 h7 i5 5:i7 7:j5 3:j6 B 1:(e7 f6 g5)"
 
/>
 
/>
  
=== [[Overlapping connections|Remaining possibilities]] for Blue ===
+
=== Remaining intrusions ===
Blue's first move must be one of the following:
+
 
 +
The only possible remaining intrusions for Blue are the following:
 
<hexboard size="7x14"
 
<hexboard size="7x14"
 
   coords="none"
 
   coords="none"
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R j2 E +:f7 +:g6 +:g7 +:h5 +:h7 +:i3 +:i4 +:i5 +:i6 +:i7 +:j3"
+
   contents="R j2  
 +
            S f7 g6 g7 h5 h7 i3 i4 i5 i6 i7 j3
 +
            E a:f7 b:g7 c:g6 d:h5 e:i4 f:i3"
 
/>
 
/>
 
+
By symmetry, if is sufficient to consider the six possible intrusions at a &ndash; f.
See
+
[[Template_VI1/Intrusion_on_the_3rd_row]],
+
[[Template_VI1/Intrusion_on_the_4th_row]],
+
[[Template_VI1/The_remaining_intrusion_on_the_fifth_row]].
+
  
 
== Specific defense ==
 
== Specific defense ==
 +
 
For the moves that intersect all the carriers, Red has to find specific answers. Let's deal with the remaining intrusions!
 
For the moves that intersect all the carriers, Red has to find specific answers. Let's deal with the remaining intrusions!
  
===One remaining intrusion on the first row (stub) ===
+
=== Intrusion at a ===
 +
 
 +
If Blue intrudes at a, Red has several winning responses. For example, Red can play at 2:
 
<hexboard size="7x14"
 
<hexboard size="7x14"
 
   coords="none"
 
   coords="none"
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R j2 B f7"
+
   contents="R j2 B 1:f7 R 2:i4 E x:i5 y:h7"
 +
/>
 +
Apart from intrusion into the bridge, which Red defends, Blue's only possible moves are at x and y. If Blue plays at x,  Red can set up a [[parallel ladder]] and connect using [[Tom's move]].
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:f7 R 2:i4 B 3:i5 R 4:g5 B 5:e6 R 6:g6 B 7:g7 R 8:h6 B 9:h7 R 10:k5"
 +
/>
 +
If Blue plays at y, Red has the following simple win, using the [[trapezoid]] template:
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:f7 R 2:i4 B 3:h7 R 4:i5 B 5:j6 R 6:g5 B 7:e6 R 8:g7"
 
/>
 
/>
  
Details to follow
+
=== Intrusion at b ===
  
===The other remaining intrusion on the first row===
+
If Blue intrudes at b, Red can respond at 2:
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:g7 R 2:i4 E x:i5 y:i6 z:i7 w:h7"
 +
/>
 +
Apart from intrusions into the bridge, which Red defends, Blue has only four possible moves x,y,z,w, because if Blue moves anywhere else, Red connects with either the [[ziggurat]] or [[edge template III1b]].
  
 +
If Blue intrudes at x, Red can set up a [[parallel ladder]] and connect using [[Tom's move]]:
 
<hexboard size="7x14"
 
<hexboard size="7x14"
 
   coords="none"
 
   coords="none"
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R j2 B g7"
+
   contents="R j2 B 1:g7 R 2:i4 B 3:i5 R 4:h5 B 5:f6 R 6:h6 B 7:h7 R 8:k5"
 +
/>
 +
If Blue intrudes at y or z (both shown simultaneously in the following diagram), Red can set up Tom's move on the opposite side:
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:g7 R 2:i4 B 3:(i6 i7) R 4:h6 B 5:h7 R 8:f5"
 +
/>
 +
Finally, if Blue intrudes at w, Red can connect by the following variant of Tom's move:
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:g7 R 2:i4 B 3:h7 R 4:h5 B 5:f6 R 6:k5"
 
/>
 
/>
  
Red should go here:
+
=== Intrusion at c ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 104: Line 142:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R 1:h5 j2 B g7"
+
   contents="R j2 B 1:g6"
 
