Difference between revisions of "User talk:Fjan2ej57w"

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(independent subgames)
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Hello.
+
==Hello :D==
<hexboard size="9x11"
+
Here is the way to make a hex board:
float="inline"
+
  [<][hexboard size=]["][*^v][x][*<>]
 +
  ["]//set size
 +
  [contents=]["][*colour] [*position]
 +
  ["]//put stones
 +
  [/>]
 +
then I will get this:
 +
  <hexboard size="7x6"
 +
  contents="R a3 b4"/>
 +
  [<][hexboard size=]["][5][x][8]["]
 +
  [contents=]["][R] [c4]["][/>]
 +
then I will get this:
 +
  <hexboard size="5x8"
 +
  contents="R c4"/>
 +
 
 +
  [<][hexboard size=]["3x4"]
 +
  [visible=]["][area(][a3,d3,d1,c1]
 +
  [)"]//set bondary by vertex
 +
  [edges=]["][bottom]["]
 +
  [coords=]["][none]["]//not coloring
 +
  [contents=]["][R c1]["]
 +
  [/>]
 +
 
 +
then I will get a [[ziggurat]]:
 +
  <hexboard size="3x4"
 +
  visible="area(a3,d3,d1,c1)"
 
   edges="bottom"
 
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="area(i1,a9,k9,k7,i5)"
+
   contents="R c1"
  contents="R h2 a3 f4 c7 b10"
+
/>
  />
+
  [<][hexboard size=]["6x7"]
 +
  [coords="none"]
 +
  [egdes="bottom"] 
 +
  [visible=]["]
 +
  [area(a6,g6,g4,f5,e4,d5,c4,b5,a4)]["]
 +
  [contents="R f5 d5 b5"]
 +
  [/>]
 +
<hexboard size="6x7"
 +
coords="none"
 +
egdes="bottom"
 +
visible="area(a6,g6,g4,f5,e4,d5,c4,b5,a4)"
 +
contents="R g4 e4 c4 a4"
 +
/>
 +
 
 +
a longer rampart :p
 +
  <hexboard size="6x15"
 +
  coords="none"
 +
  edges="bottom" visible="area(a6,o6,o1,n2,m1,l2,k1,j2,i1,h2,g1,f2,e1,d2,c1,b2,a1)"
 +
contents="R o1 m1 k1 i1 g1 e1 c1 a1 B b6 d6 f6 h6 j6 l6 n6"
 +
/>
 +
==some questions about hex==
 +
1.Is there a 2nd row ladder escape template that contains only empty cells?
 +
(It's equivalent to ask that given enough space, is a 2nd row ladder able to connect itself to the edge.
 +
2.find templates that escape ladders on both sides,e.g.,Edge template II escapes 2nd and 3rd row ladders on both size at the same time.
 +
3.is it possible to get a switchback from a 3rd row ladder?
 +
4.Does single stone 8th row edge template exist? Is there a better way to analyze such templates?
 +
5.is it possible to totally surround a single stone on an infinite board? I.e., to make the group coming from that stone remain bounded.
 +
6.Find the ways to prolong a losing game as much as possible.(or shorten a winning game as much as possible)
 +
7.Which leads to the question that how much space of an empty board would be filled if both sides play optimally.(I guess if n(approaching infinity) is the length of the board, then the game(with or without swap)would end in n^(3/2)•k moves, where k is a constant depending on whether the swap rule is used or not)
 +
8.[[Bridge ladders]] are common in actual games, so there must be some conclusions about it, comparing such "interior" bridges to an interval of edge.
 +
9.there is the concept of capture. And it can be used to determine the shape of an edge template. https://www.hexwiki.net/index.php/Theorems_about_templates
 +
more complicated and frequently occurring conditions of capturing and their corresponding strategy may be very helpful.
 +
 
 +
==independent subgames==
 +
<hexboard size="14x14"
 +
contents=" R a8 b8 c8 d8 e8 f8 g8 h8 i8 j8 k8 l8 m8 n8 B b2 d4 h2 j4 n2 b12 f10 l10 h12 i10 n12 e2 k2 i10"
 +
/>
 +
In some cases, a game is seperated into two or more independent subgames, just like the example above. In order to win, the only way for red is to connect his group to both sides. On the other hand, if red fails to connect his group to either side, blue is able to connect within that region instead. The board is actually seperated into two independent parts by red stones parallel to his own edge. No matter what happens in one part, the other part won't be affected at all.(except that the question of who would play first in a certain region)
 +
The upper part:
 +
<hexboard size="8x14"
 +
contents="R a8 b8 c8 d8 e8 f8 g8 h8 i8 j8 k8 l8 m8 n8 B b2 h2 d4 e2 k2 j4 k2 n2"
 +
/>
 +
And the lower part:
 +
<hexboard size="7x14"
 +
contents="R a1 b1 c1 d1 e1 f1 g1 h1 i1 j1 k1 l1 m1 n1 B b5 f3 h5 i3 l3 n5"
 +
/>
 +
 
 +
Sometimes, the board would be seperated into more than two independent parts. The position below is one example:
 +
 
