Equivalent patterns - Revision history

Selinger: Some copy-editing.

2023-05-09T02:17:28Z

Some copy-editing.

Demer: /* Second example game */ fixed player-name

2021-07-06T07:11:51Z

‎Second example game: fixed player-name

Demer: reverting myself because I was persuaded that "row" is accurate-enough

2021-01-24T08:29:26Z

reverting myself because I was persuaded that "row" is accurate-enough

Demer: /* Edge equivalence */ replaced references to "row"s with "line"s, since they can be columns

2021-01-19T06:11:54Z

‎Edge equivalence: replaced references to "row"s with "line"s, since they can be columns

Tompo1: /* Example 5a (Opening a1+b1 loses) */ Typo

2021-01-17T15:19:10Z

‎Example 5a (Opening a1+b1 loses): Typo

Tompo1: /* Example 5b (Opening A1+B1 loses) */ application of Example 4 seems more obvious than Example 1

2021-01-17T13:42:33Z

‎Example 5b (Opening A1+B1 loses): application of Example 4 seems more obvious than Example 1

Tompo1: Added Edge equivalence section

2021-01-17T13:24:30Z

Added Edge equivalence section

Selinger: Copy-editing.

2020-12-08T02:40:15Z

Copy-editing.

Selinger: Converted remaining diagrams.

2020-12-08T02:30:41Z

Converted remaining diagrams.

Selinger: Converted more diagrams.

2020-12-08T02:12:02Z

Converted more diagrams.

@@ Line 65: / Line 65: @@
 The equivalence is obtained by applying example 1 twice.
-Both above examples are instances of the following rule to produce equivalence pairs. Given a [[chain]] G, let the '''neighborhood''' of G, neigh(G), be the set of cells that are adjacent to G but do not belong to it. In a given pattern P<sub>1</sub>, suppose that G is a red chain, and that C is a cell in neigh(G), which may be empty or occupied by Blue. Let P<sub>2</sub> be the result of placing a red piece in C (removing a Blue stone, if necessary). Therefore P<sub>2</sub> contains a chain G' containing G and C. If neigh(G')=neigh(G) then P<sub>1</sub> and P<sub>2</sub> are equivalent. Of course the dual argument also holds for blue chains.
+Both above examples are instances of the following rule to produce equivalence pairs. Given a [[chain]] G, let the '''neighborhood''' of G, neigh(G), be the set of cells that are adjacent to G but do not belong to it. In a given pattern P<sub>1</sub>, suppose that G is a red chain, and that C is a cell in neigh(G), which may be empty or occupied by Blue. Let P<sub>2</sub> be the result of placing a red piece in C (removing a blue stone, if necessary). Therefore P<sub>2</sub> contains a chain G' containing G and C. If neigh(G')=neigh(G) then P<sub>1</sub> and P<sub>2</sub> are equivalent. Of course the dual argument also holds for blue chains.
 This rule justifies the following equivalent pairs:
@@ Line 235: / Line 235: @@
 <hex>R13 C13 Q1 Hc1 Hd6 Hd8 Hd9 Hd10 Hd11 Hd12 He6 He12 Hf5 Hf10 Hf12 Hg5 Hg6 Hg12 Hh6 Hh8 Hh9 Hh12 Hi5 Hi11 Hj4 Hj6 Hj11 Hk3 Hk11 Hl11 Vc8 Vc9 Vc10 Vc11 Vc12 Vd7 Ve7 Ve8 Ve10 Ve11 Vf4 Vf6 Vf8 Vf11 Vg11 Vh5 Vh7 Vh11 Vi6 Vi8 Vi10 Vj5 Vj10 Vk4 Vk10 Vl10 Vm2</hex>
-Clearly, m2 is strongly connected to the top, because the stone in f4 is a [[ladder escape]]. On the other hand, it is strongly connected to the bottom exactly if blue cannot connect k3 with the right, using j6 and maybe the [[group]] in h8-h9-f10 as a ladder escape. In fact he cannot do it, and it is much clearer if some patterns are locally replaced by other equivalent ones, rendering:
+Clearly, m2 has a [[virtual connection]] to the top, because the stone at f4 is a [[ladder escape]]. On the other hand, m2 is connected to the bottom exactly if Blue cannot connect k3 to the right, using j6 and maybe the [[group]] at h8-h9-f10 as a ladder escape. In fact, Blue cannot do this. This becomes much clearer when some patterns are locally replaced by other equivalent ones, yielding:
 <hex>R13 C13 Q1 Hc1 Hd6 Vd8 Vd9 Vd10 Vd11 Vd12 He6 Ve12 Hf5 Vf10 Vf12 Hg5 Hg6 Vg12 Hh6 Vh8 Vh9 Vh12 Hi5 Vi11 Hj4 Hj6 Vj11 Hk3 Vk11 Vl11 Vc8 Vc9 Vc10 Vc11 Vc12 Vd7 Ve7 Ve8 Ve10 Ve11 Vf4 Hf6 Vf8 Vf11 Vg11 Hh5 Vh7 Vh11 Vi6 Vi8 Vi10 Vj5 Vj10 Vk4 Vk10 Vl10 Vm2 Ve9 Vf7 Vf9 Vg7 Vg8 Vg9 Vg10 Vh10 Vi7 Vi9 Vb13 Vc13 Vm10 Vm11 Vi12 Vj12 Vk12 Vl12 Vm12 Vd13 Ve13 Vf13 Vg13 Vh13 Vi13 Vj13 Vk13 Vl13 Vm13</hex>
-The changes have been:
+The changes are:
-* f6 and h5 swap color, as in Example 1.
+* f6 and h5 swap colors, as in Example 1.
-* The blue group from d8 to l11 is completely wrapped by a strongly connected group belonging to the opponent, except for a narrow section (typically, 2 cells). This kind of groups typically are captured, exactly in the same reason as in Example 1.
+* The blue group from d8 to l11 is completely wrapped by a virtually connected group belonging to Red, except for a narrow section (typically, 2 cells). These kinds of groups are typically captured, for exactly the same reason as in Example 1.
-* If blue moves to i7, red moves to i9 and conversely, in both cases enclosing the blue group h8-h9-f10 as before. So, we can use the second rule for detecting equivalent patterns, capturing it.
+* If Blue moves at i7, Red moves at i9 and vice versa. In both cases, the blue group h8-h9-f10 will be enclosed. So we can use the second rule for detecting equivalent patterns, capturing it.
-* The equivalence in Example 5 can be used for the stone in c12.
+* The equivalence in Example 5 can be used for the stone at c12.
-* The equivalence in Example 4 can be used, adding a stone for Red at m11.
+* The equivalence in Example 4 can be used to add a stone for Red at m11.
-* The area in the bottom of the board is now surrounded by a red chain and a blue chain. Therefore, it may be filled as we please.
+* The area near the bottom of the board is now surrounded by a red chain and a blue chain. Therefore, the cells in this area are [[dead cell|dead]] and can be filled with any color.
-The blue stone in j6 remained, completely alone and too near to the red group to be of any use in such a small region, so it is obvious that Red has won.
+The blue stone at j6 remains, but is completely isolated and too close to the red group to be of any use in such a small region. Therefore, it is now obvious that Red has won.
 === Second example game ===

