Trabsact Sagme Diaries: March 2007

Mar 30, 2007

BEAT IT OR EAT IT

[Bill Taylor 1993 post @ r.g.a] Though this games is played with cards, and is (very vaguely) one of the trick-taking family of games, it still belongs here in rec.games.abstract. It is as much a strategy game as sprouts, chess and Go, being a 2-player game of complete information with no element of chance (apart from the starting layout, and even this is symmetric between the players).

Long ago, I played a (more complex) version of this game, for a while. The original game was a 4-suit game, each player having his own trump suit. However this 2-suit version seems just as skillful, as much fun, and "cleaner". It is even possible to play a one-suit version !!  No other changes in the rules are needed, except that the starting layout can no longer be symmetric. But the two-suit version seems neatest.

I post it here because...

(i) it deserves to be far better known than it is;
(ii) it seems to be essentially unique of its kind;
(iii) I would eventually be keen to start an email game or two.

The game has been around a fair while, but doesn't seem to have a name of its own. It was first introduced to me as "Besicovitch's game", but such a name is hardly descriptive. We sometimes used to call it "Finchley Central" as a small in-joke, indicative of the hair-trigger timing needed to decide when to strike your main blow. The game usually see-saws one way then the other, as every advance tends to leave one weaker. Thus it might fairly be called "See-Saw" or "Negative Feedback".  Until a concensus is reached, I shall call it "Beat It Or Eat It"; being descriptive of the mechanics of play.

One nice thing about the game is its almost complete freedom from *arbitrary* rules, once the basic logic of play is set. The main exception is the length of the suits.  13 is the obvious length, and feels about right. Shorter would be good for practice games, and longer (up to 26) would be possible, with one "red" suit and one "black" suit.

The core idea of the play is:- taking the lead in turns, one player leads and the other follows, until someone gets rid of all his cards, thus winning.

--------------------------------------
"BEAT IT OR EAT IT"  (Full rules)

1. Two players play with a deck of 13 hearts and 13 spades. Aces count high.

2. The initial layout is symmetric, and obtained thus:-  One player shuffles, and deals 13 cards to the other, who keeps only the red cards. The blacks are returned to the undealt cards, and the dealer gives himself black cards identical to his opponent's reds. Then each gets the remainder of the other color. The "leader", (player of the 1st card), is chosen at random.

3. The leader plays any card onto the table. The follower EITHER picks it up,  OR (if he can, & wishes to), beats it by playing a higher card of the same suit. These cards stay on the table, and the follower becomes the leader for the next play.   Continue with (3) again.

4. On any play, the follower may, if he desires, (& must, if unable to beat the card led), pick up all the cards on the table, and add them to his hand. The leader then remains as leader for the next play.  Continue with (3).

5. Whoever first plays the last card left in his hand, is the winner.
  (It is immaterial whether this occurs as a lead or a follow.)
-------------------------------------

So, the idea is to get rid of all your cards. But it is essential along the way to sometimes (voluntarily) pick up all the tabled cards, to get some of the high ones there, (even though this hinders your main goal, of course).

And often, (especially if the opponent is close to winning, or has too few low cards for comfort), you will have to lead a high card that he CAN'T beat, forcing him to pick up all the junk on the table.

Remember, all played cards, leaders and beaters, stay on the table (face up), until one player chooses to or is forced to "eat" them, i.e. pick them all up, and suffer being follower again for the start of the next series of plays. As long as no-one "eats" the stuff on the table, the lead alternates.

So there it is. It is a great game; and as I say, one of complete information. In fact it is standard for both players to keep their hands face up on the table in front of them, for convenience; (these hands are only on the physical table of course, not the "logical" table).  If you try it out, you will quickly notice the negative feedback element mentioned above.

It is usually an advantage to start, but by no means always; it depends on the nature of the starting layout, and perhaps on the parity of the suit length.

I warmly recommend everyone to give it a try.

To indicate the nature of the play, here is a sample game, with a 7-suit pack.

Initial layout: (LEFT has the opening lead)

LEFT: hearts A Q T 9 8      RIGHT: hearts K J       spades K J                   spades A Q T 9 8 8h, Jh. 8s, Js. 9h, Kh. 9s, LEFT eats all (by choice). Layout now:  (RIGHT is on lead) LEFT: hearts A K Q J T 9 8     RIGHT: hearts -       spades K J 9 8                  spades A Q T Ts, Js. 8h, RIGHT must eat. 9h (R must eat). Th (R must eat). Layout now:  (LEFT is still on lead) LEFT: hearts A K Q J      RIGHT: hearts T 9 8       spades K 9 8               spades A Q J T 8s, Ts. 8h, Jh. 9s, Js. 9h, Qh. Kh, RIGHT must eat all. Layout now:  (LEFT is still on lead) LEFT: hearts A       RIGHT: hearts K Q J T 9 8       spades K              spades A Q J T 9 8 Ah (right must eat);  Ks and LEFT WINS.

This game hardly showed much ability on either part.   RIGHT should have
struggled more actively in the last phase of play, e.g. when he led the 9h, which led to an immediate simple forced loss. But it was probably too late by then anyway. There was a very clear-cut error earlier. When RIGHT led the ten from his 3-card hand of AQT hearts, it would clearly have been uniformly better to lead the queen. (However he was probably lost anyway.)

LEFT also made a blunder in the very first round; he could have beaten the 9s with the Ks, led back the Th (compulsory eat), the Ah (ditto) & Qh (winning). (Suit length of 7 is not really enough to give the full flavor of the game.)

Mar 24, 2007

FANO NIM

[bill Taylor, 2005] Here's a cute little mathematical game. I'm surprised it hasn't been around the abstract game world before. It's only the second game I know of that's based on the Fano plane - the smallest possible 2D Projective Geometry.

Here it is - it has vertices THEY WAR, and lines YEA WHY TRY HER RAW WET HAT,
neatly compiled into a triangle with three altitudes, and an incircle which
is also a "line"...
Y /|\ / | \ / | \ / | \ / | \ / _.-"|"-._ \ /.' | `.\ H. | .E /: "-. | .-' :\ / | ";-T-:" | \ / : _-' | `-_ : \ / \' | `/ \ / ,-' `. | .' `-. \ /_-' `-..|..-' `-_\ W--------------R--------------A

Note there are no interior intersections between the altitudes and
the circle, only at the tangent points where they triply intersect
with the sides as well.

Anyway, the game is a form of Nim.

The diagram starts with a small integer at each of the seven vertices,
the number of "seeds" at that vertex, preferably a different one for each.
(These can be decided on in one of several standard ways.)

On each turn, the player to move must remove THE SAME number of seeds from
any three vertices in a line, (remembering that the circle is also a line).

Last person to make a legal move wins. (This is the standard
winning condition for CGT games. There is also the "misere" version OC.)

I will leave it as a fun exercise for fans to compile a list,
hopefully exhaustive, of all the "N positions" (Next player wins),
and all the "P positions" (Previous player wins).

And note, too, that if one too quickly learns the correct optimal play,
(though it is not as straightforward as regular Nim), one can easily
extend it to larger projective geometries; (the next one has 13 lines
comprising four vertices each and meeting four at a vertex.)

It can, OC, be played on any geometry at all; but the projective
nature of the game ensures that the winning condition is equivalent
to producing a line of zeros, which is a nice target to aim for!

BESIEGE

[by João Neto] Start with the NxN board full of stones. Then, alternating moves,
each player removes a stone (that stone is not captured) When a player makes a
square, he captures all the stones inside that square (no diagonal squares).

