User:Bobson8 - Revision history

Bobson8 at 01:54, 9 May 2025

2025-05-09T01:54:37Z

Bobson8 at 12:09, 5 May 2025

2025-05-05T12:09:19Z

Bobson8: Add image

2025-05-05T12:01:34Z

Add image

Bobson8: Some interior templates and descriptions

2025-05-05T11:54:41Z

Some interior templates and descriptions

Bobson8: Fix a wrong strategy

2024-08-26T04:42:42Z

Fix a wrong strategy

Bobson8: winning move with a smaller template

2024-04-21T18:54:40Z

winning move with a smaller template

Bobson8 at 18:41, 21 April 2024

2024-04-21T18:41:24Z

Bobson8: typo

2024-04-21T18:40:47Z

typo

Bobson8: typo

2024-04-21T10:59:48Z

typo

Bobson8: A conjecture on trapezoid boards

2024-04-21T07:00:06Z

A conjecture on trapezoid boards

@@ Line 186: / Line 186: @@
-When the dotted edge is not broken, Red saves the yellow edge by playing a red piece at the yellow spot and connect the two parts with a bridge. If Blue breaks the yellow edge by playing a blue piece on it, then Red can also automatically peep at the yellow spot. The same applies to green.
+When the dotted edge have not been played, Red can save the yellow edge by playing a red piece at the yellow spot and connect the two parts with a bridge. If Blue breaks the yellow edge by playing a blue piece on it, then Red can also automatically peep at the yellow spot. The same applies to green.
-In any other cases, If Blue plays inside the tunnel, then they are considered breaking the dotted edge. Red saves the dotted edge by playing a red piece on the cyan spot and connect to the yellow and green spots with bridges.
+In any other cases, If Blue plays inside the tunnel, then they are considered breaking the dotted edge. Red can save the dotted edge by playing a red piece on the cyan spot and connect to the yellow and green spots with bridges.
 This switching game itself is not obviously winning for Red. One has to check case by case.

← Older revision		Revision as of 12:09, 5 May 2025
Line 186:		Line 186:


−	Red saves the yellow edge by playing a red piece at the yellow spot and connect the two parts with a bridge. If Blue breaks the yellow edge by playing a blue piece on it, then Red can also automatically peep at the yellow spot. The same applies to green.	+	When the dotted edge is not broken, Red saves the yellow edge by playing a red piece at the yellow spot and connect the two parts with a bridge. If Blue breaks the yellow edge by playing a blue piece on it, then Red can also automatically peep at the yellow spot. The same applies to green.

← Older revision		Revision as of 12:01, 5 May 2025
Line 183:		Line 183:


−	(Image ~~Placeholder)~~	+	[[Image: TripleV_SSG.png\|center\|500px]]