 +
<hexboard size="14x14"
 +
contents="R a8 b8 c8 d8 e8 f8 g8 h8 i8 j8 k8 l8 m8 n8 B b13 e13 h13 k13 h1 h2 h3 h4 h5 h6 h7 n13 i11"
 +
/>
 +
Now the board is seperated into at least 3 independent parts:
 +
1.
 +
<hexboard size="14x14"
 +
visible="area(a8,a14,n14,n8)"
 +
contents="R a8 b8 c8 d8 e8 f8 g8 h8 i8 j8 k8 l8 m8 n8 B b13 e13 h13 k13 h1 h2 h3 h4 h5 h6 h7 n13 i11"
 +
/>
 +
2.
 +
<hexboard size="14x14"
 +
visible="area(a1,a8,h8,h1)"
 +
contents="R a8 b8 c8 d8 e8 f8 g8 h8 i8 j8 k8 l8 m8 n8 B b13 e13 h13 k13 h1 h2 h3 h4 h5 h6 h7 n13 i11"
 +
/>
 +
3.
 +
<hexboard size="14x14"
 +
visible="area(h8,h1,n1,n8)"
 +
contents="R a8 b8 c8 d8 e8 f8 g8 h8 i8 j8 k8 l8 m8 n8 B b13 e13 h13 k13 h1 h2 h3 h4 h5 h6 h7 n13 i11"
 +
/>
 +
Blue will achieve the final victory if and only if she wins 1 or both 2 and 3.
 +
On the other hand, red will win if and only if he wins 1 and either 2 or 3.
 +
 
 +
It actually doesn't matter what's the relative
 +
position of these subgames or how they are interconnected in the original board. What We really care is their logical relationships, that is, the winning condition for both players.
 +
 
 +
These examples looks more or less artificial, but the phenomen of independent subgames actually occurs quite often in actual game play. By merely identifying such patterns, a game would be much easier to analyze in some cases. The following are some examples with subgames  having more vague boundaries.

Revision as of 13:21, 26 April 2024

Hello :D

Here is the way to make a hex board:

 [<][hexboard size=]["][*^v][x][*<>] 
 ["]//set size
 [contents=]["][*colour] [*position] 
 ["]//put stones
 [/>]

then I will get this:

abcdef1234567
 [<][hexboard size=]["][5][x][8]["]
 [contents=]["][R] [c4]["][/>]

then I will get this:

abcdefgh12345
 [<][hexboard size=]["3x4"]
 [visible=]["][area(][a3,d3,d1,c1] 
 [)"]//set bondary by vertex
 [edges=]["][bottom]["]
 [coords=]["][none]["]//not coloring
 [contents=]["][R c1]["]
 [/>]

then I will get a ziggurat:

 [<][hexboard size=]["6x7"]
 [coords="none"]
 [egdes="bottom"]  
 [visible=]["] 
 [area(a6,g6,g4,f5,e4,d5,c4,b5,a4)]["]
 [contents="R f5 d5 b5"]
 [/>]

a longer rampart :p

some questions about hex

1.Is there a 2nd row ladder escape template that contains only empty cells? (It's equivalent to ask that given enough space, is a 2nd row ladder able to connect itself to the edge. 2.find templates that escape ladders on both sides,e.g.,Edge template II escapes 2nd and 3rd row ladders on both size at the same time. 3.is it possible to get a switchback from a 3rd row ladder? 4.Does single stone 8th row edge template exist? Is there a better way to analyze such templates? 5.is it possible to totally surround a single stone on an infinite board? I.e., to make the group coming from that stone remain bounded. 6.Find the ways to prolong a losing game as much as possible.(or shorten a winning game as much as possible) 7.Which leads to the question that how much space of an empty board would be filled if both sides play optimally.(I guess if n(approaching infinity) is the length of the board, then the game(with or without swap)would end in n^(3/2)•k moves, where k is a constant depending on whether the swap rule is used or not) 8.Bridge ladders are common in actual games, so there must be some conclusions about it, comparing such "interior" bridges to an interval of edge. 9.there is the concept of capture. And it can be used to determine the shape of an edge template. https://www.hexwiki.net/index.php/Theorems_about_templates more complicated and frequently occurring conditions of capturing and their corresponding strategy may be very helpful.

independent subgames

abcdefghijklmn1234567891011121314

In some cases, a game is seperated into two or more independent subgames, just like the example above. In order to win, the only way for red is to connect his group to both sides. On the other hand, if red fails to connect his group to either side, blue is able to connect within that region instead. The board is actually seperated into two independent parts by red stones parallel to his own edge. No matter what happens in one part, the other part won't be affected at all.(except that the question of who would play first in a certain region) The upper part:

abcdefghijklmn12345678

And the lower part:

abcdefghijklmn1234567

Sometimes, the board would be seperated into more than two independent parts. The position below is one example:

abcdefghijklmn1234567891011121314

Now the board is seperated into at least 3 independent parts: 1.

abcdefghijklmn891011121314

2.

abcdefgh12345678

3.

hijklmn12345678

Blue will achieve the final victory if and only if she wins 1 or both 2 and 3. On the other hand, red will win if and only if he wins 1 and either 2 or 3.

It actually doesn't matter what's the relative position of these subgames or how they are interconnected in the original board. What We really care is their logical relationships, that is, the winning condition for both players.

These examples looks more or less artificial, but the phenomen of independent subgames actually occurs quite often in actual game play. By merely identifying such patterns, a game would be much easier to analyze in some cases. The following are some examples with subgames having more vague boundaries.