Solutions to worst move puzzles - Revision history

Selinger: /* Puzzle 4 */ fixed a typo

2023-10-07T03:31:05Z

‎Puzzle 4: fixed a typo

Selinger: /* Puzzle 4 */ minor edit

2023-07-06T04:05:39Z

‎Puzzle 4: minor edit

Selinger: /* Approaches to solving the puzzles */ Wording.

2023-07-04T20:47:03Z

‎Approaches to solving the puzzles: Wording.

Selinger: /* Puzzle 5 */ Explicit answer for uniformity.

2023-07-02T23:04:27Z

‎Puzzle 5: Explicit answer for uniformity.

Selinger: /* Approaches to solving the puzzles */ added a link

2023-07-02T23:00:42Z

‎Approaches to solving the puzzles: added a link

Selinger: Rewrote the answers to the first 4 puzzles in terms of mustplay analysis, and added an explanation.

2023-07-02T22:59:57Z

Rewrote the answers to the first 4 puzzles in terms of mustplay analysis, and added an explanation.

Hexanna: reorder puzzles by difficulty

2023-07-02T21:17:59Z

reorder puzzles by difficulty

Selinger: /* Puzzle 2 */ Some editing, and added links to templates.

2023-07-02T15:46:01Z

‎Puzzle 2: Some editing, and added links to templates.

Hexanna: grammar

2023-07-02T12:38:20Z

grammar

Hexanna: link back to puzzles, category

2023-07-02T04:48:17Z

link back to puzzles, category

@@ Line 59: / Line 59: @@
 However, when we compute the carrier of this connection, we find that it consists of all of the empty cells on the board! So we have not yet found any losing move for Red.
-Now, consider what happens if Blue starts at B. After an optional bridge intrusion, Red must defend the upper left corner, and then Red can get B, for example like this:
+Now, consider what happens if Blue starts at B. After an optional bridge intrusion, Red must defend the upper left corner, and then Blue can get A, for example like this:
 <hexboard size="6x6"
    contents="R a5 d3 B d1 f4 B 1:e2 R 2:d2 B 3:e1 R 4:b2 B 5:b3 R 6:c2 B 7:c5 E *:a6,c4 S blue:area(e2,f1,f2)"

@@ Line 43: / Line 43: @@
 === Puzzle 4 ===
-This puzzle is a bit harder. With Blue to move, it is perhaps not immediately obvious how Blue can win. The key points for Blue are A, B, and C (and in fact, any one of those three moves is winning for Blue):
+This puzzle is a bit harder. With Blue to move, it is perhaps not immediately obvious how Blue can win. The key points for Blue are A, B, and C (and in fact, each one of these three moves is winning for Blue):
 <hexboard size="6x6"

@@ Line 3: / Line 3: @@
 There are several approaches to solving the worst move puzzles. The brute-force method is to try every possible move and check whether it is winning or losing, until a losing move is found. This method can be very labor intensive, especially since it is often hard to decide whether a particular move is winning or losing.
-A more principled approach is using the concept of the [[mustplay region]]. Let's say the puzzle is "Red to play the unique losing move". Since Red has a losing move, it is clear that [[passing]] would also be losing for Red. We may therefore start by asking: if it were Blue's turn in the puzzle, then how could Blue win? Once a Blue connection has been identified, we can then determine the [[carrier]] of that connection. The carrier of Blue's connection is the set of all cells that are required for the connection. If there is some cell that is not in the carrier of Blue's connection, then it is a losing move for Red. If the carrier consists of all the empty cells of the puzzle, then we must look for a different Blue connection.
+A more principled approach is using the concept of the [[mustplay region]]. Let's say the puzzle is "Red to play the unique losing move". Since Red has a losing move, it is clear that [[passing]] would also be losing for Red. We may therefore start by asking: if it were Blue's turn in the puzzle, then how could Blue win? Once a Blue connection has been identified, we can then determine the [[carrier]] of that connection. The carrier of Blue's connection is the set of all cells that are required for the connection. If there is some empty cell that is not in the carrier of Blue's connection, then it is a losing move for Red. If there is no such cell, then we must look for a different Blue connection.
-Once a losing move has been found, proving that it is the ''only'' losing move is harder. However, this is not usually the objective of the puzzle, which guarantees as part of the problem statement that there is only one losing move.
+Once a losing move has been found, proving that it is the ''only'' losing move is harder. However, this is not usually the objective of the puzzle.
 == Solutions to individual puzzles ==