E.g.

x x x x x 5 x . x x . 4 x x x x x 3 x x . . x 2 x . x x x 1 a b c d e

If a player removes stone e1, he will capture stones c3 and d3

When there are no capturable stones left on the board, wins the player
with more captured stones.

Area vs Territory Go Rules

[Bill Taylor] The following Go example shows a safe set of stones with two eyes:

. . . . . . . . o o . . . o o . o . . o . o o . . . o o . . . . . . . .

Would this set be called a single group even though it is composed of two disconnected halves?

Most players would call it a single group. In area rules it doesn't matter what you call it. In territory rules it could conceivably matter what you call it, (though it doesn't in practice), because territory rules, in all their artificial absurdity, have to refer to groups from time to time in order to define what is considered "dead" and what is considered "alive".

In territory rules, this matters(!). In area rules, it doesn't matter - if there is any dispute you just play it out (and with no cost) until a group or groups is removed from the board.

Logically speaking, you should call the above two separate groups, each helping to keep the other alive. But people never speak so precisely in practice.

There is even a worse situation. The following group has only one true "eye"; the other one, in the NW corner, is a so-called "false eye", and can eventually be filled and the whole lot captured.

. x x . o . x o o o o o x x x x o . x . x x o o x x x x o .

Territory rules actually have to define the concept of eyes, false eyes, and the rest. It is lunacy. (Area rules define nothing - you just play it out to the grim end if necessary). Territory rules, with their defined false eyes, come to grief in this famous sort of position:

. x x x x x | Here, black has TWO false eyes, and not a single true one!
x o o o o x | And yet, both separate groups, or parts of a group, are
x o o . o x | keeping one another alive; rather like your above example.
x o . o o x |
x o o o o x | And even the Japanese admit that black is alive, in spite of
x x x x x . | what their rule books say!

Basically, territory rules are an abortion. Computers cannot handle them because they are essentially logically flawed.

Games Operators, aka, Mutators

[Posts@rec.games.abstract ~ Dec 99, Jan 2k]

[Douglas Zare] Here are five examples of operators on games:

1) Misere: The object of the new game is to lose while playing the old game.
2) Move and switch: After the first move, the second player can choose to switch sides rather than respond.
3) Bad advice, as in Fred Galvin's "Compromise Chess": A player offers two possible moves to the opponent, who advises the player which to play. If there is only one legal move then this is simply made.
4) Use a doubling cube, as in backgammon. I have heard that this is a nice addition to speed chess, and would guess that this would make spectator sports more interesting.
5) Bughouse: Instead of playing, you can drop in a piece your partner has captured on a parallel color-reversed board. The partnership of the first winner wins.This doesn't do much for go or pente, but how does it affect checkers?

All of these take in games and produce new games, though there may be a few choices to make in the implementation. In my experience, understanding the original game carries over to some level of understanding in the new game, although there are substantial differences. Interestingly, the first three are close to involutions, i.e., applying the operator twice may produce the original game. The first three also produce legal games, albeit strange ones, and can be played on many internet game servers to the confusion of kibitzers. (I'm always willing to play backgammon misere unrated as zare_10027 on Yahoo.)

I am curious what other interesting operators have been tried, particularly those which are simple, preserve some of the structures of the original game, and are fun. I would also appreciate any pointers to annotated games or the theory of hex misere and backgammon misere. (Are there videos of grandmasters playing bughouse?) On the other hand, I would like to know how much the endgame theory of bad advice go resembles that of ordinary go.

[Tim Chow] John Conway has suggested "Whim," which is Nim except that at any point in the game a player may say "Whim" instead of moving; this decree alters the game from normal to misere. The whim may be invoked only once per game (*not* once per player per game). It turns out that Whim is just Nim with an extra "invisible" pile of counters (I forget of what size) representing the whim, but the same operator will surely have more interesting effects on other games.

[David Bush] Misere doesn't work well for chess and related games. Selfmate puzzles notwithstanding, it's usually impossible to force your opponent to checkmate you. Misere Go sounds like a disaster. Checkers might be interesting, with cumpulsory captures. Maybe Misere suicide chess, where captures are compulsory and the King is just another piece, but the last player to move WINS...? Speaking of which, suicide might be an applicable operator to battle games. There are tons of fairy chess rulesets which might apply to battle games (chess, checkers, etc.) For example, unambiguous chess requires all moves to be representable unambiguously with 3 symbols in descriptive notation, where the dash - doesn't count, but the x for captures does. E.g. you couldn't play QR-K1 if KR-K1 is also possible. Maybe some games have move notation which could be adapted for this. Or, salt shaker chess starts the game with a salt shaker on a central square. The shaker moves in the same direction & distance as whatever piece is moving (king when castling.) You may not move the shaker off the board with your move. The issue of what2do if the shaker would land on an occupied square, could be dealt with in several ways. Then there's nuclear chess, where each piece adjacent to a capture, friend or foe, is removed from the board (except the capturing piece.) Or protean chess, where each piece (except possibly the king) takes on the powers of movement of whatever piece it captures. Or et cetera...

[Fred Galvin] Here are some references on misere hex (and other variants):

Ronald Evans, A winning opening in Reverse Hex, J. Recreational Math., Vol. 7, No. 3 (Summer, 1974), 188-192.
Ronald Evans, Some variants of Hex, J. Recreational Math. Vol. 8(2) (1975-1976), 120-122.
Jeffrey Lagarias and Danny Sleator, Who wins Misere Hex?, in: Scott Kim, ed., Articles in Tribute to Martin Gardner (Atlanta International Museum of Art and Design, January 16, 1993), 146-148.

[anonymous] you can pre-give an amount of -say- money to each player. Whenever such a predefined condition occurs , both players bet from their money , whether they want the condition to be valid or cancelled , and the highest bet decides. Of course this bet-amount is substracted from the higher-bet-player's money. This usually increases (and sometimes completely changes) the structure of the game. As an example consider betting tic-tac-toe , where always the player moves , who bets most. Or betting go : after move 3n , the n-th free square is filled with a piece from the player whith the higher bet. Or betting football : all 10minutes a player is removed from the team whith the lower bet. A handicap for good-players can be applied by allowing different initial amounts of money.

[Andy Tepper] How about a HoardMoves(G,n) operator? At any time instead of moving you may delay up to n moves and then make them all at once in the future. For instance, in HoardMoves(Go,1) you could skip a turn and then play the two moves in parallel on some future turn. A group would have to have 3 eyes to be alive. (Assuming moves were made in parallel. You could always say that the two moves must be made in sequence which keep the 2 eyes rule.)

[Gerry Quinn] Sounds a bit horrid in Chess. Both players will hoard at the start, with White hoping to declare mate in (say) ten, and Black having to scramble for some tactics to prevent it. But then Black will be further behind, and White will win when he has hoarded a couple more. As Alekhine said (or was it Nimzowich) "In Chess, the threat is stronger than the execution!" Hoarding a _partial_ move might be interesting in Chess, though. Pass a move twice, and you have a free one in hand. This might be fairly balanced.

Mar 23, 2007

Nice Chess Sites

About the Origins of Chess

http://www.mynetcologne.de/~nc-jostenge
http://www.samsloan.com/origin.htm

About Tournament Chess

http://www.mark-weeks.com/chess/55571wix.htm
http://www.chesscafe.com/

Curiosities

http://www.xs4all.nl/~timkr/chess/chess.html

Go and the 'Three Games'

Go and the 'Three Games' -- A text by William Pinckard

Games-playing is one of the oldest and most enduring human traits. Disparate pieces of evidence such as dice discovered at Sumer, game-boards depicted on Egyptian frescoes, Viking chess pieces, and ball parks constructed by ancient empires deep in the Andes link up directly with contemporary phenomena such as Saturday night poker games in Kansas City and the annual go title matches in Tokyo.

