Difference between revisions of "Peep"

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A '''peep''' is a move in specific region such that
+
A '''peep''' is a move in a specific region such that
 
+
* looking just at that region, it would normally be a [[Forcing moves|forcing move]] (even if the global situation means the opponent should respond elsewhere), and
* Looking just at that region, it would normally be a [[Forcing moves|forcing move]] (even if the global situation means the opponent should not respond anywhere nearby).
+
* if the opponent does respond in that region (rather than merely close to it), then the exchange does not help the opponent and may help the player who played the peep.
 
+
and
+
 
+
* If the opponent does respond in that region (rather than merely close to it), then one can easily prove that the exchange does not help the opponent and can't easily prove that the exchange doesn't matter.
+
 
+
.
+
 
+
 
+
  
 
This situation occurs if one threatens to cut a connection between major groups or key stones of the opponent, or to create a connection between one's own major groups or key stones.
 
This situation occurs if one threatens to cut a connection between major groups or key stones of the opponent, or to create a connection between one's own major groups or key stones.
Line 15: Line 7:
 
Sometimes it's hard to tell if a move is sufficiently [[Forcing moves|forcing]], and whether the opponent can gain by resisting, rather than just responding in the obvious way. Playing correct peeps is one of the most sophisticated aspects of Hex strategy.
 
Sometimes it's hard to tell if a move is sufficiently [[Forcing moves|forcing]], and whether the opponent can gain by resisting, rather than just responding in the obvious way. Playing correct peeps is one of the most sophisticated aspects of Hex strategy.
  
Another term for peep is '''nozoki''' (borrowed from Go).
+
The term "peep" is borrowed from the game of Go. Another term for peep, also borrowed from Go but less frequently used, is '''nozoki'''.
  
 
== Examples ==
 
== Examples ==
<hexboard size="3x3"
 
  visible="-(a1 c3)"
 
  edges="none"
 
  coords="none"
 
  contents="R c1 b3 B E *:c2 S red:(b1 a3) blue:a2"
 
/>
 
  
If at least one of the shaded hexes is occupied by a piece of the corresponding color, then Blue playing * is a peep.  This is because, in such cases,
+
=== Bridge peep ===
  
 +
By far the most common form of peep is a "bridge peep".  That is, playing the correct side of a [[bridge]] which is [[Bolstered template|bolstered]] on exactly one side.
 
<hexboard size="3x3"
 
<hexboard size="3x3"
 
   visible="-(a1 c3)"
 
   visible="-(a1 c3)"
 
   edges="none"
 
   edges="none"
 
   coords="none"
 
   coords="none"
   contents="R c1 b3 2:c2 B E d:b2 S red:(b1 a3) blue:a2"
+
   contents="R c1 b3 B E *:c2 a:b2 S red:(b1 a3) blue:a2"
 
/>
 
/>
 +
If at least one of the shaded hexes is occupied by a piece of the indicated color, then Blue playing "*" is a peep.  This is because in such cases, Red playing "*" would [[Dead cell|kill]] ''a''. Therefore, if Red just defends the [[bridge]], then Blue can't do any better here than Blue playing "*" and Red responding at ''a''.
  
Red playing * would [[Dead cell|kill]] ''d'', so if Red just defends the [[Bridge|bridge]], then Blue can't do any better here than Blue playing * and Red responding with the central hex.
+
=== Ziggurat peep ===
  
This is by far the most common form of peep, a "bridge peep".
+
Similarly to the bridge peep, there can also be peeps at other templates, with suitable surrounding conditions. For example, consider the following position, where Red is connected to the bottom edge by a [[ziggurat]].
 
