Difference between revisions of "Solutions to worst move puzzles"

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(reorder puzzles by difficulty)
(Rewrote the answers to the first 4 puzzles in terms of mustplay analysis, and added an explanation.)
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== Approaches to solving the puzzles ==
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 +
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.
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 +
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.
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== Solutions to individual puzzles ==
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=== Puzzle 1 ===
 
=== Puzzle 1 ===
  
 +
With Blue to move, Blue can win as follows, using [[edge template III2e]]:
 
<hexboard size="4x4"
 
<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"
 
   />
 
   />
 +
The carrier of Blue's connection is highlighted. Since c1 is not in the carrier, c1 is a losing move for Red.
 +
 +
'''Answer:''' c1
  
 
=== Puzzle 2 ===
 
=== Puzzle 2 ===
  
 +
With Blue to move, Blue can win as follows:
 
<hexboard size="4x4"
 
<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"
 
   />
 
   />
 +
The carrier of Blue's connection is highlighted. Since d1 is not in the carrier, d1 is a losing move for Red.
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 +
'''Answer:''' d1
  
 
=== Puzzle 3 ===
 
=== Puzzle 3 ===
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 +
With Blue to move, Blue can win by playing at c4, getting a [[ziggurat]] on the right and a 3rd row [[ladder]] on the left, which b1 [[ladder escape|escapes]]:
  
 
<hexboard size="5x5"
 
<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.
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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 ===
 
=== Puzzle 4 ===
  
−
Red's unique losing move is 1, and Blue's unique winning reply is 2:
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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"
 
<hexboard size="6x6"
−
   contents="R a5 d3 B d1 f4 R 1:f3 B 2:e2 S blue:area(f1,e2,f2)"
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   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.
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 +
Consider, for example, what happens if Blue starts at A. Then Blue captures the entire highlighted upper left corner, and Red must defend at B. After this, Blue plays C and threatens both a6 and c4.
  
 
<hexboard size="6x6"
 
<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:
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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.
  
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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:
 
<hexboard size="6x6"
 
<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:
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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.
+
−
 
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−
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"
 
<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.
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Since f3 is not in the carrier, it is a losing move for Red.
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 +
In fact, after Red f3, Blue e2 is the unique winning reply.
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 +
'''Answer:''' f3
  
 
=== Puzzle 5 ===
 
=== Puzzle 5 ===

Revision as of 22:59, 2 July 2023

Approaches to solving the puzzles

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.

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.

Solutions to individual puzzles

Puzzle 1

With Blue to move, Blue can win as follows, using edge template III2e:

abcd1234213

The carrier of Blue's connection is highlighted. Since c1 is not in the carrier, c1 is a losing move for Red.

Answer: c1

Puzzle 2

With Blue to move, Blue can win as follows:

abcd1234213

The carrier of Blue's connection is highlighted. Since d1 is not in the carrier, d1 is a losing move for Red.

Answer: d1

Puzzle 3

With Blue to move, Blue can win by playing at c4, getting a ziggurat on the right and a 3rd row ladder on the left, which b1 escapes:

abcde1234554321

The carrier of Blue's connection is highlighted. Since b5 is not in the carrier, b5 is a losing move for Red.

Answer: b5

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):

abcdef123456BAC

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.

Consider, for example, what happens if Blue starts at A. Then Blue captures the entire highlighted upper left corner, and Red must defend at B. After this, Blue plays C and threatens both a6 and c4.

abcdef123456213

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:

abcdef1234563462157

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 kills f3. This means that f3 is not required for Blue's connection! Thus, the carrier of Blue's connection is the following:

abcdef123456

Since f3 is not in the carrier, it is a losing move for Red.

In fact, after Red f3, Blue e2 is the unique winning reply.

Answer: f3

Puzzle 5

Red 1 is the unique losing move, and Blue 2 is the unique winning reply:

abcdefg123456712


See also

Back to puzzles page

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