Games are undeniably a concomitant of civilization and even in their most primitive forms presuppose some degree of sophistication. Most of all, they require the ability to think in abstractions and to manipulate ideas in logical terms, thereby giving form to what is formless and creating small, recognizable patterns in the shadow of great mysteries.

From ancient times in Japan the so-called 'Three Games' were backgammon, chess and go. Chess probably comes from India, backgammon from the Near or Middle East, and go from pre-Han China. Backgammon is a gambling game which, using dice, gives luck or chance the preponderant role. Chess in one of its earlier forms also used dice, but takes its present shape from the structure of a royal society and from war maneuvers. Go is the most abstract and 'open' of the three; and with its freedom from complicated rules, its simplicity of form, its fluidity and spaciousness, it comes remarkably close to being an ideal mirror for reflecting basic processes of mentation.

Go is played with black and white 'stones' all of exactly the same value, thus somewhat resembling the binary mathematics which is the basis of the computer. The stones are played onto the board and are left as they stand throughout the game, so that the game itself takes shape as a visible record of the thinking that went into it. About three hundred years ago an eminent Chinese monk came to Japan on a visit and was shown the diagram of a game of go which a master of that time had recently played. Without knowing anything of the game save the sketchy description they gave him at the time (this was after go had more or less died out in China), the monk studied the moves as shown on the record and after a few moments remarked with much admiration and respect that the player must have been a man who had become enlightened -- which was indeed the case. (It is interesting to note that this story is told on the one hand by go players to illustrate the quality of the game and on the other hand by Buddhists to show the acuity of the monk from China.)

The great 17th century Japanese playwright Chikamatsu, in a famous passage, compares the four quarters of the go board to the four seasons, the black and white stones to night and day, the 361 intersections of the board to the days of the year, and the center point on the board to the Pole Star. It would be easy to erect a tower of fanciful theory along these lines, but that would only obscure the obvious point. In this striking analogy Chikamatsu is describing a feeling of hugeness and all-inclusiveness -- the board conceived as a complete world system in potential form. The board and pieces can be thought of as limitless: any number of lines and an endless supply of stones to play with, the game itself being the life of the players. (In Chikamatsu's play a young man becomes old and grows a long beard while watching a single game.) Only because we are human and must put practical limits to our activities, do we use just a small part of the infinite board. But this field of nineteen by nineteen is large enough to contain everything we are able to put into it. The number of possible games playable on this board has been reckoned to be more than the number of molecules in the universe.

An anonymous go player has written: 'The board is a mirror of the mind of the player as the moments pass. When a master studies the record of a game he can tell at what point greed overtook the pupil, when he became tired, when he fell into stupidity, and when the maid came by with the tea.'

Contrary to the opinion of many people, go has nothing to do with Buddhism. Because it is a valid system in itself, it offers nothing contradictory to other systems, but in fact go is an older inhabitant on this planet than is Buddhism. In China it became one of the Four Accomplishments, the others being poetry, painting and music. It reached Japan around the 6th century and for a long time remained the exclusive property of a leisured noble class. Then during the 16th century all this changed. The many great families and clans which had warred happily against each other for a thousand years were gradually brought under the hegemony of the Tokugawa Shogunate. It was during the subsequent period of the Tokugawa era (roughly from 1600 to 1868) that go, along with haiku, kendo, tea ceremony and so on, was most actively cultivated as a way of constructively channeling the mental energies of the people during the long years of peace. One formal word for go in Japanese is Kido. Ki is the old Chinese word for go, and -do is the Chinese word for Tao, which means Way -- or, more specifically, a Way to enlightenment.

All games channel mental energies, whether they lead to enlightenment or the reverse, but it is suggestive to consider the 'Three Games' in their social context because then we can see how each of them reflects certain basic characteristics of a general or regional type.

Chess, for example, the great historical game of the West, involves monarchs, armies, slaughter, and the eventual destruction of one king by another. The game appears to be entirely directed along the lines of the great myths of the West from the Mahabharata to the Song of Roland -- the overthrow of a hero and the crowning of a new hero. The pieces, from king down to pawn (peon), give a picture of a heirarchical and pyramidal society with powers strictly defined and limited.

Backgammon, the favorite game of the Near and Middle East, is preoccupied with the question of Chance and Fate (Kismet). All play is governed by the roll of dice over which the player has no control whatever. The players are matched against each other, but each tries to capture a wave of luck and ride it to victory. The loser curses his misfortune and tries again, but the individual is helpless in the grip of superior forces.

Go, the game of ancient China and modern Japan (and now popular throughout the world), is unique in that every piece is of equal value and can be played anywhere on the board. The aim is not to destroy but to build territory. Single stones become groups, and groups become organic structures which live or die. A stone's power depends on its location and the moment. Over the entire board there occur transformations of growth and decay, movement and stasis, small defeats and temporary victories. The stronger player is the teacher, the weaker is the learner, and even today the polite way to ask for a game is to say 'Please teach me.'

Things are different now, but in earlier times, when go was so much admired by painters and poets, generals and monks, the point of the game was not so much for one player to overcome another but for both to engage in a kind of cooperative dialogue ('hand conversation', they used to call it) with the aim of overcoming a common enemy. The common enemy was, of course, as it always is, human weaknesses: greed, anger and stupidity.

Every year in March department stores all over Japan present elaborate displays in connection with the Doll Festival. If one looks carefully at the miniature weapons, musical instruments and furniture of a really complete display one will find a tiny backgammon board, a Japanese chess (shogi) board and a go board.

The 'Three Games' is a useful classification because taken together they make up a coherent world view. Most of philosophy boils down to speculation centered around the three basic relationships of the human species. The first is man in his relationship to the remote gods and the mysterious forces of the universe. The second is man in the society he builds up around him. The third is man in his own self. Or, to put it another way, man the backgammon-player, man the chess-player, and man the go-player.

That we have these three shows that they answer basic needs in the human spirit. People everywhere are preoccupied with social structures, position and status; and everyone who is capable of reflection must sometimes speculate on his private relationship to fortune and fate.

But go is the one game which turns all preoccupations and speculations back on their source. It says, in effect, that everyone starts out equal, that everyone begins with an empty board and with no limitations, and that what happens thereafter is not fate or wealth or social position but only the quality of your own mind.

Seo's unknown game

[Seo Sanghyeon mathmaniac@hanmail.net] This game (name and author unknown) is played on square grid board of odd size. But don't choose smaller than 3x3. It's trivial. 5x5 with two players is interesting enough, but 7x7 is more strategic, I think.

EQUIPMENT: I strongly recommend to play by paper and pencil. It's hard to play with board and stones. Since there is no capture, no need for eraser.

INITIAL POSITION: Each player chooses start position, and drop his stone there.

DROP: On each turn, each player drops a stone on an empty cell adjacent, orthogonally or diagonally, to his last dropped stone. And board wraps, i.e. first row is adjacent to last row, etc. And draw (this is why paper and pencil is recommended) a link from his last dropped stone to his newly dropped stone. When playing diagonally adjacent cell, this link should not be crossed.

. . . . . o 1 . . x , . . . . .

If 'x' plays '1', a link from 'x' to '1' is drawn, and since it cannot be crossed, 'o' cannot play ','.

