Giga Games 1/Giga Games.iso/net/go/info/variati.ons

Text File | 1993-06-20 | 30.1 KB | 685 lines

                               Variations on GO

From: [e-mail] (Bill Taylor)

Hex go, (6 liberties), is quite fun. I have played this several times. Once on
a big board, but mostly on a 91-point board (similar to 9x9 go). It is a good
game for relaxation between rounds at go tournaments. I warmly recommend it,
for these occasions at least.

Hex go is a far more strategic game than regular go. There are no tactical
complications like cross-cuts, ko, ladders etc. But the basic ideas of aji,
kikashi, sente, 2 eyes, connecting, etc are all present. It is a little bit
more like Piet Hein's "hex" connection game than like real go, perhaps. (For
instance, you can create a bamboo joint between just two stones !) It's fun
just to investigate the minimal dead and unsettled big-eye shapes; they're
quite a bit bigger than those for regular go ! (which has 6 minimal unsettled,
7 for seki).  Seki still exists in hex go. (Maybe even a little more common).

Tri-go, (3 liberties), I've not played (I think), but have fiddled with quite
a bit. Just as hex go is far more strategic, tri-go is far more tactical !!
Ladders can begin as soon as a contact move is played ! With nothing else
nearby, such a ladder is its own ladder-break !!!!!!  Kos, double-kos, multi-
step kos and worse abound all over. Indeed, it can be *very* tricky to be sure
that a group is unconditionally alive, especially on a small board. The edges
are trickier than usual.  I would like to find a real person to play these two
games against; but it's fun to fiddle with tri-go, (playing against youself,
if you can handle that scene).  I get the feeling though, that with its
massive tactical instability, tri-go would be a little unsatisfactory.

Hex and tri go can both be played on the same board, if you don't mind
playing inside the cells (yuk!) for one of them.

Of course, quad go, (4 liberties) that we all know and love, is ideally
placed between these two variants in terms of strategy vs tactics; hence its
greatness.

I've often thought it would be fun to play go on (say) a big cube. The faces
and edges would be like the centre of the board at regular go, but the 6
corners would have 3 liberties each (a little like tiny real-go-edges,
maybe!)

I *have* fiddled a little with playing go on a torus; have even programmed up
to record/referee such a game on screen. It's like regular go with no corners
or edges of any type at all, of course. But again, I get the feeling it would
be a little unsatisfactory - takes too long for any real conflict to develop.



From: [e-mail] (Bill Taylor)

Ken Chase writes about hex-go...

>To make 2 eyes required 17 stones in the center, 8 stones on the side and
>curiously, 8 stones at the "corners".

This is not correct.  Though to make strictly 2 eyes at an edge or corner
*does* require 8 stones, one can in fact use only 7 stones to make *3* eyes !

       -------------------------
        \ / \ /O\ /O\ /O\ / \ /
       --*---*---*---*---*---*--  [ Please forgive the "dual" board
        / \ / \O/O\ /O\O/ \ / \     requiring stones to be played inside
       *---*---*---*---*---*---*    the cells:- it's much easier for ascii ]
        \ / \ / \ / \ / \ / \ /


The statement that it requires 17 in the centre is wildly wrong. It needs
only 10 for 3 eyes, (or 11 for strictly 2).

        \ / \ / \ / \ / \ / \ /
       --*---*---*---*---*---*--
        / \ /O\O/ \O/O\ / \ / \
       *---*---*---*---*---*---*
        \ / \O/ \O/ \O/ \ / \ /
       --*---*---*---*---*---*--
        / \ / \O/O\O/ \ / \ / \
       *---*---*---*---*---*---*
        \ / \ / \ / \ / \ / \ /


For hex go, (same board, but stones played on the intersections), example
minimal groups at corners, edges and centre are...