/>
 
/>
  
See more details [[Template VI1/Other Intrusion on the 1st row| here]].
+
Red may play here:
 +
 
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 2:i5 B 1:g6
 +
            S blue:area(g7 m7 m5 l5 l3 k3)
 +
            E a:k2 b:j3 c:k3 d:j4"
 +
/>
  
===The remaining intrusion on the second row (stub)===
+
Note that if Red plays at c, then in the blue area there is a strategy such that both Red 2 and c connect down without choice, unless Blue first plays at d. Also, with this strategy the paths for 2 connecting down would not pass c or d in any case.
 +
Therefore, Blue must spend one move at either a,b,c or d in order to block Red on the right side, while Red 2 is always guaranteed to connect down. Thus we have this forcing sequence:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 115: Line 163:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R j2 B g6"
+
   contents="R j2 2:i5 B 1:g6 R 4:i3 B 5:i4 R 6:h4 B 7:h5  R 8:f5
 +
            S blue:area(g7 m7 m5 l5 l3 k3)
 +
            E a:k2 b:j3 c:k3 d:j4"
 
/>
 
/>
  
===The remaining intrusion on the third row (stub)===
+
(Assume that Blue 3 is played at either a,b,c or d, and there were no extra turns in the blue area.)
 +
 
 +
=== Intrusion at d ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 124: Line 176:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R j2 B h5"
+
   contents="R j2 B 1:h5"
 
/>
 
/>
  
Red should go here:
+
Red may go here:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 133: Line 185:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R j2 1:k3 B h5"
+
   contents="R j2 2:h3 B 1:h5"
 
/>
 
/>
  
Details to follow.
+
Details to follow. See more details [[Template_VI1/Intrusion_on_the_3rd_row|here]].
  
===The remaining intrusion on the fourth row===
+
=== Intrusion at e ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 144: Line 196:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R j2 B i4"
+
   contents="R j2 B 1:i4"
 
/>
 
/>
  
Line 153: Line 205:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R h3 j2 B i4 E +:k3"
+
   contents="R 2:h3 j2 B 1:i4 E +:k3"
 
/>
 
/>
  
For more details, see [[Template VI1/Intrusion on the 4th row|this page]].
+
Now the shaded area is a [[ladder creation template]], giving Red at least a 3rd row ladder as indicated.
===The remaining intrusion on the fifth row===
+
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 163: Line 214:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R j2 B i3"
+
   contents="R h3 j2 B i4 E +:k3 S area(h3,g3,e4,c5,a7,h7) E arrow(3):(h5 h6 h7)"
 
/>
 
/>
  
First establish a [[double ladder]] on the right.
+
Red can escape both 2nd and 3rd row ladders using a [[ladder escape fork]] via "+". Specifically, Red escapes a third row ladder like this, and is connected by a [[ziggurat]] and double threat at "+":
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 172: Line 223:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R 7:i4 j2 1:j3 5:j5 3:k4 B 8:h5 i3 2:i5 6:i7 4:k5"
+
   contents="R h3 j2 B i4 E +:k3 S area(h3,g3,e4,c5,a7,h7) R 1:h5 B 2:h6 R 3:j5 E +:i5"
 +
/>
 +
 
 +
If Blue [[ladder handling|yields]], or Red starts out with a 2nd row ladder, the escape fork works anyway:
 +
 
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R h3 j2 B i4 E +:k3 S area(h3,g3,e4,c5,a7,h7) R 1:h5 B 2:h7 R 3:h6 B 4:g7 R 5:j6 B 6:i6 R 7:j5 E +:i5"
 +
/>
 +
 
 +
=== Intrusion at f ===
 +
 
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:i3"
 +
/>
 +
 
 +
First establish a [[parallel ladder]] on the right.
 +
 
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R 8:i4 j2 2:j3 6:j5 4:k4 B 9:h5 1:i3 3:i5 7:i7 5:k5"
 
/>
 
/>
  
Line 181: Line 259:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
   contents="R 3:f4 1:f5 5:h3 i4 j2 j3 j5 k4 B 2:f6 4:g5 h5 i3 i5 i7 k5"
+
   contents="R 12:f4 10:f5 14:h3 i4 j2 j3 j5 k4 B 11:f6 13:g5 h5 i3 i5 i7 k5"
 