.. o2 o3 o4 .. o1 ,, o5 .. x. o7 o6 .. .. .. ..

On the other hand, if 'o1'-'o7' is played in that order, there is no link from 'o1' to 'o7', so 'x' can play ','. So that's why playing with board and stones is ambiguous.

A variant

.. .. .. .. .. x1 x3 .. .. ,, x2 .. .. .. .. ..

According to the rule described above, now 'x' cannot play ',', because 'x1' to 'x2' link will be crossed. But there's a variant that allows crossing of his own link. And it seems this leads to more interesting game on 7x7.

EXAMPLE GAME:

'x0', 'o0' is initial position, i.e. diagonally opposite corner. Since board wraps, there's no difference between corner and center. But wrapping moves are harder to read. 'x' plays first.

x0 .. x5 o5 o6 o7 .. x4 x6 x7 x8 .. x3 .. ** .. x2 o2 o1 .. .. x1 o3 o4 o0

(o8: resign)

COMMENT:

x2-x3 prevents white to move toward upper left corner. x5 is of same line, preventing o4-x5 wrap move. And notice that, after o7, o cannot play marked ** cell, since it will cross x7-x8 link.

. . / . . . . \ x . x o o . \ o . x x x \ . x . x . . \ . . x o o . . . . x o o o . . / . . | . .

If you draw board this way, It's not that hard to see. (It can be done on TwixT board, too.) So, white has at most three cells to play, but black has five.

. . / . . . . \ x 5 x o o . \ o 3 x x x \ . x 1 x . . \ \ 2 x o o 6 . . 4 x o o o \ . / . . | . .

Also notice that

x0 ** x5 o5 o6 o7 :: x4 x6 x7 x8 .. x3 .. .. .. x2 o2 o1 .. ## x1 o3 o4 o0

o cannot play o7-**-## line, since it crosses x0-x1 link. Therefore
:: as a reply to ** is a mating move.

RingGo 80% Go 20% Hex

[(C) 2001 William I. Chang] RingGo is a variant of Go played on a hexagonal lattice with 127 points, 18 of which are removed from play in order to strike a balance between how easy or hard it is to make a group of stones live. The board inherits from Rosette, Medusa, and especially a conversation with Greg Van Patten. Most points have 4 liberties.  While it is harder to make two eyes, it is easier to connect groups so the network may have two eyes.  In this sense, I think the game achieves its goal of combining Go's intricate eye-making tactical play with the connection-making strategy of Hex (though perhaps not enough of the latter).  I'm sure it can be refined and improved if more people tried it.  The board is generalizable to odd-order lattices by repeating the Medusa pattern, although there are lots of other beautiful patterns to choose from.

           . . . . . . .                  . . . . . . . . .           . . . . . . . .                . . . . . . . . . .          . . . o . o . . .              . . . o . o . o . . .         . . o . . . . o . .            . . o . . . . . . o . .        . . . . . o . . . . .          . . . . . o . o . . . . .       . . o . o . . o . o . .        . . o . o . . . . o . o . .      . . . . . . . . . . . . .      . . . . . . . o . . . . . . .       . . o . o . . o . o . .      . . o . o . o . . o . o . o . .        . . . . . o . . . . .      . . . . . . . . . . . . . . . . .         . . o . . . . o . .        . . o . o . o . . o . o . o . .          . . . o . o . . .          . . . . . . . o . . . . . . .           . . . . . . . .            . . o . o . . . . o . o . .            . . . . . . .              . . . . . o . o . . . . .                                        . . o . . . . . . o . .              . . . . .                  . . . o . o . o . . .             . . . . . .                  . . . . . . . . . .            . . . o . . .                  . . . . . . . . .           . . o . . o . .          . . . . . . . . .           . . o . . o . .     RingGo boards of order 5/7/9, with 55/109/193            . . . o . . .      points.  The order-11 board has 301 points.             . . . . . .              . . . . .

One might play RingGo on a Go board if one can envision dividing each
square into two triangles with a diagonal line drawn top-left/bottom-right.
(There actually was a commercial version of Hex done this way!)  Or, have
the players sit adjacent and both look toward the *.

   . . . . . . .           *    . . . . . . . .    . . . o . o . . .    . . o . . . . o . .    . . . . . o . . . . .    . . o . o . . o . o . . W   . . . . . . . . . . . . .      . . o . o . . o . o . .        . . . . . o . . . . .          . . o . . . . o . .            . . . o . o . . .              . . . . . . . .                . . . . . . .                B

This game should play very differently from other hexagonal Go variants.
I tried and liked it :-)  Any suggestions or comments will be greatly
appreciated!

William Chang   Los Gatos, California   18 April 2001
email: wchang@acm.org, williamichang@hotmail.com
(C) 2001 William I. Chang

---

Other Boards: Rings board           o o   o o   o o          o   o o   o o   o           o o   o o   o o        o o   o o   o o   o o       o   o o   o o   o o   o        o o   o o   o o   o o     o o   o o   o o   o o   o o    o   o o   o o   o o   o o   o     o o   o o   o o   o o   o o        o o   o o   o o   o o       o   o o   o o   o o   o        o o   o o   o o   o o           o o   o o   o o          o   o o   o o   o           o o   o o   o o Medusa board           o o o o o o o o          o   o   o   o   o         o o o o o o o o o o        o   o   o   o   o   o       o o o o o o o o o o o o      o   o   o   o   o   o   o     o o o o o o o o o o o o o o    o   o   o   o   o   o   o   o     o o o o o o o o o o o o o o      o   o   o   o   o   o   o       o o o o o o o o o o o o        o   o   o   o   o   o         o o o o o o o o o o          o   o   o   o   o           o o o o o o o o

Revolving Games

Finding ways to spin your games
or
The stable King and his revolving servants

[December 2000] We (Bill Taylor and Joao Neto) have invented and started to play some games wth a common rule theme: if some condition is met, the moved piece changes its powers. Like a game with rotating officials.

This, if done well, can result in very dynamic games, where some pieces lead very wild lies.

Joao invented a game based on a simple idea: depending whether the turn number is even/odd, the moved piece is promoted/demoted. This basic change makes (in Joao's opinion) a very good chess variant, called Promotions and Demotions, or just ProDem.

But, is this the only possible way to use this dynamic idea?

Revolving Games

There is an old chess variant, named Revolving Chess (whose origins we don't know), whose rules are:

REVOLVING CHESS

1. Same as FIDE, except:
2. Each moved non-king piece, changes its status in the following order:
  2.1. Knight to Bishop,
  2.2. Bishop to Rook
  2.3. Rook to Queen
  2.4. Queen to Knight

--------------------------------------------------
The original game was fully "royal", but we play with stalemate
= win for stalemater, though still with castling (R changing to Q).
--------------------------------------------------

Game Sample

1. d4      d5      
2. c4      d:c4    
3. Nc3(B)  b5      
4. a4      c6      
5. a:b5    c:b5    
6. b3      a5      
7. b:c4    b:c4    
8. e3      R:a6(Q) 
9. Ne2(B)  e5      
10 B:c4(R) Bd6(R)  
11 R:c8(Q)  Qa:c8(N)
12 B:a5(R)  e:d4    
13 Rb5(Q)+  Nc6(B)  
14 Q:c6(N)  R:c6(Q) 
15 Q:d4(N)  Q:d4(N) 
16 e:d4     Nf6(B)  
17 Bf4(R)   O-O(Q)  
18 Ra6(Q)   Qc2(N)+ 
19 Kd2      N:d4(B) 
20 R:f6(Q)  B:f6(R) 
21 Q:f6(N)+ g:f6
22 Ba6(R)   Kg7
23 Rha1(Q)  Qe8(N)
24 Ke3      Ncd6(B)
25 R:d6(Q)  N:d6(B)
26 h3       Be5(R)+
27 Q:e5(N)  f:e5
28 Ke4      f3
29 Kf5      Kf7 
30 g4  1-0 [if Ke7 or Kf7 then h4]

Final Position: 

. . . . . . . .
. . . . . k . p
. . . . . p . .
. . . . p K . .
. . . . . . O .
. . . . . . . O
. . . . . O . .
. . . . . . . .