      O-------O------edge--O-------O-------O-
     / \ / \ / \ / \      / \ / \ / \ / \ / \      / \ / \ / \ / \ / \
    /---O---O---*---     *---O---O---O---O---     *---O---O---*---*---
   / \ / \ / \ / \ /      \ / \ / \ / \ / \ /      \ / \ / \ / \ / \ /
  O---O---*---*---*-     --*---*---*---*---*-     --O---*---O---O---*-
 / \ / \ / \ / \ / \      / \ / \ / \ / \ / \      / \ / \ / \ / \ / \
/---*---*---*---*---     *---*---*---*---*---     *---O---O---*---O---
 \ / \ / \ / \ / \ /      \ / \ / \ / \ / \ /      \ / \ / \ / \ / \ /
                                                  --*---*---O---O---*-
So, overall, we get the following figures          / \ / \ / \ / \ / \
for the minimal number of stones for unconditional
life...
          tri go   (quad) go   hex go

corner      7         6         6
side        7         8         7
centre     10        10        10

....so in this respect (though few others) there is little tactical
difference between the three games.



From: [e-mail] (Nici Schraudolph)

So how about Tetra-go (4 liberties), played in 3-D on a diamond lattice?
A real "Tetraban" may be a bit unwieldy to make (never mind play on), but
it shouldn't be too hard to implement with a computer graphics package.
Since this has the "right" number of liberties, but different dimensionality,
it would make an interesting comparison to regular Go.



From: [e-mail] (Karen Hunt)

There's a variant I and a couple of friends of mine have played, invented
not by me but by one of them.  The rules are exactly as in go, but not all
locations have the same number of liberties.  For example, the corner location
has exactly one liberty, while the edges have 2 and everywhere else has 4.
 It looks like this:

.   .   .   .   .
 \ / \ / \ / \ /
  .   .   .   .
 / \ / \ / \ / \
.   .   .   .   .    (Actually, the board is significantly larger, and
 \ / \ / \ / \ /      it is square, but this gives the right idea)
  .   .   .   .
 / \ / \ / \ / \
.   .   .   .   .

It has the unusual property that a stone placed on a "2-2" point gives an
immediate eye, and the smallest guaranteed alive group requires only 3
stones.

.   .   .   .   .   .
 \ / \ / \ / \ / \ /
  O   O   .   .   .
 / \ / \ / \ / \ / \
.   O   .   .   .   .   Here, for example, the 3 O's are unconditionally
 \ / \ / \ / \ / \ /    alive.
  .   .   .   .   .
 / \ / \ / \ / \ / \
.   .   .   .   .   .

Several effects come as a result:  life is very easy along the edge, and
seki is a fairly common situation.  Ladders are the same as in go, except
that they travel vertically or horizontally instead of diagonally.

Karen Hunt, approximately 10kyu.


From: [e-mail] (Frank en stijn)

 - one could allow movement. Instead of putting a stone on the board, you
   are allowed to move a stone to an neigbour intersection. Suddenly
   bad shape becomes repairable, eyes can migrate and dangos can develop eyes.
 - In a previous post hex go sounded interesting (must try it), but you could
   also play go on a Penrose tiling. Makes for weird connectivity numbers.
   You realy have to work out what every shape does, since at every place the
   connectivities can be slightly different.
                                                    Ton Hospel


From: [e-mail] (Bill Taylor)

Ton Hospel writes:

>  - one could allow movement. Instead of putting a stone on the board, you
>    are allowed to move a stone to an neigbour intersection. Suddenly
>    bad shape becomes repairable, eyes can migrate and dangos can develop eyes.

I've played this variation too!  Actually, not quite. The way we played it
was: at each turn a player could make two "moves", one was to be a new stone
placement, as in regular go, the other was to be a movement of a stone on
the board to an orthogonally adjacent free point (possibly resulting in a
capture as usual). Either (or both) of these "move" options could be waived
if desired, *and* (important) the two options could be played in *either*
order.

We called this "move go". Several of us played it for fun (after tournament
rounds) for several years. (I was NZ champion at "move go" for a long time ;-) )
We mostly played on 9x9, rarely on 13x13.