/>
 
/>
  
 +
There are two marginal cases. If Blue 3 blocks at a or b, then Red 4 can reduce the situation to "Intrusion at a" or "Intrusion at b" as before, since intruding the vertical bridge is irrelevant in these two cases.
 +
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 2:j3 4:i4 B 1:i3 E a:i7 b:g7"
 +
/>
  
 
[[category:edge templates]]
 
[[category:edge templates]]
 
[[category:theory]]
 
[[category:theory]]

Latest revision as of 05:15, 10 May 2024

Template VI1-a is a 6th row edge template with one stone.

This template is the first one stone 6th row template for which a proof of validity has been written out. The template has been verified by computer, and also verified to be minimal.

Elimination of irrelevant Blue moves

Red has a number of direct threats to connect, using smaller templates. Blue must play in the carrier of these threats in order to counter them. To prevent Red from connecting Blue must play in the intersection of Red's threats carriers.

Edge template IV1a

Edge template IV1b


Using Tom's move

6 intrusions can furthermore be discarded thanks to Tom's move, also known as the parallel ladder trick. Of course, symmetry will cut our work in half!

If Blue moves in any of the cells marked "1" on the left (and, using mirror symmetry, in the corresponding 3 cells on the right), Red can respond as follows:

214316156

At this point, Red can use Tom's move to connect:

8617214315

Remaining intrusions

The only possible remaining intrusions for Blue are the following:

fedcab

By symmetry, if is sufficient to consider the six possible intrusions at a – f.

Specific defense

For the moves that intersect all the carriers, Red has to find specific answers. Let's deal with the remaining intrusions!

Intrusion at a

If Blue intrudes at a, Red has several winning responses. For example, Red can play at 2:

2x1y

Apart from intrusion into the bridge, which Red defends, Blue's only possible moves are at x and y. If Blue plays at x, Red can set up a parallel ladder and connect using Tom's move.

24310568179

If Blue plays at y, Red has the following simple win, using the trapezoid template:

26475183

Intrusion at b

If Blue intrudes at b, Red can respond at 2:

2xy1wz

Apart from intrusions into the bridge, which Red defends, Blue has only four possible moves x,y,z,w, because if Blue moves anywhere else, Red connects with either the ziggurat or edge template III1b.

If Blue intrudes at x, Red can set up a parallel ladder and connect using Tom's move:

24385617

If Blue intrudes at y or z (both shown simultaneously in the following diagram), Red can set up Tom's move on the opposite side:

2843153

Finally, if Blue intrudes at w, Red can connect by the following variant of Tom's move:

246513

Intrusion at c

1

Red may play here:

abcd21

Note that if Red plays at c, then in the blue area there is a strategy such that both Red 2 and c connect down without choice, unless Blue first plays at d. Also, with this strategy the paths for 2 connecting down would not pass c or d in any case. Therefore, Blue must spend one move at either a,b,c or d in order to block Red on the right side, while Red 2 is always guaranteed to connect down. Thus we have this forcing sequence:

a4bc65d8721

(Assume that Blue 3 is played at either a,b,c or d, and there were no extra turns in the blue area.)

Intrusion at d

1

Red may go here:

21

Details to follow. See more details here.

Intrusion at e

1

Red should move here (or the equivalent mirror-image move at "+"):

21

Now the shaded area is a ladder creation template, giving Red at least a 3rd row ladder as indicated.

Red can escape both 2nd and 3rd row ladders using a ladder escape fork via "+". Specifically, Red escapes a third row ladder like this, and is connected by a ziggurat and double threat at "+":

132

If Blue yields, or Red starts out with a 2nd row ladder, the escape fork works anyway:

1736542

Intrusion at f

1

First establish a parallel ladder on the right.

128493657

Then use Tom's move:

1412101311

There are two marginal cases. If Blue 3 blocks at a or b, then Red 4 can reduce the situation to "Intrusion at a" or "Intrusion at b" as before, since intruding the vertical bridge is irrelevant in these two cases.

124ba