--------------------------------------------------

Since it's the older game, we may think of it as the standard revolving positional game in this text, despite the fact that Promotions and Demotions was an independent discovery. In fact, ProDem was the first game we played, and it was then that many different and yet related ideas appeared.

Firstly, the rules of ProDem:
-----------------------------------------------------
PROMOTIONS & DEMOTIONS [aka "even-up, odd-down"]

1. The FIDE rules apply except in the following:
2. On even turns, a moved (non king) man is promoted after move completion.
3. On odd turns, a moved (non king) man is demoted after move completion.
4. The Promotion/Demotion system has this ordering: P < N or B < R < Q.
5. Pawns on the 1st rank may move 1 or 2 squares.
6. Pawns on the 8th rank cannot move, but may be captured.
7. There is no En-Passant, Mate, Check or Castling.
8. The winner is whoever first captures the opponent's King.
9. White does not play on turn 1.

notes:

* A Pawn can promote to Bishop or Knight at the mover's choice.
* A Rook can demote to Bishop or Knight at the mover's choice.
* Queens cannot be promoted, so they cannot move on even turns.
* Pawns cannot be demoted, so they cannot move on odd turns.
* Since every pawn promotes when moving, there is no FIDE promotion.
* Black must start, with a Knight's demotion.
(helps neutralize 1st move advantage).
----------------------------------------------------

Here goes two sample games:

1.   --     Nf6(P) 
2.   c4(B)   e6(B) 
3. B:e6(P)  Qe7(R) 
4. e:f7(B)  R:B(Q) 
5.  Qc2(R)  Nc6(P) 
6.   g3(B)   h5(N) 
7.  Bg2(P) N:g3(P) 
8. f:g3(B)   d6(B) 
9. B:d6(P) B:d6(P) 
10  Nf3(R)   d5(B) 
11.  Rc5(B)  Rh4(N)
12.  Nc3(R) N:g2(R)
13.  Rb1(B) B:a2(P)
14.  Bh7(R)  Bc6(R)
15. Rhf1(B)  Rg6(N)
16.  Bg2(R)   a6(N)
17. R:g6(B) Q:g6(R)
18. Rh8(Q)+  Kf7
19. Qxa8(R) Nxc5(P)
20.   b3(B)   b5(B)
21. B:e6(p)+  Kd7
22. R:c5(Q)+  K:e6
23. Rf5(B)+   Kf7 
24. Rf8(Q)+   1-0

Final Position: 
. . . . . Q . .
. . p . . k p .
. . p . . p r .
. b Q . . B . .
. . . . . . . .
. . . . . . . .
p . . O O . . O
. . B . K . . .

[note: even if these games have non royal Kings, we still use the + symbol which means some piece is attacking the King]

1. --        Nc3(P)      
2. b3(B)     d5(B)       
3. Bb2(P)    Bf5(P)      
4. c4(B)     e6(B)        
5. Ne3(P)    Bc4:B(P)    
6. h4(B)     c4:B(B)     
7. Bh4:Q(P)  Bb3:Q(P)    
8. Nc3(R)    Ra8:P(Q)    
9. Rb1(B)    g6(B)       
10.Bb1xB(P)  Be6:P(P)    
11. g6:f7(B)+  K:f7         
12. Rh5(B)+    Ke7          
13. d3(N)      a5(B)        
14. Kd2        Ba:Rc3(P)+   
15. b2:c3(B)   Nf6(R)       
16. Ke1        Rg8(B) 
17. g4(N)      Rf5(Q) 
18. Bf6(P)+    Qxf6(R) 
19. Ne5(R)+    Re6(Q) 
20. R:Qe6(N)   K:e6 
21. Be8(R)+    Kd7 
22. Re8e5(N)+  Ke6 
23. Bh3(R)     a1(B) 
24. Kf1        B:e5(P)
25. resigns    0-1

Final Position: 

. . . q . b b . 
. p p . . . . p 
. . p . k . . . 
. . . . p . . . 
. . . . . p N . 
. . . . . O . R 
. . . . O O . . 
. . . p . K . .

Other remarks:

* The Rook is probably the strongest piece. It may move any turn, and still transforms into a strong piece.
* A (possibly good) method of play would be to always move your king on odd moves, so your piece strength constantly went up and never down. Hwever, this would waste so much time it probably wouldn't pay off anyway, since a player doing that increases his army, but slows by half its efficiency.
* Of course, making the other player moves his King on a even turn, makes him lose a promoting turn.
* A pawn on the last rank cannot move. That is especially bad, since the other player can use it as a protecting wall.

**********************

Well, taking different behavior given a turn number is one possibility, others exist:

which colour square the piece is on
which colour square the piece goes to
whether the piece changes square colour when it moves
whether the piece moves forward or back (allowing no promotion for sideways)
whether the piece has any immediate neighbours or not
whether the piece is making a capture or not

These options can be divided into two groups, where the game is completely defined by presenting:
a) merely the board and the next player
b) the board and the next player and some extra information (like the turn number as in ProDem)

Notice that in this classification, FIDE Chess belongs to group b), as it may be necessary to state that a King is moved (for castling) or if some pawn was moved (to make en passant capturing). We both feel that group a) games are more elegant, (but that does not mean dropping the others!! :)

With that last point as motivation, Bill invented the next game:

