Ladder puzzle 1/Solution - Revision history

Selinger: Terminology: prefer "pivot" to "break"

2025-05-26T02:29:35Z

Terminology: prefer "pivot" to "break"

Selinger: Numbered solutions.

2021-06-16T03:43:29Z

Numbered solutions.

Selinger: /* Correct: e5 then c7 */ Corrected sequence of forcing moves.

2021-06-16T03:41:04Z

‎Correct: e5 then c7: Corrected sequence of forcing moves.

Selinger: /* Alternative Solution: f6 */ Edited.

2021-06-16T03:36:54Z

‎Alternative Solution: f6: Edited.

Selinger: Added another alternative solution: switchback

2020-05-21T01:58:46Z

Added another alternative solution: switchback

Selinger: /* Alternative: e5 then f5 */ Refined solution.

2020-03-01T20:52:46Z

‎Alternative: e5 then f5: Refined solution.

Selinger: Edited headers

2020-03-01T20:43:28Z

Edited headers

Selinger: /* Correct: e5 */ Improved the solution. The problem with the old solution is that Blue has other options besides 4.f6

2020-03-01T20:39:52Z

‎Correct: e5: Improved the solution. The problem with the old solution is that Blue has other options besides 4.f6

GCRhoads: /* Alternative Solution: f6 */

2009-11-09T15:59:19Z

‎Alternative Solution: f6

Roland Illig: /* Alternative Solution: f6 */ typo

2009-02-16T13:57:58Z

‎Alternative Solution: f6: typo

@@ Line 9: / Line 9: @@
 <hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Ve5 Hd8 Vd7 Hc8 Vf7 He6 Vf5 Hf6 Vh5</hex>
-Therefore, Blue must play 2.e6. Red can now play the ladder at c7, break the ladder at g7, and win:
+Therefore, Blue must play 2.e6. Red can now play the ladder from c7, pivot at g7, and win:
 <hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Ve5 He6 Vc7 Hc8 Vd7 Hd8 Ve7 He8 Vg7 Hf7 Vg6 Hf6 Vg4</hex>
@@ Line 43: / Line 43: @@
 == Solution 4: switchback ==
-Red can also use h4 to do a [[switchback]]. To set this up, Red starts playing the 2nd row ladder and breaks the ladder at f7:
+Red can also use h4 to do a [[switchback]]. To set this up, Red starts playing the 2nd row ladder and pivots at f7:
 <hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 V1c7 H2c8 V3d7 H4d8 V5f7 H6e7</hex>

@@ Line 1: / Line 1: @@
-== Correct: e5 then c7 ==
+== Solution 1: e5 then c7 ==
 Red's main threat is the ladder starting at c7. If Red plays out this ladder, Blue can block it. So Red needs a helping stone somewhere on the right and on the second line from the bottom. Red 1.e5 threatens the follow-up moves at *, which Blue must defend.
@@ Line 13: / Line 13: @@
 <hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Ve5 He6 Vc7 Hc8 Vd7 Hd8 Ve7 He8 Vg7 Hf7 Vg6 Hf6 Vg4</hex>
-== Alternative: e5 then f5 ==
+== Solution 2: e5 then f5 ==
 Alternatively, after 1.e5 2.e6, Red could have continued the 4th row ladder with 3.f5, to which Blue may reply 4.f6. This allows Red to play a double threat 5.g7. This stone is the ladder helper, and it also threatens to connect along the top. Either way, Red wins.
@@ Line 31: / Line 31: @@
 Again, e7 is connected upwards via *.
-== Alternative Solution: f6 ==
+== Solution 3: f6 ==
 <hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Vf6 He7</hex>
@@ Line 41: / Line 41: @@
 Now Red plays f4 completing the win. f4 is connected to the bottom edge via a [[trapezoid]], and to Red's central group via double threat at f3 and e5.  Therefore, all of Red's pieces form a single group which is connected to both the top and bottom.
-== Alternative Solution: switchback ==
+== Solution 4: switchback ==
 Red can also use h4 to do a [[switchback]]. To set this up, Red starts playing the 2nd row ladder and breaks the ladder at f7:

@@ Line 7: / Line 7: @@
 The potential connection at d7 passes through e6, c7, c8, and d8. The potential connection at f6 passes through e6, f5, and [[Template IIIa]]. These two potential connections only overlap at the two hexes marked +, so Blue must play there. However, if Blue plays 2.d8, Red wins with the following sequence of forcing moves:
-<hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Ve5 Hd8 Vd7 Hc8 Vf7 He7 Vf6</hex>
+<hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Ve5 Hd8 Vd7 Hc8 Vf7 He6 Vf5 Hf6 Vh5</hex>
 Therefore, Blue must play 2.e6. Red can now play the ladder at c7, break the ladder at g7, and win:

@@ Line 35: / Line 35: @@
 <hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Vf6 He7</hex>
-The stone at f6 is a ladder breaker for the ladder starting at c7, and it also threatens to connect via e5.
+The stone at f6 (essentially [[Tom's move]], but requiring a bit less space than usual due to h4) is a ladder escape for the ladder starting at c7, and it also threatens to connect via e5. Blue has no choice but to defend at e7. Red continues like this:
-This looks very strong, but Blue can still defend at e7, which is a ladder breaker and threatens a connection at the lower side.
 <hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 Vf6 He7 V3g6 H4f7 V5g7 H6e6</hex>
-Now Red plays f4 completing the win.  f4 is connected to the lower-right group in two non-overlapping ways, through f5 and g4.  f4 is also connected to the central group in two non-overlapping ways, through f3 and e5.  Therefore, all of Red's pieces form a single group which is connected to both the top and bottom.
+Now Red plays f4 completing the win. f4 is connected to the bottom edge via a [[trapezoid]], and to Red's central group via double threat at f3 and e5.  Therefore, all of Red's pieces form a single group which is connected to both the top and bottom.
 == Alternative Solution: switchback ==

← Older revision		Revision as of 01:58, 21 May 2020
Line 41:		Line 41:

	Now Red plays f4 completing the win. f4 is connected to the lower-right group in two non-overlapping ways, through f5 and g4. f4 is also connected to the central group in two non-overlapping ways, through f3 and e5. Therefore, all of Red's pieces form a single group which is connected to both the top and bottom.		Now Red plays f4 completing the win. f4 is connected to the lower-right group in two non-overlapping ways, through f5 and g4. f4 is also connected to the central group in two non-overlapping ways, through f3 and e5. Therefore, all of Red's pieces form a single group which is connected to both the top and bottom.
		+
		+	== Alternative Solution: switchback ==
		+
		+	Red can also use h4 to do a [[switchback]]. To set this up, Red starts playing the 2nd row ladder and breaks the ladder at f7:
		+
		+	<hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 V1c7 H2c8 V3d7 H4d8 V5f7 H6e7</hex>
		+
		+	Now the killer move is f5, which connects to Red's main group by double threat, and is also connected to h4. Blue is forced to play at f6, but Red reconnects with h5 via [[Fourth_row_edge_templates#IV-2-d\|edge template IV-2-d]].
		+
		+	<hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 Vc7 Hc8 Vd7 Hd8 Vf7 He7 V7f5 H8f6 V9h5</hex>
		+
		+	This method works no matter the size of the board or how far away the 5th row stone is. If h5 is further to the right, the 2nd row ladder will just turn into a 4th row ladder going the opposite way, eventually connecting with d5.

	[[category:ladder puzzle]]		[[category:ladder puzzle]]

@@ Line 19: / Line 19: @@
 <hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Ve5 He6 Vf5 Hf6 Vg7</hex>
-However, this play is more complicated, because Blue may also respond in a number of other places instead of f6. If Blue plays 4.e8, the winning sequence looks like this:
+However, this play is more complicated to analyze, because Blue may also respond in a number of other places instead of f6. If Blue plays 4.e8, then 5.g7 still works, albeit for slightly different reasons:
-<hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Ve5 He6 Vf5 He8 Vc7 Hc8 Vg7 Hg6 Ve7 *d7 *f6 +d8 +f7</hex>
+<hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 N:on Ve5 He6 Vf5 He8 Vg7 Hg6 Vc7 Hc8 Ve7 *d7 *f6 +d8 +f7</hex>
 Note that in the final position, the red stone at e7 is connected up via * and down via +, so it is a winning position.

@@ Line 22: / Line 22: @@
 <hex>R8 C8 Q1 Vc6 Vd5 Vd4 Ve3 Vh4 Hd6 Hb8 Hc1 He4 Hg3 Vf6 He7 V3g6 H4f7 V5g7 H6e6</hex>
-Now Red pushes to the right, making both f4 and f6 winning points. Blue cannot defend both.
+Now Red plays f4 completing the win.  f4 is connected to the lower-right group in two non-overlapping ways, through f5 and g4.  f4 is also connected to the central group in two non-overlapping ways, through f3 and e5.  Therefore, all of Red's pieces form a single group which is connected to both the top and bottom.
 [[category:ladder puzzle]]