It is great fun. I warmly recommend you to try it.

As Ton Hospel says, the most dango-ish and hopelessly dead-looking group can
come alive if there is an inch of room for it ! It becomes a matter of great
skill to spot how closely such a group must be constrained, to keep it dead.
The two-"move" turn (i.e. one place and one move) together with the optional
order give one great flexibilty of action, it's even possible it could be a
better game than regular go, (what heresy ! throw him out!). It certainly
simulates warfare better than regular go, (so what). We never did figure out
how to record it easily !

Anyway; if you're into variations at all, do give this one a try. Great fun.



From: [e-mail] (The Technicolour Throw-up)

I've had a few requests to explain save-go.  The game was described in article
on this group last year by Wilfred someone.  I appear to have lost the headers
on the article so I don't know his last name or address.  Here's what I still
have (followed by various comments of mine):
-----------------------------------------------------------------------------
Subject: Go variants

A variant of Go for you to think about -

In addition to the normal moves of Go, add a new move "save-move". Each saved
move can be used any time in the future.  So the player who has saved N moves
will be able to make upto N+1 moves as a single move.  Groups will still be
life with two eyes.  The rule for ko will not change either (that you may not
go through a previously occuring position).  Though if you have extra moves
saved up, you can make one move somewhere else to alter the board first and
then take back the ko and fill it if you still have extra moves left.
However, you can't take back the ko first and then play the second move since
this moves through a previous position.  Thus taking back a ko and
immediately filling it will not be allowed --- hence you need at least three
stones to take back a ko.  It seems to me that all rules in Go can be
'preserved' in some interesting way this game.

Here is an argument that this game will not degenerate into either normal Go
or one where both players keep saving moves and never play the moves out.
The value of an extra move is certainly greater than (with one move saved,
you cannot be caught in a ladder, for example, or you can threaten to cut
through bamboo joints) going first.  So, if nothing else, black should be
willing to play save-move on his first move.  Degeneration in the other
direction is also not possible, because there will be a point when a player
can use all of his saved moves and make live groups everywhere.  So optimal
play occurs somewhere in between.

I played this acouple of times and found it very hard.  Anyone else had tried
this?

I wonder what's the computational complexity of this game?

Wilfred
-----------------------------------------------------------------------------

I hinted in my previous message that its possible to generalise this.  In
ordinary Go you can't save any moves at all while in save-Go you can save any
number of moves at all.  Define n-th order Go to be like save-go except that
there is an upper bound of n on the number of moves that any player may have
saved at any one point.  Ordinary Go is then just 0-th order while save-go is
infinitieth-order Go.  (Note that by the argument in the second paragraph of
Wilfred's article it follows that for a fixed size board there is some k such
that for all m >= k  m-th order Go is effectively the same game as save-go.

I'm personally inclined to suspect that save-go may be too hard for humans to
play but 1st, 2nd or even 3rd order versions may be playable.



From: [e-mail] (Thomas Kettenring)

I sometimes play a Go version (surprise go) where you need three boards, four
players, and an umpire.  It's the funniest game I ever played, but it's
difficult to find four others who want to do it...

    Board 1       Board 3       Board 2

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
O                                           O
O     +---+         +---+         +---+     O
O  B1 |   | W1      |   |      B2 |   | W2  O
O     +---+         +---+         +---+     O
O                     U                     O
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO


Rules:
------
The players are B1, B2, W1, W2.  U is the umpire, O onlookers.  B1 and B2
are a team, using the black stones, and W1 and W2 are a team, using the
white stones.  B1 and W1 can see only Board 1, B2 and W2 can see only Board 2.
U can see all three boards.

The thing is just a normal game of go played on board 3, only that each
"player" consists of two persons with limited (and different) knowledge of
the board - e.g. B1 and W1 don't know what B2 and W2 do.

