Theorems about templates - Revision history

Selinger: Fixed numbering of theorems

2024-12-17T18:21:38Z

Fixed numbering of theorems

Demer: fixed typo

2024-04-21T05:51:35Z

fixed typo

Selinger: /* Alternative connection up */ more grammar and copy-edits

2024-04-21T02:19:18Z

‎Alternative connection up: more grammar and copy-edits

Selinger: /* Alternative connection up */ copy-edit

2024-04-21T02:17:59Z

‎Alternative connection up: copy-edit

Selinger: /* Alternative connection up */ Extended Theorem 5 to the case where the bridge is filled. This actually simplifies the proof slightly.

2024-04-21T02:16:26Z

‎Alternative connection up: Extended Theorem 5 to the case where the bridge is filled. This actually simplifies the proof slightly.

Selinger: typos

2024-04-21T00:04:35Z

typos

Selinger: /* Alternative connection up */ Copy-editing

2024-04-20T22:07:15Z

‎Alternative connection up: Copy-editing

Selinger: Re-numbered the theorems.

2024-04-20T21:43:53Z

Re-numbered the theorems.

Selinger: /* Crescenting */ Added a theorem for "alternative connection up"

2024-04-20T21:41:25Z

‎Crescenting: Added a theorem for "alternative connection up"

Selinger: Re-numbered the theorems.

2024-04-15T21:15:03Z

Re-numbered the theorems.

@@ Line 747: / Line 747: @@
 '''Proof.''' This is very similar to Theorem 10. By the previous observation, y might as well be blue, contradicting the minimality of the template. □
-Theorems 7–9 explain why the rightmost few columns of edge templates frequently have one of these shapes
+Theorems 9–11 explain why the rightmost few columns of edge templates frequently have one of these shapes
 <hexboard size="5x3"
    float="inline"

@@ Line 383: / Line 383: @@
    />
-To see that B is at least as good for Red as A, note that if Red plays first in the region, then Red play z in B, which [[captured cell|captures]] w and is at least as good as anything Red could do in A. If Blue plays first in the region, A and B become identical. To see that C is at least as good for Red as B, note that Red can get at least one of x and y by defending the [[bridge]]. No matter which way the bridge goes, the result is identical or better for Red than B.
+To see that B is at least as good for Red as A, note that if Red plays first in the region, then Red plays z in B, which [[dead cell|kills]] w and is at least as good as anything Red could do in A. If Blue plays first in the region, A and B become identical. To see that C is at least as good for Red as B, note that Red can get at least one of x and y by defending the [[bridge]]. No matter which way the bridge goes, the result is identical or better for Red than B.
 '''Theorem 5 (alternative connection up).''' Suppose some (edge or interior) template has a piece of boundary of the form

@@ Line 411: / Line 411: @@
    />
-are also connected. (It may fail to be a template only because it may fail to be minimal).
+are also connected. (They may fail to be templates only because they may fail to be minimal).
 '''Proof.''' Consider the template containing A. Since the shaded hexes are outside the carrier, they may as well be blue stones, except that we need to connect the stone marked "↑" to something, which we can without loss of generality do like this:
@@ Line 435: / Line 435: @@
    contents="B a3 b2 d1 R b4 c2 arrow(12):c1"
    />
-Therefore, B and C are connected, proving the theorem. □
+Therefore, the patterns using B and C are connected, proving the theorem. □
 '''Example'''

@@ Line 395: / Line 395: @@
    />
-Here, as usual, the blue-shaded cells are not part of the template, but the stone marked "↑" must of course be connected up. Then the pattern where this area has been replaced by
+Here, as usual, the blue-shaded cells are not part of the template, but the stone marked "↑" must of course be connected up. Then the patterns where this area has been replaced by
 B: <hexboard size="4x4"
@@ Line 411: / Line 411: @@
    />
-is also connected. (It may fail to be a template only because it may fail to be minimal).
+are also connected. (It may fail to be a template only because it may fail to be minimal).
 '''Proof.''' Consider the template containing A. Since the shaded hexes are outside the carrier, they may as well be blue stones, except that we need to connect the stone marked "↑" to something, which we can without loss of generality do like this:

@@ Line 361: / Line 361: @@
 === Alternative connection up ===
-Some templates, such as [[Tom's move]], have an "alternative connection up". There is a general theorem about this. We begin by observing that B is at least as good for Red as A:
+Some templates, such as [[Tom's move]], have an "alternative connection up". There is a general theorem about this. We begin by observing that from Red's point of view, C is at least as good as B, and B is at least as good as A:
 A: <hexboard size="3x4"
@@ Line 368: / Line 368: @@
    coords="none"
    visible="area(a2,b3,d1,c1)"
-   contents="R c1 b2 b3 B a2 d1 E w:c2"
+   contents="R c1 b2 b3 B a2 d1 E c2"
    /> B: <hexboard size="4x4"
    float="inline"
    edges="none"
@@ Line 377: / Line 383: @@
    />
-Indeed, if Red is first to play in the region, then Red plays z in B, which captures the whole region and is therefore at least as good as anything Red could do in A. Now suppose Blue plays first. If Blue plays x or z in B, Red responds with y, [[dead cell|killing]] x and leaving B at least as good for Red as A with no moves played. If Blue plays y, Red responds at x. We claim that B with these two moves played is still at least as good for Red as A with no moves played. Indeed, if Blue plays next, A and B become identical, and if Red plays next, Red z captures the whole region and is therefore at least as good as anything Red could do in A.
+To see that B is at least as good for Red as A, note that if Red plays first in the region, then Red play z in B, which [[captured cell|captures]] w and is at least as good as anything Red could do in A. If Blue plays first in the region, A and B become identical. To see that C is at least as good for Red as B, note that Red can get at least one of x and y by defending the [[bridge]]. No matter which way the bridge goes, the result is identical or better for Red than B.
 '''Theorem 5 (alternative connection up).''' Suppose some (edge or interior) template has a piece of boundary of the form
 <hexboard size="4x4"
    float="inline"
    edges="none"
@@ Line 387: / Line 394: @@
    contents="S blue:all none:b4,c3 R arrow(12):b4"
    />
 Here, as usual, the blue-shaded cells are not part of the template, but the stone marked "↑" must of course be connected up. Then the pattern where this area has been replaced by
 <hexboard size="4x4"
    float="inline"
    edges="none"
@@ Line 395: / Line 410: @@
    contents="S blue:all none:b4,c3,b3,d2,c2 R b4 arrow(12):c2"
    />
 is also connected. (It may fail to be a template only because it may fail to be minimal).
-'''Proof.''' Since the shaded hexes are outside the carrier, they may as well be blue stones, except that we need to connect the stone marked "↑" to something, which we can without loss of generality do like this:
+'''Proof.''' Consider the template containing A. Since the shaded hexes are outside the carrier, they may as well be blue stones, except that we need to connect the stone marked "↑" to something, which we can without loss of generality do like this:
 <hexboard size="4x4"
    float="inline"
@@ Line 405: / Line 421: @@
    contents="B a3 b2 d1 d2 R a:b4 b:b3 c:c2 arrow(12):c1"
    />
-The fact that the stones "b" and "c" are adjacent to cells in the template does not matter, because "a" is adjacent to the same cells anyway. Due to the above observation, the following is at least as good for Red, and therefore also connected:
+The fact that the stones "b" and "c" are adjacent to cells in the template does not matter, because "a" is adjacent to the same cells anyway. By the above observation, each of the following is at least as good for Red, and therefore also connected:
 <hexboard size="4x4"
    float="inline"
    edges="none"
@@ Line 413: / Line 435: @@
    contents="B a3 b2 d1 R b4 c2 arrow(12):c1"
    />
-This proves the theorem. □
+Therefore, B and C are connected, proving the theorem. □
 '''Example'''
@@ Line 424: / Line 446: @@
    contents="R arrow(12):b2 a3 a4 B b3"
    />
-By Theorem 5, the following is therefore also connected:
+By Theorem 5, the following are therefore also connected:
 <hexboard size="6x8"
    edges="bottom"
    coords="none"