← Older revision		Revision as of 11:54, 5 May 2025
Line 150:		Line 150:
	contents="+:(d7 e7 f7) *:h1 a:f5 b:f4 B a6 b5 c4 d3 e2 f1 j6 j5 j4 j3 j2 j1 R 1:g3"		contents="+:(d7 e7 f7) *:h1 a:f5 b:f4 B a6 b5 c4 d3 e2 f1 j6 j5 j4 j3 j2 j1 R 1:g3"
	/>		/>
		+
		+	== Some interior templates ==
		+
		+	These templates assume that one has to connect all the red pieces on the boundaries. (The middle pieces are also gauranteed to be connected, though insignificant in practice.)
		+
		+	<hexboard size="5×5"
		+	coords="hide"
		+	edges="none"
		+	visible="area(a5,c5,e3,e1,c1,a3)"
		+	contents="R a3 a5 c1 c3 c5 e1 e3"
		+	/>
		+
		+	This template can be verified by the disjoint spanning tree theorem (the criterion being used in the Shannon switching game).
		+
		+	<hexboard size="5×5"
		+	coords="hide"
		+	edges="none"
		+	visible="area(a5,c5,d4,e2,e1,c1,a3)"
		+	contents="R a3 a5 c1 c3 c5 e1"
		+	/>
		+
		+	Removing a Red terminal on a corner from the previous template still gives a template.
		+
		+	<hexboard size="7×7"
		+	coords="hide"
		+	edges="none"
		+	visible="area(a5,e5,g3,g2,f1,d1,b2,a3)"
		+	contents="R a3 a5 c3 c5 e3 e5 g3"
		+	/>
		+
		+	A strategy for this template could be intepreted as the Shannon switching game on this graph of 7 vertices and 12 edges. There is an exceptional rule for this game: Red can save the dotted edge only after the yellow and green edges have been played by any player.
		+
		+
		+	(Image Placeholder)
		+
		+
		+	Red saves the yellow edge by playing a red piece at the yellow spot and connect the two parts with a bridge. If Blue breaks the yellow edge by playing a blue piece on it, then Red can also automatically peep at the yellow spot. The same applies to green.
		+
		+
		+	In any other cases, If Blue plays inside the tunnel, then they are considered breaking the dotted edge. Red saves the dotted edge by playing a red piece on the cyan spot and connect to the yellow and green spots with bridges.
		+
		+
		+	This switching game itself is not obviously winning for Red. One has to check case by case.

@@ Line 142: / Line 142: @@
     />
-(a,b)=(3,7): Blue edges form a pre-template. There are two template within it, removing cells either with * or with + (due to Quasar). One possible defence for Red:
+(a,b)=(3,7): Blue edges form a pre-template. There are two template within it, removing cells either with * or with + (due to Quasar). When Red plays 1, Blue can play at a for the * case, or b for the + case:
 <hexboard size="7x10"
@@ Line 148: / Line 148: @@
     edges="none"
     visible="area(a6,a7,i7,j6,j1,f1)"
-    contents="+:(d7 e7 f7) *:h1 B a6 b5 c4 d3 e2 f1 j6 j5 j4 j3 j2 j1 2:f5 R 1:g3"
+    contents="+:(d7 e7 f7) *:h1 a:f5 b:f4 B a6 b5 c4 d3 e2 f1 j6 j5 j4 j3 j2 j1 R 1:g3"
     />

@@ Line 139: / Line 139: @@
     edges="none"
     visible="area(a7,a8,g8,h7,h1,g1)"
-    contents="R h1 g1 a8 b8 c8 d8 e8 f8 g8 1:f3 S gray:area(c5,c6,b7,b8,g8,h7,h1,g1)"
+    contents="R h1 g1 a8 b8 c8 d8 e8 f8 g8 1:f4 S gray:area(a7,a8,f8,g7, g4,h3,h1,g1)"
     />

@@ Line 40: / Line 40: @@
 (1) "Red always loses regardless of who goes first" if and only if b >= 2a+1
-(In other words, Red's average edge length <= Blue's+1).
+(In other words, Red's average edge length <= Blue's-1).
 (2) "Red always wins regardless of who goes first" if and only if b <= 2a-3
-(In other words, Red's average edge length >= Blue's-1).
+(In other words, Red's average edge length >= Blue's+1).
 Indeed, this conjecture is an analog of the results on parallelogram boards. I have verified this conjecture for a=1,2,3, and KataHex suggested that a=4,5,6 are possibly true.

@@ Line 3: / Line 3: @@
     contents="R e4 R h4 R c7 R g7 R c8 R d8 R e8 R f8"
     />
 The following pattern guarantees a stone on 7th row to connect to the edge. It was verified by me using HexProver developped by mirefek. Whether this connection is minimal is not known. This partially answers the single-stone 7th row edge template problem (The template exists but the shape is unknown).
@@ Line 32: / Line 34: @@
 For Blue playing a,b,c,d, Red can respond with X,Y,Y,Z respectively. For the rest Red can respond with b.