B1 makes the first move on board 1.
U puts a black stone at the same place on board 3.
W1 makes the second move on board 1.
U puts a white stone at the same place on board 3.
B2 makes the third move on board 2.
U looks if the same place on board 3 is occupied.  If yes, he says, "There
  is already a black stone" or "There is already a white stone," and B2
  places a stone of that color on board 2.  B2 is in a new situation and
  chooses another move, and so on until he has found an allowed one.
  U puts a black stone at the same place on board 3.
W2 makes the fourth move on board 2.  Again the umpire tests if the move is
  allowed as above.
B1 makes the fifth move, and the umpire test it.
etc...

Here are the sentences the umpire is allowed to say, or has to say, when B1
makes a move (modulo paraphrasing):

 B1 made a forbidden move:
  "There is already a black stone." (A black stone is added on board 1)
  "There is already a white stone." (A white stone is added on board 1)
  "That is suicide." (B1 will try the adjacent points and find out why)
  "That is not allowed because of the ko rule." (as above)

 B1 made an allowed move:
  "W1 to play." (This is important because it's very common that someone
     doesn't know it's his move)
  "Black puts White in atari.  W1 to play." (optional.  W1 will find out why)
  "Black captures n stones." (U shows where these stones were on both boards)

Atari on own stones is NOT announced because it would be unfair.

Onlookers are allowed to laugh and talk but they should of course make
only remarks that are cryptic enough so that the players can't extract info
from them.

Tactics:
--------
As a player, just try to play a normal game.  Normal fuseki, normal joseki.
The umpire and the onlookers will laugh, as your shimari is part of a silly
bulk on board 3.

Sometimes you will get a nasty surprise: the ladder does not work; your
bamboo joint has been cut; one of your two eyes is filled in by your partner
(who too thinks he has two eyes, but in other places.  You will become aware
of that when the neutral points are filled and you are suddenly in atari);
there is already a living group in what you took for your moyo.


Example:
--------
The best occurrence I encountered yet: From the fuseki of a game five or
six years ago.

B2 has just intercepted at the 4-5 point, and W2 has answered with an atari.

B1 played in another corner.

 Board 1              Board 3              Board 2

 . . . . . .         . . . . . .           . . . . . .
 . . . . . .         . . . . . .           . . . . . .
 . . # . . .         . . # # . .           . . . # . .
 . . . . . .         . . . . . .           . . . . . .
 . . . . . .         . . O # O .           . . O # O .
 . . . . . .         . . . O . .           . . . O . .
 . . . . . .         . . . # . .           . . . # . .
 . . . . . .         . . . . . .           . . . . . .


W1 plays the shoulder-hit at 4-4.  The umpire (that was me) says: "White
captures one stone."  Imagine B1's feelings: You have a stone on 3-3, the
corner is yours, and your opponent plays a ponnuki next to it.
Imagine B2's feelings: You try to cut a one-point jump because you speculate
other black stones will help you, you are put in atari, and you are captured
before you can do anything.



From: [e-mail] (Thomas Kettenring)

Ken Blake writes:
>This is very similar to the game "crazy go" which I first saw at the
>previous U.S. Go congress, organized by Bob High.  The only difference
>is that instead of 3 boards there are 5 boards - 1 for each of the 4
>players and 1 for the umpire.  Each player only gets to look at his own
>board of course.  Other than this the game is identical.

I know that one too (it is also played at European Go Congresses), and it
is a TOTALLY DIFFERENT GAME.  You see only your own stones.  That allows
for strategies which are not at all go-like, for example first making a
living group and then spreading tentacles over the board from there.  In my
version you have to play "real go" with real strategy, and the random element
plays not *such* a big role.  It is much closer to go than to crazy go.
The players have the impression of playing a normal game with occasional
surprises, while crazy go is more like walking around blind.

>Its hilarious to watch and very difficult to play.  In fact, I've only
>seen it played on smaller board sizes.  I highly recommend it as an
>interesting diversion.