+
<hexboard size="4x5"
 
+
  visible="-(d1 c1 b2 b1 a2 a1)"
 
+
  edges="bottom right"
Similarly, with suitable surrounding conditions, there can also be peeps at templates.
+
  coords="bottom right"
 
+
  contents="B a3 R c2 S area(d4,a4,c2,d2)"
<hexboard size="5x4"
+
  visible="-(d2 d3 c4 d4 c5 d5)"
+
  edges="left top"
+
  contents="R b5 B c3"
+
 
/>
 
/>
 
+
If Red just defends the ziggurat, then Blue can't do any better here than Blue getting the outside of the ziggurat and c2 connecting down.
Here, C3 has a [[Ziggurat|ziggurat]] to the left edge,
+
<hexboard size="4x5"
 
+
  visible="-(d1 c1 b2 b1 a2 a1)"
<hexboard size="5x4"
+
  edges="bottom right"
  visible="-(d2 d3 c4 d4 c5 d5)"
+
  coords="bottom right"
  edges="left top"
+
  contents="B a3 R c2 S area(d4,a4,c2,d2) B d2--d4 R b4"
  contents="R b5 B c3 S area(a2,a5,c3,c2)"
+
 
/>
 
/>
 
+
Blue d2 [[captured cell|captures]] the entire corner, turning the corner into this:
so if Blue just defends the ziggurat, then Red can't do any better here
+
<hexboard size="4x5"
 
+
  visible="-(d1 c1 b2 b1 a2 a1)"
<hexboard size="5x4"
+
  edges="bottom right"
  visible="-(d2 d3 c4 d4 c5 d5)"
+
  coords="bottom right"
  edges="left top"
+
  contents="B a3 R c2 S area(d4,a4,c2,d2) B area(d2,d4,e4,e1) 1:d2 E *:b4"
  contents="R b5 a2 b2 c2 B c3 a4"
+
 
/>
 
/>
 +
So Blue d2 is a peep at Red's ziggurat.
  
than Red getting the outside of the ziggurat and C3 connecting left.
+
Note that d2 [[dominated cell|dominates]] d3 by [[Dominated cell#Capture-domination|capture-domination]].
 
+
Conversely, Blue d3 dominates d2, because d3 also [[captured cell|captures]] the entire corner:
Red C2 [[Captured cell|captures]] C1 and D1, turning the corner into
+
<hexboard size="4x5"
 
+
  visible="-(d1 c1 b2 b1 a2 a1)"
<hexboard size="5x4"
+
  edges="bottom right"
  visible="-(d2 d3 c4 d4 c5 d5)"
+
  coords="bottom right"
  edges="left top"
+
  contents="B a3 R c2 S area(d4,a4,c2,d2) B area(d2,d4,e4,e1) 1:d3 E *:b4"
  contents="R b5 1:c2 c1 d1 B c3 a4 E *:a4"
+
 
/>
 
/>
 
+
Namely, d3 first captures e2 and e3, then d4 and e4, and finally d2 and e1 by this capture pattern:
[[Dominated_cell#Star_decomposition_domination|a region where whoever moves connects]].
+
<hexboard size="4x4"
 
+
  coords="none"
[[Dominated_cell#Star_decomposition_domination|Giving Red A2 and B2 does not change that]],
+
  edges="none"
 
+
  visible="area(d1,a3,b4,d2)"
<hexboard size="5x4"
+
  contents="B d1--d2--b4 R a3 S blue:(b3 c2)"
  visible="-(d2 d3 c4 d4 c5 d5)"
+
  edges="left top"
+
  contents="R b5 1:c2 c1 d1 a2 b2 B c3 a4 E *:a4"
+
 
/>
 
/>
 +
Therefore, the moves d2 and d3 are [[Dominated_cell#Mutually_dominating_moves|equivalent]] for Blue, and both are peeps at Red's ziggurat.
 +
== Automatic peep ==
  
so Red's 1 is a peep at Blue's ziggurat.
+
If Red just completed a bridge and Blue has a bridge peep such that cutting off the bridge would kill the stone Red just played, then Blue should play the bridge peep. This is called an ''automatic bridge peep''. More precisely, if Blue was winning before Red's move, then Blue will still be winning after playing the automatic bridge peep.
  