@@ Line 379: / Line 379: @@
 Indeed, if Red is first to play in the region, then Red plays z in B, which captures the whole region and is therefore at least as good as anything Red could do in A. Now suppose Blue plays first. If Blue plays x or z in B, Red responds with y, [[dead cell|killing]] x and leaving B at least as good for Red as A with no moves played. If Blue plays y, Red responds at x. We claim that B with these two moves played is still at least as good for Red as A with no moves played. Indeed, if Blue plays next, A and B become identical, and if Red plays next, Red z captures the whole region and is therefore at least as good as anything Red could do in A.
-'''Theorem 4x (alternative connection up).''' Suppose some (edge or interior) template has a piece of boundary of the form
+'''Theorem 5 (alternative connection up).''' Suppose some (edge or interior) template has a piece of boundary of the form
 <hexboard size="4x4"
    float="inline"
@@ Line 424: / Line 424: @@
    contents="R arrow(12):b2 a3 a4 B b3"
    />
-By Theorem 4x, the following is therefore also connected:
+By Theorem 5, the following is therefore also connected:
 <hexboard size="6x8"
    edges="bottom"
@@ Line 459: / Line 459: @@
 An interesting application of this is getting a 2nd row [[ladder creation template]] from an ordinary template.
-'''Theorem 5 (ladder creation template from template).''' Suppose an edge template has a corner of the form
+'''Theorem 6 (ladder creation template from template).''' Suppose an edge template has a corner of the form
 A: <hexboard size="3x4"
    edges="bottom"
@@ Line 528: / Line 528: @@
    contents="S blue:c1--c3,d1--d3"
    />
-then the left-to-right direction of Theorem 5 is still valid: Given an edge template, we still obtain a ladder creation template in two different ways. However, in this case, the latter may not be minimal.
+then the left-to-right direction of Theorem 6 is still valid: Given an edge template, we still obtain a ladder creation template in two different ways. However, in this case, the latter may not be minimal.
 == Overlapping templates ==
@@ Line 536: / Line 536: @@
 === Edge template II in the overlap ===
-'''Theorem 6.''' If the region in which two edge templates overlap is [[edge template II]], then both templates remain valid.
+'''Theorem 7.''' If the region in which two edge templates overlap is [[edge template II]], then both templates remain valid.
 <hexboard size="3x4"
    coords="hide"
@@ Line 580: / Line 580: @@
 The following theorem is due to Eric Demer.
-'''Theorem 7 (ziggurat theorem).''' Consider a [[ziggurat]].
+'''Theorem 8 (ziggurat theorem).''' Consider a [[ziggurat]].
 <hexboard size="3x4"
    visible="area(a3,d3,d1,c1)"
@@ Line 654: / Line 654: @@
 Indeed, if Red plays in one of these cells, Blue can play in the other, [[dead cell|killing]] Red's stone. From this, we immediately get the following theorem:
-'''Theorem 8.''' There is no edge template with an empty corner of height 2, i.e., with a corner of this shape:
+'''Theorem 9.''' There is no edge template with an empty corner of height 2, i.e., with a corner of this shape:
 <hexboard size="3x2"
    edges="bottom"
@@ Line 693: / Line 693: @@
 To see why, first assume that Blue plays first in the region. Then Blue x captures the entire region, so the stone at y no longer matters. Next, assume that Red plays first in the region. If Red plays anywhere other than x, then Blue x [[dead cell|kills]] the red stone. If Red plays at x, then y is [[dead cell|dead]], so the stone at y no longer matters. As a consequence, we get the following theorem:
-'''Theorem 9.''' There is no edge template with a piece of boundary of this shape:
+'''Theorem 10.''' There is no edge template with a piece of boundary of this shape:
 <hexboard size="3x3"
    float="inline"
@@ Line 704: / Line 704: @@
 '''Proof.''' Since the shaded hexes are outside the carrier, they may as well be blue stones. Then by the previous observation, y might as well be blue, contradicting the minimality of the template. □
-Once again, we remark that Theorem 9 presupposes that there are no stones in the relevant portion of the template. For example, the [[ziggurat]] is an edge template that ends in two columns of height 3, but this does not contradict Theorem 9 due to the presence of a red stone in one of these columns.
+Once again, we remark that Theorem 10 presupposes that there are no stones in the relevant portion of the template. For example, the [[ziggurat]] is an edge template that ends in two columns of height 3, but this does not contradict Theorem 10 due to the presence of a red stone in one of these columns.
-'''Theorem 10.''' There is no edge template with a corner of this shape:
+'''Theorem 11.''' There is no edge template with a corner of this shape:
 <hexboard size="4x3"
    float="inline"
@@ Line 716: / Line 716: @@
 As always, the blue-shaded cells indicate the outside of the template, i.e., they are not part of the carrier.
-'''Proof.''' This is very similar to Theorem 9. By the previous observation, y might as well be blue, contradicting the minimality of the template. □
+'''Proof.''' This is very similar to Theorem 10. By the previous observation, y might as well be blue, contradicting the minimality of the template. □
 Theorems 7–9 explain why the rightmost few columns of edge templates frequently have one of these shapes