Me too, but of course I think surprise go is much better :-)

>One question I have is should the teams be allowed to discuss strategy
>before hand or not?  The strategy might involve a certain number of
>fixed moves so both team members would know each others moves.  However,
>this could be foiled by the other teams plays, and they would also have
>their own strategy.

At surprise go, they should not be allowed to do that.  I think that would
spoil the fun.  Crazy go, on the other side... I don't know it well enough
to say anything but it could get a bit better by this.



From: [e-mail] (Bill Taylor)

(1) The first is the matter of the two variants played on a hexagonal grid
board; one played in the cells; one played on the intersections.  "Tri-go" (3
liberties) has been least regarded, being full of irritating (but fascinating)
petty tactical concerns.  "Hex go" (6 liberties) has had more coverage, and
Nicol N. Schraudolph, as well as going to the trouble of sending postscript
version of the boards, (thanks Nici !), has made a few comments on the
smallest-board versions of this game.

Nici says.....

>it looks like for board sizes up to 2, black can kill white completely.

This seems clear enough indeed. It's interesting to look at the "size 2"
board (i.e. 19 points) again though; and investigate the variant where
black is forbidden from playing the centre point on his first move (a kind
of komi). This is not nearly such an easy win for black ! At first I thought
it was a fairly simple-minded win for black, 10-9, (Chinese counting), but
then I noticed a clever variant play which leads to quite a different
result indeed (which I *think* is optimal). I will not give it here, as it is
rather fun to seek for yourself, and also I'd like "confirmation without
misinformation". So, all you small-board and variant investigators; have
a go at this one, and let me know your conclusions. (Incidentally, it never
fails to amaze me, how this type of 1st-move-restriction komi can make
such a large change in the length of analysis !)

Nici also mentions that

>On a size 3 board (37 intersections), black wins but white lives.

Yes, this seems clear too. I make it that the optimal result is a win
to black of 22-15. This is achieved, e.g. by the simple-minded play whereby
(black starting in the centre, and white attaching) both players make
a solid line across the board. Though variations are possible at different
stages, it doesn't seem as if either can improve on the 22-15 result.

It would be nice to have confirmation of this result !

Nici also says...

>I believe that this is also the case on the size 4 board, but would
>like to see others confirm or refute this, since it's a close call --
>it seems like white can *almost* invade successfully.

...but I'm not quite sure what this means. Surely if Black can win but white
can live on size 3, then both these will obtain on size 4 ? I would have
thought this was clear;   still, I have not investigated in detail what the
optimal result should be - it looks quite hard.


(2) My second lot of remarks follow up an interesting article by
Michael Chisnall about "save-go" and related topics.

The chief idea of the save-go game, was that one may either play one's move
normally, or save it up (and so on cumulatively), so as to play some larger
number of "simultaneous" moves later on, using part or all of one's "saved"
moves. Michael correctly points out that it is an advantage to save *some*
moves, but not indefinitely many. So somewhere in between there is a "best"
but changing optimum number of saved moves to keep, which will also depend
on the size of the board.

It should be noted, that in this game, and others mentioned below (where
more than one stone may be played at once), the multiple moves are not
really "simultaneous". The moves should be thought of as being played
CONSECUTIVELY, in that after each single play, a legal postion must
be left, even if more plays are about to be made. This is a vital
restriction in variants like "save-go" and the "progressive" types (below),
otherwise a game could never finish, as even very small eyses could be filled
up with sufficiently many moves.

So; in multi-move go variants, NO liberty-less groups may be left after any
of the separate moves. So two (small) eyes is always sufficient for life.

Also; regarding Ko-like situations, the logical alternative seems to me to
be that a seeming ko *can* be retaken on the first of a sequence of moves,
(allowing immediate retake-and-fill), provided, (as always), that the final
position you leave is *not* one you have left (as a final position) before.