← Older revision		Revision as of 23:04, 2 July 2023
Line 83:		Line 83:
	/>		/>

		+	'''Answer:''' d1

	== See also ==		== See also ==

@@ Line 3: / Line 3: @@
 There are several approaches to solving the worst move puzzles. The brute-force method is to try every possible move and check whether it is winning or losing, until a losing move is found. This method can be very labor intensive, especially since it is often hard to decide whether a particular move is winning or losing.
-A more principled approach is using the concept of the [[mustplay region]]. Let's say the puzzle is "Red to play the unique losing move". Since Red has a losing move, it is clear that passing would also be losing for Red. We may therefore start by asking: if it were Blue's turn in the puzzle, then how could Blue win? Once a Blue connection has been identified, we can then determine the [[carrier]] of that connection. The carrier of Blue's connection is the set of all cells that are required for the connection. If there is some cell that is not in the carrier of Blue's connection, then it is a losing move for Red. If the carrier consists of all the empty cells of the puzzle, then we must look for a different Blue connection.
+A more principled approach is using the concept of the [[mustplay region]]. Let's say the puzzle is "Red to play the unique losing move". Since Red has a losing move, it is clear that [[passing]] would also be losing for Red. We may therefore start by asking: if it were Blue's turn in the puzzle, then how could Blue win? Once a Blue connection has been identified, we can then determine the [[carrier]] of that connection. The carrier of Blue's connection is the set of all cells that are required for the connection. If there is some cell that is not in the carrier of Blue's connection, then it is a losing move for Red. If the carrier consists of all the empty cells of the puzzle, then we must look for a different Blue connection.
 Once a losing move has been found, proving that it is the ''only'' losing move is harder. However, this is not usually the objective of the puzzle, which guarantees as part of the problem statement that there is only one losing move.

@@ Line 1: / Line 1: @@
 === Puzzle 1 ===
 <hexboard size="4x4"
-   contents="R a1 b1 b3 B a2 a3 a4 R 1:c1 B 2:b4 R 3:d3 B 4:c2"
+   contents="R a1 b1 b3 B a2 a3 a4 B 1:c2 R 2:b2 B 3:b4 S blue:area(d1,b2,c3,b4,d4),a3,a4"
    />
 === Puzzle 2 ===
 <hexboard size="4x4"
-   contents="R a3 B d2 R 1:d1 B 2:c2 R 3:b2 B 4:a4"
+   contents="R a3 B d2 B 1:c2 R 2:b2 B 3:a4 S blue:all-a3,d1"
    />
 === Puzzle 3 ===
 <hexboard size="5x5"
-   contents="R d2 d1 B e1 *:b1 S gray:area(c4,e2,e5,c5) red:area(d1,e1,d2)"
+   contents="R d2 d1 B e1 b1 B 1:c4 R 2:b4 B 3:c3 R 4:b3 B 5:c2 S blue:all-d1,d2,e1,b5"
    />
-Red [[captured cell|captures]] the cells shaded red, leaving room for only a [[ziggurat]] (shaded gray) below to connect. Blue has a 3rd row ladder escape on the left with (*). In order for Blue to win, Red should try to intrude neither the ladder escape nor the ziggurat. Red 1 below achieves this, and it's in fact the unique losing move; Blue 2 is the unique winning reply.
+The carrier of Blue's connection is highlighted. Since b5 is not in the carrier, b5 is a losing move for Red.
-<hexboard size="5x5"
+'''Answer:''' b5
-  contents="R d2 d1 B e1 b1 R 1:b5 B 2:c4 S gray:area(c4,e2,e5,c5) red:area(d1,e1,d2)"
-  />
 === Puzzle 4 ===
-Red's unique losing move is 1, and Blue's unique winning reply is 2:
+This puzzle is a bit harder. With Blue to move, it is perhaps not immediately obvious how Blue can win. The key points for Blue are A, B, and C (and in fact, any one of those three moves is winning for Blue):
 <hexboard size="6x6"
-   contents="R a5 d3 B d1 f4 R 1:f3 B 2:e2 S blue:area(f1,e2,f2)"
+   contents="R a5 d3 B d1 f4 E A:b3 B:e2 C:c5"
    />
-Note that Blue captures the shaded region, hence [[Captured cell#Captured cells and dead cells|killing Red 1]]. Here is a likely continuation:
+Note that A forms [[edge template IV2d]] with d1; B forms [[edge template II]] and a [[bridge]] to d1, and C forms [[edge template IV2h]] with f4.
 <hexboard size="6x6"
-   contents="R a5 d3 B d1 f4 R f3 B e2 R 3:d2 B 4:e1 R 5:b2 B 6:b3 R 7:c2 B 8:c5"
+   contents="R a5 d3 B d1 f4 B 1:b3 R 2:e2 B 3:c5 E *:a6,c4 S blue:area(a1,a4,d1)"
    />
-How do we approach this puzzle? You could solve it with lots of trial and error, but here is one attempt at motivating the answer. Some of Blue's strongest threats are at A, B, and C below:
+However, when we compute the carrier of this connection, we find that it consists of all of the empty cells on the board! So we have not yet found any losing move for Red.
 <hexboard size="6x6"
-   contents="R a5 *:d3 B d1 f4 E A:b3 B:e2 C:c5 S blue:area(a1,d1,a4)"
+   contents="R a5 d3 B d1 f4 B 1:e2 R 2:d2 B 3:e1 R 4:b2 B 5:b3 R 6:c2 B 7:c5 E *:a6,c4 S blue:area(e2,f1,f2)"
    />
+Note that after 7, Blue is again connected by [[edge template IV2h]] and a double threat at a6 and c4. Moreover, Blue 1 captures the highlighted triangle and [[dead cell|kills]] f3. This means that f3 is not required for Blue's connection! Thus, the carrier of Blue's connection is the following:
-In particular, A forms [[edge template IV2d]] and captures the entire corner region (shaded blue); B is a strong move in combination with d1, analogous to [[Openings on 11 x 11#a9|the combination of a9 and b10]] in 11&times;11; and C forms [[edge template IV2h]] with f4. This makes the entire bottom half of the board unlikely candidates for Red's losing first move, because any red stone in area allows (*) to connect to bottom right even after Blue plays C.
-−
-The moves a1&mdash;c1 on the first row intrude on Blue's plan to play A, so they are also unlikely. Note that a1 is particularly tempting, because Blue b2 kills a1. However, Red still wins after Blue b2:
 <hexboard size="6x6"
-   contents="R a5 d3 B d1 f4 R 1:a1 B 2:b2 R 3:c2 B 4:c1 R 5:e2 B 6:c5 R 7:b5"
+   contents="R a5 d3 B d1 f4 S blue:all-a5,d3,f3"
    />
-The moves e1/f1 also block Blue's plan to play B, so they seem too strong. We're now a bit stumped, so we refer again to [[Openings on 11 x 11#a9]]. We realize that not only is Red b10 strong (for Red) in combination with Red a9, but ''Blue'' b10 is also strong (for ''Blue'') against Red a9 &mdash; in other words, Red a9 is weak against Blue b10! The analogous statement in our puzzle is that Red f3 is weak against Blue e2. So we check if f3 is losing, and indeed, we come to the surprising fact that Red's unique losing move isn't on the top or bottom row.
+Since f3 is not in the carrier, it is a losing move for Red.
 === Puzzle 5 ===