By [[Dominated_cell#Star_decomposition_domination|star decomposition domination]], red C2 is at least as good for Red as red B2.
+
For example, consider the following situation, where Red just played at 1:
 +
<hexboard size="6x6"
 +
  coords="none"
 +
  edges="bottom left"
 +
  contents="B d5 d4 f1 R f3 d2 1:e3 E a:d3 b:e2 S (e2,e3,d2,d3)"
 +
  />
  
Also, red B2 [[Captured_cell#Examples|captures]] D1 and C2, so
+
Because the shaded bridge is [[bolstered template|bolstered]] on the b-side, a is a peep. Moreover, if Blue had both a and b, the red stone at 1 would be [[dead cell|dead]]. Therefore, this is an automatic peep and Blue should play at a. (More precisely, if Blue was winning before Red played at 1, then Blue playing at a preserves the win. If Blue was in fact losing before Red played at 1, then Blue playing at a may be losing and Blue may potentially have a winning move elsewhere. In any case, the situation after Blue plays at a is no worse for Blue than before Red played at 1.)
  
<hexboard size="4x4"
+
There is an analogous notion of automatic peep for ziggurat peeps.
  visible="c1 d1 c2 b2 a3 b3 b4"
+
  edges="none"
+
  coords="none"
+
  contents="R a3 b2 c1 d1 B b4"
+
/>
+
 
+
red B2 is at least as good for Red as red C2.
+
 
+
Thus red B2 is [[Dominated_cell#Mutually_dominating_moves|equivalent]] to red C2, so
+
 
+
<hexboard size="5x4"
+
  visible="-(d2 d3 c4 d4 c5 d5)"
+
  edges="left top"
+
  contents="R b5 1:b2 1:c2 B c3 a4 E *:a4"
+
/>
+
  
Red playing B2 instead of C2 is also a peep at Blue's ziggurat.
 
  
 
== Crucial peep ==
 
== Crucial peep ==
Line 132: Line 102:
 
   /><br/>
 
   /><br/>
  
== Bad Peeps ==
+
== Bad peeps ==
  
There are at least several ways a peep can be bad:
+
There are several possible reasons a peep can be bad. Here are some of them:
  
* Due to circumstances outside of the region, the apparent cutting threat does not actually work, in which case the peep is close to a wasted move.
+
* Due to circumstances outside of the region, the apparent cutting threat does not actually work, in which case the peep is close to a [[irrelevant move|wasted move]].
 
+
* While the apparent threat does work, it's not a big enough threat for the opponent to respond by defending against it.
* While the apparent threat does work, it's not big-enough for the opponent to respond by defending against that threat.
+
* The opponent benefits by resisting the peep.
 
+
* The opponent can [[Minimax|minimax]], rather than answering more locally.
 
+
* The opponent can [[Minimax|minimax]], rather than answering more-locally.
+
  
 +
For example, consider the following situation, with Blue to move:
 
<hexboard size="8x8"
 
<hexboard size="8x8"
   contents="R a3 e4 e5 d7 g7 B e2 d3 d5 f6"
+
  coords="none"
 +
   contents="R a3 e4 e5 d7 g7 B e2 d3 d5 f6 E a:g1 b:f3 *:d6 c:e6 d:g4"
 
/>
 
/>
 
+
Here, Blue can win fairly easily by playing at ''a'' or ''b''. But suppose that Blue instead decides to play the bridge peep at "*". If Red defends the bridge at ''c'' as Blue expects, then Blue still wins by playing ''a'' or ''b''. However, if Red instead responds to the peep with a [[Minimax|minimaxing]] move at ''d'', Red wins.
Here, Blue wins fairly easily with G1 or F3.
+
 
+
 
<hexboard size="8x8"
 
<hexboard size="8x8"
   contents="R a3 e4 e5 d7 g7 B e2 d3 d5 f6 1:g1 1:f3"
+
  coords="none"
 +
   contents="R a3 e4 e5 d7 g7 2:g4 B e2 d3 d5 f6 1:d6"
 
/>
 
/>
 +
Thus, the bridge peep was a bad peep in this situation.
  