@@ Line 288: / Line 288: @@
       R 11:h9 B 12:g9 R 13:h7"
    />
-This time, Black's 13 acts as a ladder stone for a 2nd row ladder stone in the opposite direction, which the ziggurat again escapes.
+This time, Red's 13 acts as a ladder stone for a 2nd row ladder stone in the opposite direction, which the ziggurat again escapes.
 [[Category: Strategy]]
 [[Category: Computer Hex]]
 [[Category: Theory]]

@@ Line 3: / Line 3: @@
 == Edge equivalence==
-Edges are equivalent to edge-connected lines of pieces on a larger board.
+Edges are equivalent to edge-connected rows of pieces on a larger board.
 === Example 0 ===
-Any position on an <em>nxm</em> board is equivalent to the same position on the interior (i.e. the cells non-adjacent to the edges) of an <em>(n+2)x(m+2)</em> board with the lines next to each edge filled with friendly pieces and [[dead cell]]s at the vertex cells (or equivalently either red or blue pieces at any of the vertex cells)
+Any position on an <em>nxm</em> board is equivalent to the same position on the interior (i.e. the cells non-adjacent to the edges) of an <em>(n+2)x(m+2)</em> board with the first rows next to each edge filled with friendly pieces and [[dead cell]]s at the vertex cells (or equivalently either red or blue pieces at any of the vertex cells)
 <hexboard size="4x4"
@@ Line 19: / Line 19: @@
    />
-This allows the examples below to be applied at the edges: either formally, by converting to the larger board, applying the equivalence in the example, then converting back to the smaller board; or informally, by considering an edge to be a line of friendly pieces.
+This allows the examples below to be applied at the edges: either formally, by converting to the larger board, applying the equivalence in the example, then converting back to the smaller board; or informally, by considering an edge to be a row of friendly pieces.
 == Equivalence by capture ==