------------------
MOVE UP, TAKE DOWN
~~~~~~~~~~~~~~~~~~
1. All men move as in chess, except there is no castling or en passant.

2. Movement is compulsory, and once his king is captured a player loses.

3. Once a move is made, that piece immediately changes into the next one up this cycle -  P to N to B to R to Q to P; except if it was a capture,
then the order is the opposite.

4. A pawn on the 1st rank may move 1 or 2 spaces if not capturing.
  A pawn on the 8th rank may never move again, but can be captured.

5. If there are no pawns on the board before a move is to be made,
  the order changes to N B R Q N, (or its reverse for captures).
------------------

We would like you to specially notice rule 5. Its motivation was to ensure that a board with no pawns would no longer require knowledge of its *orientation*, similar to the "no-external-info" mentioned above. However, it has resulted in a new idea:- that when a certain condition is true (no pawns), the game dynamics changes. This is not a common feature in Chess Variants, but may be an excellent concept to extend.

And here goes a sample game

1. g4(N)   d5(N)  
2. e3(N)   f6(N)  
3. Ne:N(P) Q:d5(R)
4. b4(N)   b6(N)  
5. c4(N)   a6(N)  
6. N:f6(P) e:f6(Q)
7. N:b6(P) c:b6(Q)
8. Be2(R)+ Be7(R) 
9. N:d5(P) Q:a1(R)
10 Nc3(B)  Nf6(B) 
11 B:a1(N) B:a1(N) 
12 R:e7(B) K:e7    
13 Ba3(R)  Bd7(R)  
14 Nf3(B)  Rd6(Q)  
15 Rd3(Q)  Nc5(B)  
16 Be4(R)+ Kd8     
17 Rh4(Q)+ Kc7     
18 Q:a1(R) B:f2(N) 
19 Q:f2(R) Q:f2(R) 
20 K:f2    Ra7(Q)+ 
21 Ke2     Re8(Q)+ 
22 Kd1     Na6(B)  
23 Rf1(Q)  B:d3(N) 
24 Q:d3(R) g5(N)   
25 h4(N)   Ne4(B)  
26 Rc3(Q)+ Kd8     
27 Nf3(B)  B:d5(N) 
28 B:d5(N) Q:d5(R)
29 a3(N)   h6(N)
30 Nc4(B)  Q:a1(R)+
31 Q:a1(R) Ng4(B)+
32 Kc2     R:d2(B)
33 K:d2    Qe4(N)+
34 Kd3     Nd6(B)?
35 Bg8(R)+ Bf8(R)
36 R:g4(B) Rf6(Q)
37 Ra5(Q)+ 1-0

Final Position: 

. . . k . . . .
. . . . . . . .
. . . . . q . .
Q . . . . . . .
. . . . . . B .
. . . K . . . .
. . . . . . . .
. . . . . . . .

Mar 22, 2007

SloPro LISBON

Winner is who first makes a connected group touching three non-adjacent
sides; or one touching two opposite sides; or closed circuit surrounding at least one space or opponent stone. 122234445666... moving with different-group restriction. Corners belong to both sides.

Sample Game:

OOO starts 1. p7 2. m6 m8 3. n7 n9 4. l9 q8 5. p5 o8 n11 6. o6 r7 m10 q10 7. j7 p3 k12 o10 8. m4 j5 q6 s10 9. g4 g6 k7 h10 q12 10. h3 k4 f5 j11 t11 t13 11. h5 j3 k6 m2 n5 r9 12. f2 k2 o2 v13 l5 t9 13. u8 i4 n3 v11 p9 s12 k10 14. h1 l3 h7 l11 v7 u12 v15 x9 15. n1 r5 u4 k14 u6 q14 s8 y8 16. j1 s2 i8 t15 z7 p11 z9 m12 17. c6 i6 w6 w10 A8 s14 A10 w14 A6 18. e4 t5 f7 t7 B7 a8 d9 B9 g10 o12 19. b9 j9 o4 t3 w8 y6 x11 e10 h11 u14 20. s4 e6 x7 c8 i10 y10 d11 g12 y12 j13 21. C8 d7 f9 r3 y4 e12 i12 i14 n13 n15 x13 22. f3 x3 v5 z5 c10 z11 f13 l13 o14 h15 l15 r15 23. e8 v9 h13 p15 & resign Final position: abcdefghijklmnopqrstuvwxyzABC x x . o . . . . 1 x . x o x . x . . 2 x x o x o o o o . x 3 x o o x x o . x o . o 4 . x o x x o o o x x . x 5 o x o o o x x x . o o o o 6 . o x x o o o o x x x x x x 7 x x o . x : x o x o o o o o o 8 o x o o o x o o o x o x x x 9 x o x x o x o x x . o x o 10 x . o x x o x . x o o x 11 o x o o x x o o x - x 12 x o x x o . . x x o 13 . o o - x o o o o 14 x . x oo x x x 15 abcdefghijklmnopqrstuvwxyzABC

XXX wins at k8 and then either f11/m14 next turn

Mar 20, 2007

DISPATCH GAMES

Dispatch is a game, or set of games, where starting with a stone on board, the player orthogonally dispatches a seed to another empty cell and then grows a pattern (usually a mino, like a tetrominoe or a pentominoe). Check the rules here.

Here are two games:

TETROMINOE DISPATCH XXX OOO 1. c3-h1234 c9-j8h8910 2. h4-h567,g78 j3-j4567 3. g8-g9,10,11,f11 c9-b234c4 4. h5-d4bcd5 c9-de9e1011 5. g8-cdef8 c9-b789a7 6. c8-abc6c7 j3-j12kl1 7. c3-abc1c2 j8-k891011 8. a1-a2345 resigns a b c d e f g h j k l 1 x x x . . . . x o o o 2 x o x . . . . x o . . 3 x o x . . . . x o . . 4 x o o x . . . x o . . 5 x x x x . . . x o . . 6 x x x . . . . x o . . 7 o o x . . . x x o . . 8 . o x x x x x o o o . 9 . o o o o . x o x o . 10 . . . . o . x o . o . 11 . . . . o x x . . o . a b c d e f g h j k l PENTOMINOE DISPATCH XXX OOO 1. c3-l12345 m3-m456kl6 2. l5-jk5j6jk7 c13-l13,lmno12 3. c3-abcde12 l12-cdef11f12 4. k7-lmn7n56 c11-bd3bcd4 5. c3-cde2e34 f11-e5f2345 6. e2-efg1g23 f5-f6789,f10 7. b12-b7891011 f4-g4,h1234 8. c2-a2345 b4-a678b56 9. l1:m1mn2n34 c11-c10,9876 10. k7-k11121314l1 f10-jklm10m9 11. k11-hj11h8910 f12-gh12h131415 12. e12-efg13g1415 f6-gh5gh6h7 13. h9-jklmn9 o12-no10o98p8 14. h8-jklmn8 l12-l1415m14n1314 15. n4-o4567p7 c13-d13def14f15 16. b12-b1314c1415d15 h4-j4k1234 draw 30-30 (!) a b c d e f g h j k l m n o p 1 . . . . x x x o . o x x . . . 2 x x x x x o x o . o x x x . . 3 x o x o x o x o . o x . x . . 4 x o o o x o o o o o x . x x . 5 x o . . o o o o x x x . x x . 6 o o o . . o o o x . . . x x . 7 o x o . . o . o x x x x x x x 8 o x o . . o . x x x x x x o o 9 . x o . . o . x x x x x x o . 10 . x o . . o . x o o o o o o . 11 . x o o o o . x x x x o . . . 12 x x x x x o o o . x o o o o . 13 . x o o x x x o . x o . o . . 14 . x x o o o x o . x o o o . . 15 . . x x . o x o . . o . . . . a b c d e f g h j k l m n o p

Sacrifice Mutator

Sacrifice Reversi (by Patrick Duff) is a Reversi variant with an extra rule: Instead of making a regular Reversi move, a player can choose to flip one of his own stones on board. There's also a KO rule to avoid repetitions.

This idea is a game mutator, it can be extended to modify many other games. Reversi Draughts or Reversi Chess could make a difference in some positions. Other games, like Moku or Hex would not produce interesting variants, since there's no position where an enemy piece is better than your own stones. I'm not sure about Reversi Go. Could it be possible to make a position where an enemy stone is better than a friendly one?

Meta-Game

Nick Bentley sent me an idea for a meta-game with an automatic balacing mechanism, which is called Mind Ninja:

Take any boad which begins empty. The game proceeds in 5 steps

Player 1 decides three things, which he must convey to player 2:

what the pattern will be;
whether the builder or blocker will receive free moves in step 3;
how many free moves that player will receive.

Then, player 2 decides which player is the builder, and which is the blocker.
Either the builder or blocker takes free moves as specified in step 1.
Starting with the builder, the players alternate moves.
The game ends either when the board is completely full or the pattern has been built. If the pattern has been built, the builder wins. Otherwise, the blocker wins.

Mar 16, 2007

More hex-mokus

