Difference between revisions of "Edge template VI1a"
(→Remaining possibilities for Blue: trying to fix diagram) |
(wording) |
||
(18 intermediate revisions by 4 users not shown) | |||
Line 1: | Line 1: | ||
− | + | Template VI1-a is a 6th row [[edge template]] with one stone. | |
− | < | + | <hexboard size="7x14" |
− | + | coords="none" | |
− | + | edges="bottom" | |
− | + | visible="area(a7,n7,n5,k2,i2,c5)" | |
− | + | contents="R j2" | |
− | + | /> | |
− | + | ||
− | + | 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 == | == Elimination of irrelevant Blue moves == | ||
− | Red has a | + | 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" |
− | + | coords="none" | |
− | + | edges="bottom" | |
− | + | visible="area(a7,n7,n5,k2,i2,c5)" | |
− | + | 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" | |
− | + | /> | |
− | + | ||
− | + | ||
− | + | <hexboard size="7x14" | |
− | + | coords="none" | |
− | + | 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" | |
− | + | coords="none" | |
− | + | edges="bottom" | |
− | + | visible="area(a7,n7,n5,k2,i2,c5)" | |
− | + | 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 [[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" | |
+ | coords="none" | ||
+ | edges="bottom" | ||
+ | visible="area(a7,n7,n5,k2,i2,c5)" | ||
+ | contents="R 4:h5 6:h6 2:i4 j2 B 5:g7 6:h7 3:i5 B 1:(e7 f6 g5)" | ||
+ | /> | ||
− | + | At this point, Red can use [[Tom's move]] to connect: | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | <hexboard size="7x14" | |
− | + | coords="none" | |
− | + | edges="bottom" | |
− | + | visible="area(a7,n7,n5,k2,i2,c5)" | |
− | + | 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)" | |
+ | /> | ||
− | + | === Remaining intrusions === | |
− | < | + | The only possible remaining intrusions for Blue are the following: |
− | + | <hexboard size="7x14" | |
− | + | coords="none" | |
− | + | edges="bottom" | |
− | + | visible="area(a7,n7,n5,k2,i2,c5)" | |
− | + | 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 – 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: | |
− | [[ | + | <hexboard size="7x14" |
− | [[ | + | coords="none" |
− | [[ | + | edges="bottom" |
+ | visible="area(a7,n7,n5,k2,i2,c5)" | ||
+ | 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" | ||
+ | /> | ||
− | == | + | === Intrusion at b === |
− | + | ||
− | === | + | If Blue intrudes at b, Red can respond at 2: |
− | + | <hexboard size="7x14" | |
− | + | coords="none" | |
− | 1: | + | 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" |
+ | 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" | ||
+ | /> | ||
− | + | === Intrusion at c === | |
− | === | + | <hexboard size="7x14" |
− | + | coords="none" | |
− | + | edges="bottom" | |
− | 1: | + | visible="area(a7,n7,n5,k2,i2,c5)" |
− | + | contents="R j2 B 1:g6" | |
− | + | /> | |
− | + | ||
− | + | ||
− | + | ||
− | + | 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" | ||
+ | /> | ||
− | + | 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" | |
− | + | coords="none" | |
+ | edges="bottom" | ||
+ | visible="area(a7,n7,n5,k2,i2,c5)" | ||
+ | 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" | ||
+ | /> | ||
− | + | (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" | |
− | </ | + | coords="none" |
+ | edges="bottom" | ||
+ | visible="area(a7,n7,n5,k2,i2,c5)" | ||
+ | contents="R j2 B 1:h5" | ||
+ | /> | ||
− | + | Red may go here: | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | <hexboard size="7x14" | |
− | </ | + | coords="none" |
+ | edges="bottom" | ||
+ | visible="area(a7,n7,n5,k2,i2,c5)" | ||
+ | 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" | |
− | </ | + | coords="none" |
+ | edges="bottom" | ||
+ | visible="area(a7,n7,n5,k2,i2,c5)" | ||
+ | contents="R j2 B 1:i4" | ||
+ | /> | ||
− | + | Red should move here (or the equivalent mirror-image move at "+"): | |
− | === | + | <hexboard size="7x14" |
− | + | coords="none" | |
− | + | edges="bottom" | |
− | 1: | + | visible="area(a7,n7,n5,k2,i2,c5)" |
− | + | contents="R 2:h3 j2 B 1:i4 E +:k3" | |
− | + | /> | |
− | + | ||
− | + | ||
− | + | ||
− | + | Now the shaded area is a [[ladder creation template]], giving Red at least a 3rd row ladder as indicated. | |
− | + | ||
− | + | <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) E arrow(3):(h5 h6 h7)" | ||
+ | /> | ||
+ | |||
+ | 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" |
− | + | coords="none" | |
− | 1: | + | 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: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)" | |
− | 1: | + | 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" | ||
+ | /> | ||
Then use [[Tom's move]]: | Then use [[Tom's move]]: | ||
− | < | + | <hexboard size="7x14" |
− | + | coords="none" | |
− | + | edges="bottom" | |
− | + | visible="area(a7,n7,n5,k2,i2,c5)" | |
− | + | 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.
Contents
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.