@@ Line 3: / Line 3: @@
 == Edge equivalence==
-Edges are equivalent to edge-connected rows of pieces on a larger board.
+Edges are equivalent to edge-connected lines of pieces on a larger board.
 === Example 0 ===
-Any position on an <em>nxm</em> board is equivalent to the same position on the interior (i.e. the cells non-adjacent to the edges) of an <em>(n+2)x(m+2)</em> board with the first rows next to each edge filled with friendly pieces and [[dead cell]]s at the vertex cells (or equivalently either red or blue pieces at any of the vertex cells)
+Any position on an <em>nxm</em> board is equivalent to the same position on the interior (i.e. the cells non-adjacent to the edges) of an <em>(n+2)x(m+2)</em> board with the lines next to each edge filled with friendly pieces and [[dead cell]]s at the vertex cells (or equivalently either red or blue pieces at any of the vertex cells)
 <hexboard size="4x4"
@@ Line 19: / Line 19: @@
    />
-This allows the examples below to be applied at the edges: either formally, by converting to the larger board, applying the equivalence in the example, then converting back to the smaller board; or informally, by considering an edge to be a row of friendly pieces.
+This allows the examples below to be applied at the edges: either formally, by converting to the larger board, applying the equivalence in the example, then converting back to the smaller board; or informally, by considering an edge to be a line of friendly pieces.
 == Equivalence by capture ==

@@ Line 138: / Line 138: @@
 In response to a1+b1 by Red, Blue can play at b2. This captures a2 and a3 by Example 5, and then a1 and b1 by applying Example 4 twice (use of these examples at the edges is justified by Example 0 above). Thus, the left position is equivalent to the right one. This includes the cells a1 and a2, so the original position is losing for Red by a strategy stealing argument.
-In fact, making the opening play b2 is equivalent to playing 5 stones at a1, a2, b1, b1, and c1. However this is known to produce a losing position on boards of size 4, 7 & 8 (see [[small boards]]).
+In fact, making the opening play b2 is equivalent to playing 5 stones at a1, a2, b1, b2, and c1. However this is known to produce a losing position on boards of size 4, 7 & 8 (see [[small boards]]).
 This example also shows that the opening moves a1 and b1 should never be swapped. Playing at b2 is always at least as good for Blue as swapping (but there are even better moves for Blue available elsewhere on the board).

@@ Line 122: / Line 122: @@
 This pattern, along with Example 4, can be used to show that the 2-move opening a1 + b1 (or either single move) for Red loses on a board 3x3 or bigger:
-=== Example 5b (Opening A1+B1 loses) ===
+=== Example 5a (Opening a1+b1 loses) ===
 <hexboard size="4x3"
@@ Line 136: / Line 136: @@
    />
-In response to a1+b1 by Red, Blue can play at b2. This captures a2 and a3 by Example 5, and then a1 and b1 by applying Example 1 twice. Thus, the left position is equivalent to the right one. This includes the cells a1 and a2, so the original position is losing for Red by a strategy stealing argument.
+In response to a1+b1 by Red, Blue can play at b2. This captures a2 and a3 by Example 5, and then a1 and b1 by applying Example 4 twice (use of these examples at the edges is justified by Example 0 above). Thus, the left position is equivalent to the right one. This includes the cells a1 and a2, so the original position is losing for Red by a strategy stealing argument.
 In fact, making the opening play b2 is equivalent to playing 5 stones at a1, a2, b1, b1, and c1. However this is known to produce a losing position on boards of size 4, 7 & 8 (see [[small boards]]).