@@ Line 5: / Line 5: @@
    />
-Red [[captured cell|captures]] the cells shaded red, leaving room for only a [[ziggurat]] (shaded gray) below to connect. Blue has a 3rd row ladder escape on the left with (*). In order for Blue to win, Red should try not to intrude either the ladder escape or the ziggurat. Red 1 below achieves this, and it's in fact the unique losing move; Blue 2 is the unique winning reply.
+Red [[captured cell|captures]] the cells shaded red, leaving room for only a [[ziggurat]] (shaded gray) below to connect. Blue has a 3rd row ladder escape on the left with (*). In order for Blue to win, Red should try to intrude neither the ladder escape nor the ziggurat. Red 1 below achieves this, and it's in fact the unique losing move; Blue 2 is the unique winning reply.
 <hexboard size="5x5"
@@ Line 25: / Line 25: @@
    />
-How do we approach this puzzle? You could solve it with lots of trial and error, but here is one attempt at motivating the answer. Some of Blue's strongest threats at A, B, and C below:
+How do we approach this puzzle? You could solve it with lots of trial and error, but here is one attempt at motivating the answer. Some of Blue's strongest threats are at A, B, and C below:
 <hexboard size="6x6"
@@ Line 62: / Line 62: @@
-== See Also ==
+== See also ==
 [[Worst move puzzles|Back to puzzles page]]
 [[category:Puzzle]]

← Older revision		Revision as of 04:48, 2 July 2023
Line 60:		Line 60:
	contents="R a6 e3 B b6 g4 R 1:d1 B 2:d2"		contents="R a6 e3 B b6 g4 R 1:d1 B 2:d2"
	/>		/>
		+
		+
		+	== See Also ==
		+
		+	[[Worst move puzzles\|Back to puzzles page]]
		+
		+	[[category:Puzzle]]