M u l t i m o k u Go Moku on a 3.4.6.4. tiling There are 6 rows through each hexagon, 4 through each square and 3 through each triangle. The ratio of Hexagons:Squares:Triangles = 1:3:2, so the average number of rows through a cell is 4 as in standard Go Moku. (The 4.6.12 tiling gives the identical game as the 3.4.6.4. tiling. Each dodecagon has 6 rows, hexagons have 3 rows, squares have 4. The topology of the game is the same.) In this ascii representation, 'H' is the center of a hexagon, 'S' is the center of a square and 'T' is the center of a triangle: . . . . . . . . . . . . . . . . . H T . . . . . S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of rows --> Each hexagon 6 rows through it: . 1 . . . . . 6 1 2 . . 5 6 2 3 . 5 6 1 2 3 . . 5 3 . 4 4 4 H 4 4 4 . 3 5 . . 3 2 1 6 5 . 3 2 6 5 . . 2 1 6 . . . . . 1 . Each square has 4 rows through it: . . 2 1 . . . . 2 . 3 . 4 1 3 . . 4 2 3 . . . . S . . . 3 2 4 . . . 3 1 4 . . 2 . . . 4 . . 1 . 4 2 . . . . . . 1 . . . . Each triangle has 3 rows through it: . . 2 . . . . 1 2 . . . . 1 . . 2 2 T 2 2 2 2 . 2 . . . . 2 1 . . . . . . . . 2 . . 1 . . . . 1 . 2 . . . 1 . . . . . . . The 4.6.12 tiling gives the identical game as the 3.4.6.4. tiling. Each dodecagon has 6 rows, hexagons have 3 rows, squares have 4. The topology of the game is the same. Proposed board for Multi-Moku Three move equalization No 3-3 or other placement restrictions abc d efg h ijk l mno p qrs t uvw x yzA B CDE 1 . . . 2 . . 1)l13 p15 3 . . . . . . . 2)t17 swap or play 4 . . . . 5 . . . . . . . . . . . 6 . . . . . . 7 . . . . . . . . . . . . . . . 8 . . . . . . . . 9 . . . . . . . . . . . . . . . 10 . . . . . . . . 11 . . . . . . . . . . . . . . . 12 . . . . . . . . 13 . . . . . X . . . . . . . . . 14 . . . . . . . . 15 . . . . . . . O . . . . . . . 16 . . . . . . . . 17 . . . . . . . . . X . . . . . 18 . . . . . . . . 11 . . . . . . . . . . . . . . . 20 . . . . . . . . 21 . . . . . . . . . . . . . . . 22 . . . . . . . . 23 . . . . . . . . . . . . . . . 24 . . . . . . 25 . . . . . . . . . . . 26 . . . . 27 . . . . . . . 28 . . 29 . . .

Hex-mokus

Moku games (ie, achieve n in-a-row pattern to win) are usually played on square boards with four directions (horizontal, vertical and two diagonals). When translating to a hex board there are one problem, hex tillings only have three directions, which is too few, so the direct translation of hex-moku is a very drawish game, since there are no space to create winning positions with multiple threats. One way to prevent this is to use six directions to win, extending the types of lines inside the board. Here are some possible examples with sample games, including a standard three directions (which ended in a not so surprinsing draw):

Triangular Go Moku on the vertices of a hexagonal grid. Winning rows in six directions. AB CD EF GH IJ KL MN OP QR ST UV WX YZ ab 1 . . 2 . . . . 3 . . . . . . 4 . . . . . . . . 5 . . . . . . . . . O 6 . . . . . . . . . . . . 7 . . . . . . . . . . X . . . 8 . . . . . . . O . . X . . . 9 . . . . . . . . X O O . . . 10 . . . . . . O . X X . . . . 11 . . . . . . O X X X X O . . 12 . . . . X . X X O . . X . . 13 . . . . . O X O O . . . O . 14 . . . . X O O O . . . . . . 15 . . . . . O . O . . . . . . 16 . . . . X . . . O . . . . . 17 . . . . . . . . X X . . . . 18 . . . . . . . . . . . . . . 19 . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . 21 . . . . . . . . . . . . . . 22 . . . . . . . . . . . . 23 . . . . . . . . . . 24 . . . . . . . . 25 . . . . . . 26 . . . . 27 . . "O" won with a row at M10, N11, O13, P14, Q16 HEX Go Moku I (winning rows in 3 directions) Sample Game: XXX OOO 1. i9 j10 2. i10 j11 3. j9 k9 4. i8 i7 5. i11 i12 6. h7 k10 7. g6 f5 8. h8 j12 9. h9 j13 10. j14 h6 11. j8 g8 12. k8 l8 (f) 13. f9 g9 (f) 14. h11 h10(f) 15. h12 i13 16. k13 i5 17. g5 g3 18. j6 i3 19. i4 f6 20. f4 k5 21. j5 j7 (forced) 22. j4 n10 Draw . . . . . . . . . 1 . . . . . . . . . . 2 . . . . . . O . O . . 3 . . . . . X . . X X . . 4 . . . . . O X . O X O . . 5 . . . . . O X O . X . . . . 6 . . . . . . . X O O . . . . . 7 . . . . . . O X X X X O . . . . 8 . . . . . X O X X X O . . . . . . 9 . . . . . . O X O O . . O . . . 10 a . . . . . X X O . . . . . . . 11 b / . . . . X O O . . . . . . . 12 c . . . . O O X . . . . . . 13 d / . . . . X . . . . . . . 14 e . . . . . . . . . . . 15 f . . . . . . . . . . 16 g . . . . . . . . . 17 h / / / i j k l m n o p q HEX Go Moku II with 3 move equalization (winning rows in 6 directions) Sample Game: XXXX OOOO 1 g11 m9 2 s7 m7 3 m11 j8 4 p10 p6 5 g9 n8 6 l10 o7 7 q5 (f) k7 wins in 3 8 i7 or q7 l8 (double 3) abcdefghijklmnopqrstuvwxy - 1 - - - - 2 - - - - - - - 3 - - - - - - - - - - 4 - - - - - - - - X - - - - 5 - - - - - - - O - - - - 6 - - - - -(O)O O - X - - - 7 - - - - O - O - - - - - 8 - - - X - - O - - - - - - 9 - - - - - X - X - - - - 10 - - - X - - X - - - - - - 11 - - - - - - - - - - - - 12 - - - - - - - - - - - - - 13 - - - - - - - - - - 14 - - - - - - - 15 - - - - 16 - 17 I see now that 5) ... n8 wins in 4 always ending the same double three at l8. Since 5) g9 (or s5) was forced, the 4) ... p6 wins in 5. The first two 'X' moves I suggested didn't even slow 'O' down.

More information at http://www.di.fc.ul.pt/~jpn/gv/hexgomoku.htm.

SCALA

Scala is a little-known abstract game published in 1986 by Skill Games.
It has features reminiscent of Halma, Camelot, and Lines of Action.

RULES. The game is played with the following setup:

14           [o] 13          . . . 12       . . . . . 11      . . . . . . . 10    . . . X X X . . . 09  . . X X X X X X X . . 08  . . X X       X X . . 07  . . O O       O O . . 06  . . O O O O O O O . . 05    . . . O O O . . . 04      . . . . . . . 03        . . . . . 02          . . . 01           [x]    a b c d e f g h i j k

* GROUP - A set of connected (orthogonally or diagonally) stones.
* TURN - On each turn, each player moves or jumps one stone.
  + MOVE - A stone may move to any adjacent (orthogonal
           and diagonal) empty cell.
  + JUMP - A stone may also jump over any stone (friend or foe)
           landing on the opposite empty cell (it must be empty).
           A player may make on the same turn, multiple jumps with
           the same stone, and may change direction after each jump.
  + It is not valid to move or jump to its own first cell.
* CONNECTION - After each move or jump, any stones not connected
              (orthogonally or diagonally) to another stone of the
               group is captured.
* CAPTURE - If, after a move or jump, the group is divided, the
            larger of the remaining groups containing pieces of both
            colors survives.
           The smaller group, or the group containing pieces of a
            single color are removed from the board.
           There is only one connected group on the board, after
           each move.
           Some other restrictions:  It's not valid
           to produce two groups with the same number of stones,