← Older revision		Revision as of 13:24, 17 January 2021
Line 1:		Line 1:
	We say that two Hex [[pattern]]s (subsets of a board) are '''equivalent patterns''' if, when one of them occurs embedded in ''any'' Hex board, it could be replaced by the other and the side who has winning strategy does not change.		We say that two Hex [[pattern]]s (subsets of a board) are '''equivalent patterns''' if, when one of them occurs embedded in ''any'' Hex board, it could be replaced by the other and the side who has winning strategy does not change.
		+
		+	== Edge equivalence==
		+
		+	Edges are equivalent to edge-connected rows of pieces on a larger board.
		+
		+	=== Example 0 ===
		+
		+	Any position on an <em>nxm</em> board is equivalent to the same position on the interior (i.e. the cells non-adjacent to the edges) of an <em>(n+2)x(m+2)</em> board with the first rows next to each edge filled with friendly pieces and [[dead cell]]s at the vertex cells (or equivalently either red or blue pieces at any of the vertex cells)
		+
		+	<hexboard size="4x4"
		+	float="inline"
		+	coords="none"
		+	contents="S area(a1,d1,d4,a4)"
		+	/> and <hexboard size="6x6"
		+	float="inline"
		+	coords="none"
		+	contents="S area(b2,e2,e5,b5) R line(b1,e1) line(b6,e6) B line(a2,a5) line(f2,f5) E :a1 :f1 :f6 :a6"
		+	/>
		+
		+	This allows the examples below to be applied at the edges: either formally, by converting to the larger board, applying the equivalence in the example, then converting back to the smaller board; or informally, by considering an edge to be a row of friendly pieces.

	== Equivalence by capture ==		== Equivalence by capture ==

@@ Line 1: / Line 1: @@
-We say that two hex [[pattern]]s (subsets of a board) are '''equivalent patterns''' if, when one of them occurs embedded in ''any'' hex board, it could be replaced by the other and the side who has winning strategy does not change.
+We say that two Hex [[pattern]]s (subsets of a board) are '''equivalent patterns''' if, when one of them occurs embedded in ''any'' Hex board, it could be replaced by the other and the side who has winning strategy does not change.
 == Equivalence by capture ==
@@ Line 24: / Line 24: @@
 The blue stone in the left pattern is [[captured cell|captured]] by Red. Therefore, the position is equivalent to one where this stone is actually red.
-The knowledge of equivalent patterns turns out to be very useful to play Hex, because it allows you to disregard some pieces in the board, or prune the analysis tree. In my opinion, some patterns lead to positions that are much more clear than other equivalent patterns. My intention here is to write always in second place such pattens, and the use that I make of equivalent patterns is that in my games, mentally I always replace the first patterns by the equivalent counterparts.
+The knowledge of equivalent patterns turns out to be very useful for playing Hex, because it allows the player to disregard some pieces in the board, or prune the analysis tree. Some positions are much clearer than other equivalent positions. In the examples below, the simpler pattern is usually written on the right. One can make use of these equivalent patterns by mentally always replacing the left pattern with the right one.
 Here are several examples of pairs of equivalent positions, and a short explanation on the way to prove them so.
@@ Line 43: / Line 43: @@
    />
-The equivalence is obtained by applying two times the example 1.
+The equivalence is obtained by applying example 1 twice.
-Both examples before are instances of the following rule to produce equivalence pairs. Given a [[chain]] G, let the '''neighborhood''' of G, neigh(G), be the set of cells next to one of those in G but not belonging to it. In a given pattern P<sub>1</sub>, suppose that G is a chain owned by the player A, and that C is a cell in neigh(G) such that any cell of A next to C belongs to G. Let P<sub>2</sub> be the pattern that results when A occupies C (removing an opponent stone, if necessary), therefore P<sub>2</sub> contains a chain G' containing G and C. If neigh(G')=neigh(G) then P<sub>1</sub> and P<sub>2</sub> are equivalent.
+Both above examples are instances of the following rule to produce equivalence pairs. Given a [[chain]] G, let the '''neighborhood''' of G, neigh(G), be the set of cells that are adjacent to G but do not belong to it. In a given pattern P<sub>1</sub>, suppose that G is a red chain, and that C is a cell in neigh(G), which may be empty or occupied by Blue. Let P<sub>2</sub> be the result of placing a red piece in C (removing a Blue stone, if necessary). Therefore P<sub>2</sub> contains a chain G' containing G and C. If neigh(G')=neigh(G) then P<sub>1</sub> and P<sub>2</sub> are equivalent. Of course the dual argument also holds for blue chains.
 This rule justifies the following equivalent pairs:
@@ Line 79: / Line 79: @@
    />
-Another rule producing equivalent patterns: If there are two empty cells C<sub>1</sub> and C<sub>2</sub> in a pattern, such that if the opponent of the player A occupies one of them, A can occupy the other capturing the latter, then an equivalent position is obtained if both C<sub>1</sub> and C<sub>2</sub> are occupied by A.
+Another rule producing equivalent patterns: If there are two empty cells C<sub>1</sub> and C<sub>2</sub> in a pattern, such that if Blue occupies one of them, Red can occupy the other capturing the former, then an equivalent position is obtained if both C<sub>1</sub> and C<sub>2</sub> are occupied by Red.
-Equivalent pairs obtained with such rule:
+Equivalent pairs obtained with this rule:
 === Example 5 ===
@@ Line 98: / Line 98: @@
    />
-This is a very instructive pattern because it shows that playing 2 rows out from a [[friendly]] edge with 2 free cells on the first row below the play is equivalent to playing at all three cells simultaneously, and hence at least as good as playing at either cell on the first row.  So as a rule, never play on the first row in such a situation.  We can see this by viewing the 3 upper red stones as part of a friendly edge.
+This is a very instructive pattern because it shows that playing 2 rows out from a [[friendly]] edge with 2 free cells on the first row below the play is equivalent to playing at all three cells simultaneously, and hence at least as good as playing at either cell on the first row.  So as a rule, one should never play on the first row in such a situation.  We can see this by viewing the 3 upper red stones as part of a friendly edge.
-This pattern, along with Example 4 can be used to show that the two move start A1 + B1 (or either single move) for Red loses on a board 3x3 or bigger:
+This pattern, along with Example 4, can be used to show that the 2-move opening a1 + b1 (or either single move) for Red loses on a board 3x3 or bigger:
 === Example 5b (Opening A1+B1 loses) ===
@@ Line 116: / Line 116: @@
    />
-So, in response to a1+b1 by Red, Blue can play at b2. This captures a2 and a3 by Example 5, and then a1 and b1 by applying Example 1 twice. Thus, the left position is equivalent to the right one. This includes the cells a1 and a2, so the original position is losing for Red by a strategy stealing argument.
+In response to a1+b1 by Red, Blue can play at b2. This captures a2 and a3 by Example 5, and then a1 and b1 by applying Example 1 twice. Thus, the left position is equivalent to the right one. This includes the cells a1 and a2, so the original position is losing for Red by a strategy stealing argument.
 In fact, making the opening play b2 is equivalent to playing 5 stones at a1, a2, b1, b1, and c1. However this is known to produce a losing position on boards of size 4, 7 & 8 (see [[small boards]]).
@@ Line 183: / Line 183: @@
    />
-Third rule for equivalent patterns (rather obvious) rule: Any area surrounded by a single chain of each enemy may be randomly filled. This happens because the outcome of the game does not depend on it at all.
+Here is a third (rather obvious) rule for equivalent patterns: Any area surrounded by a single chain of the opponent may be randomly filled. This happens because the outcome of the game does not depend on it at all.
 == Equivalence for reasons other than capture ==

