Difference between revisions of "Edge template VI1a"

Revision as of 02:31, 9 March 2021 (view source) Selinger (Talk | contribs) (Converted to new hexboard diagrams) ← Older edit		Latest revision as of 05:15, 10 May 2024 (view source) Bobson8 (Talk | contribs) (wording)
(9 intermediate revisions by 2 users not shown)
Line 1:		Line 1:
−	This page is devoted to details on how to [[Defending against intrusions in template VI1|defend against intrusions in template VI]]. This page explores what are the possibilities for Red to defend the template when Blue intrude on the 4th row.	+	Template VI1-a is a 6th row [[edge template]] with one stone.
	<hexboard size="7x14"		<hexboard size="7x14"
Line 5:		Line 5:
	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"
	/>		/>
−	Red should move here (or the equivalent mirror-image move at "+"):	+	This template is the first one stone 6th row [[edge template|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]] ===
	<hexboard size="7x14"		<hexboard size="7x14"
Line 14:		Line 20:
	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 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"
	/>		/>
−	== Elimination of irrelevant Blue moves ==	+	<hexboard size="7x14"
−	This gives Red several immediate threats:	+	coords="none"
−	From III1a:	+	edges="bottom"
		+	visible="area(a7,n7,n5,k2,i2,c5)"
		+	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]] ===
	<hexboard size="7x14"		<hexboard size="7x14"
Line 25:		Line 36:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R g5 h3 j2 B i4 E +:e7 +:f6 +:f7 +:g4 +:g6 +:g7 +:h4 +:h5 +:h6 +:h7"	+	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"
	/>		/>
−	From III1a again:	+
		+	=== 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:
	<hexboard size="7x14"		<hexboard size="7x14"
Line 34:		Line 50:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R g5 h3 j2 B i4 E +:d7 +:e6 +:e7 +:f5 +:f6 +:f7 +:g4 +:g6 +:g7 +:h4"	+	contents="R 4:h5 6:h6 2:i4 j2 B 5:g7 6:h7 3:i5 B 1:(e7 f6 g5)"
	/>		/>
−	From III1b :	+	At this point, Red can use [[Tom's move]] to connect:
	<hexboard size="7x14"		<hexboard size="7x14"
Line 43:		Line 59:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R g5 h3 j2 B i4 E +:d7 +:e6 +:e7 +:f5 +:f6 +:g4 +:g6 +:g7 +:h4 +:h5 +:h6 +:h7"	+	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)"
	/>		/>
−	From IV1a:	+	=== Remaining intrusions ===
		+	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 g4 h3 j2 B i4 E +:b7 +:c6 +:c7 +:d5 +:d6 +:d7 +:e5 +:e6 +:e7 +:f4 +:f5 +:f6 +:f7 +:g5 +:g6 +:g7 +:h5 +:h6 +:h7"	+	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.
−	From IV1b:	+	== 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:
	<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 g4 h3 j2 B i4 E +:b7 +:c6 +:c7 +:d5 +:d6 +:d7 +:e5 +:e6 +:e7 +:f4 +:f5 +:f7 +:g5 +:g6 +:g7 +:h4 +:h5 +:h6 +:h7 +:i5 +:i6 +:i7"	+	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"
	/>		/>
−	The intersection of all of these leaves:	+	=== Intrusion at b ===
		+	If Blue intrudes at b, Red can respond at 2:
	<hexboard size="7x14"		<hexboard size="7x14"
−	coords="full bottom right"	+	coords="none"
	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 +:e7 +:g4 +:g5 +:g6 +:g7"	+	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]].
−	== Specific defense ==	+	If Blue intrudes at x, Red can set up a [[parallel ladder]] and connect using [[Tom's move]]:
−	So we must deal with each of these responses.  (Which will not be too hard!)	+	<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: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"
		+	/>
−	=== Bg4 ===	+	=== Intrusion at c ===
	<hexboard size="7x14"		<hexboard size="7x14"
Line 82:		Line 142:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R h3 2:h4 4:h5 j2 B 1:g4 3:g6 i4"	+	contents="R j2 B 1:g6"
	/>		/>
−	And now either	+
		+	Red may play here:
	<hexboard size="7x14"		<hexboard size="7x14"
Line 90:		Line 151:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R h3 h4 h5 j2 2:j5 B g4 g6 1:h6 i4 E +:i5 +:k3"	+	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"
	/>		/>
−	or	+	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 99:		Line 163:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R h3 h4 h5 2:h6 j2 6:j5 4:j6 B g4 g6 3:g7 1:h7 i4 5:i6 E +:i5 +:k3"	+	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"
	/>		/>
−	=== Bg5 ===	+	(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"
−	coords="full bottom right"	+	coords="none"
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R 2:f4 h3 j2 B 1:g5 i4"	+	contents="R j2 B 1:h5"
	/>		/>
−	Threatening:	+
