Difference between revisions of "Peep"

Revision as of 02:35, 2 January 2021

A peep is a move in 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 not respond anywhere nearby).

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.

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.

Another term for peep is nozoki (borrowed from Go).

Examples

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,

Red playing * would kill d, so if Red just defends the bridge, then Blue can't do any better here than Blue playing * and Red responding with the central hex.

This is by far the most common form of peep, a "bridge peep".

Similarly, with suitable surrounding conditions, there can also be peeps at templates.

Here, C3 has a ziggurat to the left edge,

so if Blue just defends the ziggurat, then Red can't do any better here

than Red getting the outside of the ziggurat and C3 connecting left.

Red C2 captures C1 and D1, turning the corner into

a region where whoever moves connects.

Giving Red A2 and B2 does not change that,

so Red's 1 is a peep at Blue's ziggurat.

By star decomposition domination, red C2 is at least as good for Red as red B2.

Also, red B2 captures D1 and C2, so

red B2 is at least as good for Red as red C2.

Thus red B2 is equivalent to red C2, so

Red playing B2 instead of C2 is also a peep at Blue's ziggurat.

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:

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

If Red starts by playing d6 instead, Red wins.

Bad Peeps

There are at least several ways a peep can be bad:

• 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 big-enough for the opponent to respond by defending against that threat.

• The opponent can minimax, rather than answering more-locally.

Here, Blue wins fairly easily with G1 or F3.

If Blue plays the bridge peep, then * would give the win back to Blue

, but Red wins by instead 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.

For example, Blue can play 2 instead of *.

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.

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

C2 before pushing the ladder, whereas if Red had started with C2

, then Blue would presumably just defend the bridges.

The consequences of resisting a peep can also be much harder to assess.

Here, Blue chose to resist with 2, rather than just play *.