@@ Line 104: / Line 104: @@
 === Example 5b (Opening A1+B1 loses) ===
-<hex>R5 C9 1:*VVV**VVV 2:HVV_*BBB_ 3:H_H_*HHH_ 4:H___*HH__ 5:H****H***</hex>
+<hexboard size="4x3"
-So, in response to A1+B1 by Red, Blue can play at B2, reaching a position equivalent to just blue stones at A1, A2, A3, B1 & B2.  This includes the cells A1 & A2, so the original position is lost for Red by a strategy stealing argument.
+So, in response to a1+b1 by Red, Blue can play at b2. This captures a2 and a3 by Example 5, and then a1 and b1 by applying Example 1 twice. Thus, the left position is equivalent to the right one. This includes the cells a1 and a2, so the original position is losing for Red by a strategy stealing argument.
-In fact, making the opening play B2 is equivalent to playing 5 stones at A1, A2, B1, B2 & C1, however this is known to produce a losing position on boards of size 4, 7 & 8 (see [[small boards]]).
+In fact, making the opening play b2 is equivalent to playing 5 stones at a1, a2, b1, b1, and c1. However this is known to produce a losing position on boards of size 4, 7 & 8 (see [[small boards]]).
 === Example 6 ===
-<hex>R3 C5 Va3 Va1 Vb1 Vb3 Ve1 Vd1 Vd2 Vd3 Ve2 Ve3 Sc2 Sc3 Sc1</hex>
+<hexboard size="3x2"
 === Example 7 ===
-<hex>R2 C9 Va2 Vb1 Vc1 Vd1 Vd2 Sa1 Se1 Se2 Sf1 Vf2 Vg1 Vg2 Vh1 Vh2 Vi1 Vi2</hex>
+<hexboard size="2x4"
+  float="inline"
 Using both rules together:
 === Example 8 ===
-<hex>R3 C9 Vc1 Vb3 Ha2 Hd2 Vh1 Vg3 Vg2 Vh2 Hf2 Hi2 Sa1 Sb1 Sd1 Se1 Sf1 Sg1 Si1 Se2 Sa3 Sc3 Sd3 Se3 Sf3 Sh3 Si3</hex>
+<hexboard size="3x4"
 === Example 9 ===
 ''(generalization of the Example 5, for any horizontal length)''
-<hex>R3 C11 Va3 Vb3 Vc3 Vd3 Ve3 Vc1 Vd1 Ve1 Hc2 Hd2 Vg3 Vh2 Vh3 Vi1 Vi2 Vi3 Vj1 Vj2 Vj3 Vk1 Vk2 Vk3 Sa1 Sa2 Sb1 Sf1 Sf2 Sf3 Sg1 Sg2 Sh1</hex>
+<hexboard size="3x5"
 Third rule for equivalent patterns (rather obvious) rule: Any area surrounded by a single chain of each enemy may be randomly filled. This happens because the outcome of the game does not depend on it at all.
@@ Line 135: / Line 192: @@
 The following two patterns are equivalent:
-<hexboard size="3x9"
+<hexboard size="3x4"
-   coords="hide"
+  float="inline"
-   contents="E *:e1 *:e2 *:e3
+  coords="none"
-   E *:a1 *:a2 *:b3 *:c3 *:d3 R b1 c1 d1 d2 B a3 B b2 E a:c2
+  edges="none"
-   E *:f1 *:f2 *:g3 *:h3 *:i3 R g1 h1 i1 i2 B f3 B h2 E a:g2"
+  visible="-a1 a2 b3--d3"
-   />
+  contents="R b1--d1 d2 B a3 c2 E a:b2"
-This is easily seen by noticing that if the cells marked "a" are both blue, the patterns become equal, whereas if the cells marked "a" are both red, then the blue piece next to each of them is dead and therefore the patterns become equivalent. So whenever one player's strategy calls for playing in the cell marked "a" in one of these two patterns, the same player can play in the cell marked "a" in the other pattern.
+  /> and <hexboard size="3x4"
 == Practical examples ==