In his article, Michael mentions various "CONSTRUCTORS", that make new
games out of old ones. I haven't heard this term before, but it is a nice
term for a widespread idea among game-variers, though perhaps "TRANSFORMER"
would be more accurate. I shall use this latter term. They apply to almost
any game; go, chess, and the rest.

Michael mentions the well-known "misere transformer", much handled by Conway
Guy & Berkelamp in "Winning ways", (but not really sensible in go); and also
the "-spiel transformer", where the players play blind to their opponent's
moves, a referee being needed to keep track of legalities.  (One might also
mention the "stake-doubling" transformer of backgammon, though this really only
applies to money games; but could be used in long go matches!).  Then there is
Michael's "save" transformer, which has great possibilities.

Another class of transformers are the "multi-move" transformers, and it is
these I want to mainly talk about.     ~~~~~~~~~~

The simplest is the 1-2-2-2-.. transformer.
That is, black starts with one move, then each has two consecutive moves
from them on. I have seen a chess game recorded between Alekhine and Euwe
of this type; and have played 6-in-a-row this way (5 is too easy), and
also small-board go.  9x9 go in this style is quite fun; the tendency is
for the two moves to be played in a little line of two, but not always.
There are still hanes, and some fascinating timing problems arise when
there are two active areas. Give it a try some time.

The main idea of the 1-2-2-2-... transformer is to eliminate 1st-move
advantage, and it seems to achieve this very well; as an analysis of very
small boards indicates, and play on 9x9 seems to confirm. For those who
like mathematical over-exactitude though, it may be argued that it doesn't
*quite* achieve this, in that *first* black is a move up, *then* white is
a move up, *then* black is,  *then* white,  and so on.  But of course,
black is still the *first* to be up; so if the game were interrupted "at
random", black would be *slightly* more likely to be a move ahead (though
nowhere near so overwhelmingly as in standard play).
                                                       To cancel this tiny
effect, a 1-3-4-4-4-4-4-.... multi-move transformer is suggested. This has
the effect of having black lead first by one move, then white by 2, black
by 2, white by 2, and so on; and now seems exactly fair, (though keen
mathematical bush-lawyers might want to iterate this procedure indefinitely!)

I have never played 1-3-4-4-4-4-.... at go, but have analysed some small
board versions; it seems quite fun, but not greatly more different than
the simpler 1-2-2-2-2-... type, and so hardly worth bothering with.

Another variant along these lines is the "PROGRESSIVE" transformer.
This is multi-move 1-2-3-4-5-...., where each player has one more move in
his series than his opponent just had. This is quite popular among chess
variant players, (where, unlike most variants, the concept of "check" must
be retained rather than just "king-capture"; otherwise it is trivial).

Progressive go does not seem to get so much attention; but (of course!) I have
played it from time to time in after-tournament sessions, and it really is
quite fascinating and well worth a look. It should not be attempted on a 19x19
board (!!), I strongly suspect. I have played it mainly on 9x9, and once on
13x13 which is also quite fun. As the number of moves per turn mounts up, all
but completely invulnerable groups come under threat; some amazing effects
become possible, and it becomes vital to have a good understanding of the
concept of "hypercontrol", that appeared in this newsgroup for some time, a
while ago (also available at Milton).  Hypercontrol and progressive go are true
bedfellows !

As a final little topic, I will mention still another multi-move transformer
which I think has some merit, especially for go (rather than say, chess).
Just as considerations of "who has played more moves than the other", led
to the introduction of the 1-3-4-4-4- over the 1-2-2-2-2- transformer, the
same ideas applied to the "progressive" concept lead to the intoduction
of the "ODD-PROGRESSIVE" transformer. This is progressive go where each
player has an increasing number of ODD moves; that is 1-3-5-7-9-11-...
multi-move go.  The advantage of this over progressive, is that each
player is an increasing number of "moves-ahead" at the end of his turn,
namely 1,2,3,4,...; (rather than 1,1,2,2,3,3,.. as at ordinary progressive,
which clearly still favours black slightly). I have *not* played this
variant, (nor do I ever expect to really!), but again, I *have* analysed small
board examples, and it seems a quite worthwhile fun variant.