If Blue plays the bridge peep, then * would give the win back to Blue
+
== Resisting a peep ==
 
+
<hexboard size="8x8"
+
  contents="R a3 e4 e5 d7 g7 2:g4 B e2 d3 d5 f6 1:d6 E *:e6"
+
/>
+
 
+
, but Red wins by instead [[Minimax|minimaxing]] with 2.
+
 
+
 
+
* The opponent benefits by resisting the peep.
+
 
+
== Resisting a Peep ==
+
  
 
Rather than responding in the obvious way, the opponent can play a move that mitigates against the peep's threat, while also getting something else.  The most common example of this is [[Foiling|foiling]].
 
Rather than responding in the obvious way, the opponent can play a move that mitigates against the peep's threat, while also getting something else.  The most common example of this is [[Foiling|foiling]].
  
For example, Blue can play 2 instead of *.
+
For example, in response to Red's peep 1 in the following diagram, Blue can play 2 instead of *:
  
 
<hexboard size="5x6"
 
<hexboard size="5x6"
Line 177: Line 136:
 
/>
 
/>
  
Red could've gotten a height-2 ladder towards the right, but after 1 and 2, a red ladder towards the right would be height-3.
+
Red could have gotten a 2nd row ladder towards the right, but after 1 and 2, a red ladder towards the right would be a 3rd row ladder.
 
+
Note that this does not necessarily mean the peep was bad.
+
 
+
For example, Red could be fine with the ladder being height-3, and Red can now get
+
  
 +
Note that this does not necessarily mean the peep was bad. For example, Red could be fine with a 3rd row ladder, and Red can now get c2 before pushing the ladder:
 
<hexboard size="5x6"
 
<hexboard size="5x6"
 
   edges="left bottom"
 
   edges="left bottom"
Line 188: Line 144:
 
   contents="R e1 d2 1:e3 3:c2 5:e2 B f2 d3 b2 2:d4 4:b4"
 
   contents="R e1 d2 1:e3 3:c2 5:e2 B f2 d3 b2 2:d4 4:b4"
 
/>
 
/>
 
+
Whereas if Red had started with c2, then Blue would presumably just defend the bridges.
C2 before pushing the ladder, whereas if Red had started with C2
+
 
+
 
<hexboard size="5x6"
 
<hexboard size="5x6"
 
   edges="left bottom"
 
   edges="left bottom"
Line 197: Line 151:
 
/>
 
/>
  
,  then Blue would presumably just defend the bridges.
+
The consequences of resisting a peep can also be much harder to assess. In the following example, Red chose to resist Blue's peep 1 by playing at 2, rather than just playing *.
 
+
<hexboard size="6x8"
 
+
The consequences of resisting a peep can also be much harder to assess.
+
 
+
<hexboard size="8x6"
+
 
   visible=""
 
   visible=""
 
   edges="left bottom"
 
   edges="left bottom"
 
   coords="none"
 
   coords="none"
   contents="R f4 d2 1:f2 B arrow(2):f1 e3 2:d5 E *:e2"
+
   contents="B e1 g3 1:g1 R arrow(12):h1 f2 2:d3 E *:g2"
 
/>
 
/>
  
Here, Blue chose to resist with 2, rather than just play *.
+
[[category: Definition]]
 +
[[category: Advanced Strategy]]

Latest revision as of 00:27, 16 August 2022

A peep is a move in a specific region such that

  • looking just at that region, it would normally be a forcing move (even if the global situation means the opponent should respond elsewhere), and
  • if the opponent does respond in that region (rather than merely close to it), then the exchange does not help the opponent and may help the player who played the peep.

This situation occurs if one threatens to cut a connection between major groups or key stones of the opponent, or to create a connection between one's own major groups or key stones.

Sometimes it's hard to tell if a move is sufficiently forcing, and whether the opponent can gain by resisting, rather than just responding in the obvious way. Playing correct peeps is one of the most sophisticated aspects of Hex strategy.

The term "peep" is borrowed from the game of Go. Another term for peep, also borrowed from Go but less frequently used, is nozoki.