		+	Red may go here:
	<hexboard size="7x14"		<hexboard size="7x14"
−	coords="full bottom right"	+	coords="none"
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R 4:d5 f4 h3 j2 B g5 i4 E +:a7 +:b6 +:b7 +:c5 +:c6 +:c7 +:d6 +:d7 +:e4 +:e5"	+	contents="R j2 2:h3 B 1:h5"
	/>		/>
		+
		+	Details to follow. See more details [[Template_VI1/Intrusion_on_the_3rd_row|here]].
		+
		+	=== Intrusion at e ===
	<hexboard size="7x14"		<hexboard size="7x14"
−	coords="full bottom right"	+	coords="none"
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R 4:e6 f4 h3 j2 B g5 i4 E +:d7 +:e5 +:e7 +:f5"	+	contents="R j2 B 1:i4"
	/>		/>
		+
		+	Red should move here (or the equivalent mirror-image move at "+"):
	<hexboard size="7x14"		<hexboard size="7x14"
−	coords="full bottom right"	+	coords="none"
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R 4:e5 f4 h3 j2 B g5 i4 E +:b7 +:c6 +:c7 +:d5 +:d6 +:e6 +:e7 +:f5 +:f6 +:f7"	+	contents="R 2:h3 j2 B 1:i4 E +:k3"
	/>		/>
−	So the only hope for Blue lies in the intersection of the threats, Be5, but it is unsufficient:	+
		+	Now the shaded area is a [[ladder creation template]], giving Red at least a 3rd row ladder as indicated.
	<hexboard size="7x14"		<hexboard size="7x14"
−	coords="full bottom right"	+	coords="none"
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R f4 2:f5 4:f6 6:g6 h3 j2 8:j5 B 1:e5 3:e7 5:f7 g5 7:g7 i4 E +:i5 +:k3"	+	contents="R h3 j2 B i4 E +:k3 S area(h3,g3,e4,c5,a7,h7) E arrow(3):(h5 h6 h7)"
	/>		/>
−	=== Bg6 ===	+
		+	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 146:		Line 223:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R 2:g5 h3 4:h5 j2 B 3:f6 1:g6 i4 E +:e7"	+	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"
	/>		/>
−	3 could be played at + with the same effect; in any case	+	If Blue [[ladder handling|yields]], or Red starts out with a 2nd row ladder, the escape fork works anyway:
−	now either	+
	<hexboard size="7x14"		<hexboard size="7x14"
Line 156:		Line 232:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R g5 h3 h5 j2 2:j5 B f6 g6 1:h6 i4 E +:i5 +:k3"	+	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"
	/>		/>
−	or	+	=== Intrusion at f ===
	<hexboard size="7x14"		<hexboard size="7x14"
Line 165:		Line 241:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R g5 h3 h5 2:h6 j2 6:j5 4:j6 B f6 g6 3:g7 1:h7 i4 5:i6 E +:i5 +:k3"	+	contents="R j2 B 1:i3"
	/>		/>
−	=== Be7 ===	+	First establish a [[parallel ladder]] on the right.
−	Either this	+
	<hexboard size="7x14"		<hexboard size="7x14"
Line 175:		Line 250:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R 2:g5 h3 4:h5 j2 6:j5 B 1:e7 3:g6 5:h6 i4 E +:i5 +:k3"	+	contents="R 8:i4 j2 2:j3 6:j5 4:k4 B 9:h5 1:i3 3:i5 7:i7 5:k5"
	/>		/>
−	or a minor variation	+	Then use [[Tom's move]]:
	<hexboard size="7x14"		<hexboard size="7x14"
Line 184:		Line 259:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R 2:g5 h3 4:h5 6:h6 j2 10:j5 8:j6 B 1:e7 3:g6 7:g7 5:h7 i4 9:i6 E +:i5 +:k3"	+	contents="R 12:f4 10:f5 14:h3 i4 j2 j3 j5 k4 B 11:f6 13:g5 h5 i3 i5 i7 k5"
	/>		/>
−	=== Bg7 ===	+	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"		<hexboard size="7x14"
Line 193:		Line 268:
	edges="bottom"		edges="bottom"
	visible="area(a7,n7,n5,k2,i2,c5)"		visible="area(a7,n7,n5,k2,i2,c5)"
−	contents="R 2:g5 h3 4:h6 j2 8:j5 6:j6 B 3:f6 1:g7 5:h7 i4 7:i6 E +:i5 +:k3"	+	contents="R j2 2:j3 4:i4 B 1:i3 E a:i7 b:g7"
	/>		/>
	[[category:edge templates]]		[[category:edge templates]]
		+	[[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:

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

Remaining intrusions

The only possible remaining intrusions for Blue are the following:

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:

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.

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

Intrusion at b

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

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:

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:

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

Intrusion at c

Red may play here:

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:

(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

Red may go here:

Details to follow. See more details here.

Intrusion at e

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

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 "+":

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

Intrusion at f

First establish a parallel ladder on the right.

Then use Tom's move:

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.