From: [e-mail] (Thomas Kettenring)

There is a rather funny game invented by Ralf Gering, called 1000 Volt Go...
It is really playable, and you can think a lot.

The rules are the same as with go, but every move has an effect on the
stones on the same line.  The stones are thought of as having electric charge,
meaning that a black stone attracts white stones and repels black ones, but
only in the moment it is put on the board.

Example:

. . . . . . . . . . X . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . X . . . . . . . .
. . X . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . X . .
. O . O . . 1 . . . X . .
. O . . . O . . . . . . .
. . . . . . . . . . . . .
. . . . . . O . . . . . .
. . . . . . . . . . . O .
. O . . . . . . . . . . .

If in this situation White puts a stone on the point marked 1, the black
stone right of it will move as far as possible to the left, the white stone
left of it will move to the left, and the white stone below will move to the
bottom.  This is how the board looks afterwards:

. . . . . . . . . . X . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . X . . . . . . . .
. . X . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . X . .
. O O . . . O X . . . . .
. O . . . O . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . O .
. O . . . . O . . . . . .

Only those stones move that are connected with the new one by a straight
line with no stones in between, that is, never more than four.

This is what can happen now during a move:

1) Stone is put on the board
2) Up to four stones are moving
3) Opponent's stones without liberties are removed
4) Own stones without liberties are removed

These are the rules.
Here is a beginners' problem.
White to kill:
4. . . . . . . . . . . . .
3. . . O O O O O O O . . .
2. . . O X X X X X O . . .
1. . . O X . . . X O . . .
 a b c d e f g h i k l m n



White 1 at g1 is bad, as two black stones move and the white stone is captured:
4. . . . . . . . . . . . .
3. . . O O O O O O O . . .
2. . . O X X X X X O . . .
1. . . O . X . X . O . . .
 a b c d e f g h i k l m n

Black plays 2 at g1 again and the two stones move back.  Black is alive.
4. . . . . . . . . . . . .
3. . . O O O O O O O . . .
2. . . O X X X X X O . . .
1. . . O X . X . X O . . .
 a b c d e f g h i k l m n

Correct is White 1 at f1 (or h1):  Only one stone moves.
4. . . . . . . . . . . . .
3. . . O O O O O O O . . .
2. . . O X X X X X O . . .
1. . . O . . X . X O . . .
 a b c d e f g h i k l m n

If Black tries to make two eyes at e1, the g1 stone moves away.  Black is dead.
4. . . . . . . . . . . . .
3. . . O O O O O O O . . .
2. . . O X X X X X O . . .
1. . . O X . . X X O . . .
 a b c d e f g h i k l m n

But White has to be careful with capturing them.  Don't start at f1...
Correct: W c1, W g1, W e1, W g1.  (c1 is necessary because if you omit it,
the d1 stone will go away when you play e1.)



From: <[e-mail]>

Here's a variation of GO to play when you're not feeling serious.
The only difference from regular GO is that, instead of placing
one stone per turn, you throw a die to determine how many.  Thus
each player places from one through six stone on each turn.  The
stones are placed one after the other, as if the opponent were passing
several times.  (i.e. if black rolls a three, white has to pass twice.)
The game continues until both players agree that the board cannot be
changed.  (Yes, this does eventually happen!)



From: "Jonathan R. Ferro" <[e-mail]>

A variation in R. Wayne Schmittberger's _New Rules for Classic Games_ is
to allow each player up to 4 stones per "move", with the condition that
*all stones played must form an orthogonally-connected group*.  This
small change means that two eyes are still alive, ko is still possible
(although much more rare), tesuji are possible, and the game still
requires a worthy amount of intellectual input, but makes the game much
faster and easier to evaluate at the end.  An interesting sample game is
diagrammed in the book.