Examples

Bridge peep

By far the most common form of peep is a "bridge peep". That is, playing the correct side of a bridge which is bolstered on exactly one side.

a

If at least one of the shaded hexes is occupied by a piece of the indicated color, then Blue playing "*" is a peep. This is because in such cases, Red playing "*" would kill a. Therefore, if Red just defends the bridge, then Blue can't do any better here than Blue playing "*" and Red responding at a.

Ziggurat peep

Similarly to the bridge peep, there can also be peeps at other templates, with suitable surrounding conditions. For example, consider the following position, where Red is connected to the bottom edge by a ziggurat.

abcde1234

If Red just defends the ziggurat, then Blue can't do any better here than Blue getting the outside of the ziggurat and c2 connecting down.

abcde1234

Blue d2 captures the entire corner, turning the corner into this:

abcde12341

So Blue d2 is a peep at Red's ziggurat.

Note that d2 dominates d3 by capture-domination. Conversely, Blue d3 dominates d2, because d3 also captures the entire corner:

abcde12341

Namely, d3 first captures e2 and e3, then d4 and e4, and finally d2 and e1 by this capture pattern:

Therefore, the moves d2 and d3 are equivalent for Blue, and both are peeps at Red's ziggurat.

Automatic peep

If Red just completed a bridge and Blue has a bridge peep such that cutting off the bridge would kill the stone Red just played, then Blue should play the bridge peep. This is called an automatic bridge peep. More precisely, if Blue was winning before Red's move, then Blue will still be winning after playing the automatic bridge peep.

For example, consider the following situation, where Red just played at 1:

ba1

Because the shaded bridge is bolstered on the b-side, a is a peep. Moreover, if Blue had both a and b, the red stone at 1 would be dead. Therefore, this is an automatic peep and Blue should play at a. (More precisely, if Blue was winning before Red played at 1, then Blue playing at a preserves the win. If Blue was in fact losing before Red played at 1, then Blue playing at a may be losing and Blue may potentially have a winning move elsewhere. In any case, the situation after Blue plays at a is no worse for Blue than before Red played at 1.)

There is an analogous notion of automatic peep for ziggurat peeps.


Crucial peep

Playing peeps can be very useful. In certain situations, playing a peep can make the difference between winning and losing. Consider the following example:

abcdefghi123456789

The only winning move for Red is the peep at d6. If Red misses it, the game proceeds as follows and Blue wins:

abcdefghi123456789132546

If Red starts by playing d6 instead, Red wins.

abcdefghi123456789231

Bad peeps

There are several possible reasons a peep can be bad. Here are some of them:

  • Due to circumstances outside of the region, the apparent cutting threat does not actually work, in which case the peep is close to a wasted move.
  • While the apparent threat does work, it's not a big enough threat for the opponent to respond by defending against it.
  • The opponent benefits by resisting the peep.
  • The opponent can minimax, rather than answering more locally.

For example, consider the following situation, with Blue to move:

abdc

Here, Blue can win fairly easily by playing at a or b. But suppose that Blue instead decides to play the bridge peep at "*". If Red defends the bridge at c as Blue expects, then Blue still wins by playing a or b. However, if Red instead responds to the peep with a minimaxing move at d, Red wins.

21

Thus, the bridge peep was a bad peep in this situation.

Resisting a peep

Rather than responding in the obvious way, the opponent can play a move that mitigates against the peep's threat, while also getting something else. The most common example of this is foiling.

For example, in response to Red's peep 1 in the following diagram, Blue can play 2 instead of *:

12

Red could have gotten a 2nd row ladder towards the right, but after 1 and 2, a red ladder towards the right would be a 3rd row ladder.

Note that this does not necessarily mean the peep was bad. For example, Red could be fine with a 3rd row ladder, and Red can now get c2 before pushing the ladder:

abcdef1234535142

Whereas if Red had started with c2, then Blue would presumably just defend the bridges.

abcdef12345146325

The consequences of resisting a peep can also be much harder to assess. In the following example, Red chose to resist Blue's peep 1 by playing at 2, rather than just playing *.

12