← Older revision		Revision as of 21:41, 20 April 2024
Line 358:		Line 358:

	Often, the crescent-like shape can be more generally replaced with any capped [[flank]]. See also [[Flank#Edge_templates_from_capped_flanks\|edge templates from capped flanks]].		Often, the crescent-like shape can be more generally replaced with any capped [[flank]]. See also [[Flank#Edge_templates_from_capped_flanks\|edge templates from capped flanks]].
		+
		+	=== Alternative connection up ===
		+
		+	Some templates, such as [[Tom's move]], have an "alternative connection up". There is a general theorem about this. We begin by observing that B is at least at good for Red as A:
		+
		+	A: <hexboard size="3x4"
		+	float="inline"
		+	edges="none"
		+	coords="none"
		+	visible="area(a2,b3,d1,c1)"
		+	contents="R c1 b2 b3 B a2 d1 E w:c2"
		+	/> B: <hexboard size="4x4"
		+	float="inline"
		+	edges="none"
		+	coords="none"
		+	visible="area(a2,b3,d1,c1)"
		+	contents="R c1 b3 B a2 E x:b2 y:c2 z:d1"
		+	/>
		+
		+	Indeed, if Red is first to play in the region, then Red plays z in B, which captures the whole region and is therefore at least as good as anything Red could do in A. Now suppose Blue plays first. If Blue plays x or z in B, Red responds with y, [[dead cell\|killing]] x and leaving B at least as good for Red as A with no moves played. If Blue plays y, Red responds at x. We claim that B with these two moves played is still at least as good for Red as A with no moves played. Indeed, if Blue plays next, A and B become identical, and if Red plays next, Red z captures the whole region and is therefore at least as good as anything Red could do in A.
		+
		+	'''Theorem 4x (alternative connection up).''' Suppose some (edge or interior) template has a piece of boundary of the form
		+	<hexboard size="4x4"
		+	float="inline"
		+	edges="none"
		+	coords="none"
		+	visible="area(a3,b4,d2,d1,c1)"
		+	contents="S blue:all none:b4,c3 R arrow(12):b4"
		+	/>
		+	Here, as usual, the blue-shaded cells are not part of the template, but the stone marked "↑" must of course be connected up. Then the pattern where this area has been replaced by
		+	<hexboard size="4x4"
		+	float="inline"
		+	edges="none"
		+	coords="none"
		+	visible="area(a3,b4,d2,d1,c1)"
		+	contents="S blue:all none:b4,c3,b3,d2,c2 R b4 arrow(12):c2"
		+	/>
		+	is also connected. (It may fail to be a template only because it may fail to be minimal).
		+
		+	'''Proof.''' Since the shaded hexes are outside the carrier, they may as well be blue stones, except that we need to connect the stone marked "↑" to something, which we can without loss of generality do like this:
		+	<hexboard size="4x4"
		+	float="inline"
		+	edges="none"
		+	coords="none"
		+	visible="area(a3,b4,d2,d1,c1)"
		+	contents="B a3 b2 d1 d2 R a:b4 b:b3 c:c2 arrow(12):c1"
		+	/>
		+	The fact that the stones "b" and "c" are potentially next to cells in the template does not matter, because "a" neighbors those cells anyway. Due to the above observation, the following is at least at good for Red, and therefore also connected:
		+	<hexboard size="4x4"
		+	float="inline"
		+	edges="none"
		+	coords="none"
		+	visible="area(a3,b4,d2,d1,c1)"
		+	contents="B a3 b2 d1 R b4 c2 arrow(12):c1"
		+	/>
		+	This proves the theorem. □
		+
		+	'''Example'''
		+
		+	[[Tom's move]] ensures that the following is connected:
		+	<hexboard size="5x7"
		+	edges="bottom"
		+	coords="none"
		+	visible="-(a2 a1 b1 f1 g1 g2)"
		+	contents="R arrow(12):b2 a3 a4 B b3"
		+	/>
		+	By Theorem 4x, the following is therefore also connected:
		+	<hexboard size="6x8"
		+	edges="bottom"
		+	coords="none"
		+	visible="-g2 h2 h3 f1--h1 -area(a1,a6,c1)"
		+	contents="R b4 b5 c3 B a6 c4 R arrow(12):d1"
		+	/>
		+	This is exactly the "[[Tom's_move#Alternative_connection_up\|alternative connection up]]" of Tom's move.

	== Ladder creation templates from templates ==		== Ladder creation templates from templates ==