@@ Line 29: / Line 29: @@
 === Example 2 ===
-<hex>R3 C9 Va2 Vb1 Vc1 Vd1 Vd2 Vc3 Hb2 Hc2 Vf2 Vg1 Vh1 Vi1 Vi2 Vh3 Vg2 Vh2 Sa1 Sa3 Sb3 Sd3 Se3 Sf3 Sg3 Si3 Se1 Sf1 Se2</hex>
+<hexboard size="3x4"
 The equivalence is obtained by applying two times the example 1.
@@ Line 38: / Line 50: @@
 === Example 3 ===
-<hex>R3 C6 Va2 Vb1 Vc1 Hb2 Hb3 Vd2 Ve1 Vf1 Ve2 He3 Sa1 Sa3 Sc2 Sc3 Sd1 Sd3 Sf2 Sf3</hex>
+<hexboard size="3x3"
 === Example 4 ===
-<hex>R3 C6 Ha3 Hb3 Vb1 Vc1 Hb2 Hd3 He3 Ve1 Vf1 Ve2 Sa1 Sa2 Sc2 Sc3 Sd1 Sd2 Sf2 Sf3</hex>
+<hexboard size="3x3"
 Another rule producing equivalent patterns: If there are two empty cells C<sub>1</sub> and C<sub>2</sub> in a pattern, such that if the opponent of the player A occupies one of them, A can occupy the other capturing the latter, then an equivalent position is obtained if both C<sub>1</sub> and C<sub>2</sub> are occupied by A.
@@ Line 48: / Line 84: @@
 === Example 5 ===
-<hex>R3 C7 Va1 Vb1 Vc1 Va3 Ve1 Vf1 Vg1 Ve3 Ve2 Vf2 Sc2 Sd2 Sb3 Sc3 Sd3 Sf3 Sg2 Sg3 Sd1</hex>
+<hexboard size="3x3"
 This is a very instructive pattern because it shows that playing 2 rows out from a [[friendly]] edge with 2 free cells on the first row below the play is equivalent to playing at all three cells simultaneously, and hence at least as good as playing at either cell on the first row.  So as a rule, never play on the first row in such a situation.  We can see this by viewing the 3 upper red stones as part of a friendly edge.