           if both groups have stones of both colors.
           It's not valid to separate the two colors completely.
* GOAL - Wins the player who advances one stone into the opponent's
        first cell (the cell marked in the first diagram with a
         color dot).

More information at http://www.di.fc.ul.pt/~jpn/gv/scala.htm

Game Sample       Final Position 14          [O]             1. f5-h5-j7       d9-b7-d5-f5 13         . O .            2. d7-d9-f11      f9-d9-d7-d5 12       . O X . .          3. d6-b8-d10      f10-h10-j8-j6 11     . . . O . . .        4. h6-j8-h10-f10  d8-b6-d6-f4 10   . . . X O . . . .      5. f6-d6-b6-d8    c9-c8 09 . . . . X . O X . . .    6. c7-c9-e11      g10-h10 08 . . . .       O O . O    7. f10-g10        h10-f10-f12 07 . . . .       . O O .    8. d8-f10(:bc8)   g9-h10 06 . . . . . . . O X . .    9. g5-h6          h10-j8 05   . . . O . X X . .     10. i7-g9          i9-i7-g5 04     . . . X . . .       11. g6-i8          f5-h5 03       . . X X .         12. h6-i7          j8-h6 02         . X .           13. e6-e4-g4-g6    d5-f5-f3(:c6) 01          [o]            14. g10-e12        h6-h4   a b c d e f g h i j k    15. d10-d11        h4-h3                            16. i6-k8          j6-i6                            17. g6-h6          h8-j6                            18. h7-h8          j6-i5                            19. e11-e13        i5-h4                            20. d11-f13        h4-h2                            21. e13-f14        resigns, 1-0

SCORING HEX-MOKU

Played on a 8 sided hex board, each player gets 1 point for each 4 in-a-row made. The player that gets 7 points or a 5 in-a-row wins the game. Initially, one player drops 3 stones (2 blacks and 1 white) and the adversary decides color (black starts the game).

Sample Game: XXX OOO 1) k2 o8 2) m2 q8 3) o2 q2 4) i2+ g 5) k5 s8 6) m8 u8+ 7) w8 k4 8) m4 n3 9) n5 l3 10) j5 p5 11) h5 f5 12) k6 l7 13) j7 i8 14) i6 g4 15) k8 l9 16) g6 e6 17) i4 j3 (forced) 18) f7 e8 (forced) 19) m6 o6 (forced) 20) h3 & wins 7-1 Final Position: abcdefghijklmnopqrstuvwxyzABC 1 . . . . . . . . 2 O X X X X O . . . 3 . x o O O . . . . . 4 . O x O X . . . . . . 5 . O X X X X O . . . . . 6 . O X X X x o . . . . . . 7 . . x . X O . . . . . . . . 8 . . o . O X X O O O O X . . . 9 . . . . . O . . . . . . . . 10 . . . . . . . . . . . . . 11 . . . . . . . . . . . . 12 . . . . . . . . . . . 13 . . . . . . . . . . 14 . . . . . . . . . 15 . . . . . . . .

Projective Hex

[Bill Taylor: This article is chiefly for rec.games.abstract; but I cross-post to sci.math for the possible interest in tilings of Projective Planes]

One of the great blessings of connection games like Hex and Bridgit is, that victory is certain for one or other side, AND the structure of the game ensures that a victory for one is *automatically* a defeat for the other, with no special rule needed to say so.  So there is no element of a mere "race" to do something first, where both players might achieve this goal almost simultaneously.

Although there is no "social" defect in such races, (e.g. even chess can be so viewed - a race to capture the opponent's king before he captures yours), it is mathematically and game-theoretically slightly unaesthetic, compared to the Hexlike feature of   [win = not(loss)]   by structure.

Hex and Bridgit both suffer from another slight unaestheticity though, to wit, that the two players have (slightly) different tasks; one must make a North/South connection, and the other an East/West one.   Indeed, in Bridgit they even play on different points!  Again, this is no barrier to playing the game or to its being a jolly good game, but again it seems a very slight aesthetic defect.

One game that achieves both goals, i.e. (1) complementary winning conditions and (2) identical tasks; is the excellent "Y" version of Hex, which really deserves to be better known.  However, I introduce yet a new variant here.

------

Some while ago, Dan Hoey and myself jointly invented a game we called PROJECTIVE HEX, invented in this newsgroup, in fact.

It was Dan who, partly inspired by "Y", first ventured onto Projective Planar boards for Hex-like games, but couldn't find a nice winning condition, surprisingly. My contribution was to observe that the condition of making a GLOBAL LOOP, (i.e. a closed path that crossed the boundary an odd number of times) was "THE ONE" - and that it stood out "like a sore thumb". Dan agreed about the sore thumb, and kicked himself for not having seen it before. Dan also constructed a program to print out beautiful Hex-like boards based on the Projective Plane, and thus having 6 pentagons amongst a variable number of hexagons.

My latest contribution has been to change the pattern of the boards slightly, to make them more homogeneous-looking (though not fully homogeneous in fact), and thereby arrange it so that games can easily be played at the keyboard, i.e. by email etc.

For the new Projective Hex, now probably the best abstract board game in the world (ha-ha!), the boards are similar to this as follows...

   A B C        As you see I've had to insert a 27th alphabet letter!   D E F G       Interior cells and interior-edge cells each have 6 H I J K L      neighbours, as in Hex; but the 6 corner cells have 5. M N O # P Q R S T U V      The side dimensions are always n and n+1.   Each edge   W X Y Z       is flipped end-to-end and laid alongside its opposite.

In this 3-&-4-sided board, there are 15 edge cells which thus connect to their opposite cells via Projective connections as shown here...

   z_y_x_w   z/A B C\w     Each of the original edge/corner cells "re-appears" v/D E F G\r    on the opposite side, in lower case letters. q/H I J K L\m q|M N O # P Q|m Each corner still has 5 neighbors. l\R S T U V/h g\W X Y Z/d   c~c~b~a~a

So on the original board, cell  H  is connected to D I N M Q V (in order). Whereas  M  is connected only to  N R L Q H.

The whole collection of 21 hexagons and 6 pentagons makes up a "standard" tiling of the Projective Plane.

To play the game, "Projective Hex", one merely plays as at Hex, filling any one cell your own colour on your turn; and whoever makes a global loop of adjacent cells of their own colour, is the winner.  And, as mentioned above, it is only possible for ONE colour to do so, and at least one of them must always do so, by the time all cells have been coloured. So complementary winning conditions, and equal tasking have both been achieved.

Example: here is a completed game, with both having played 7 moves, and the 2nd player (white) has won, despite his opening disadvantage.

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

The loop might be more visible if "ghost" edge cells are entered as well...

    ___o_x    /. X O\x   /. . X O\x /. . O O .\ <. X O X . .> \X O X . ./ o\X O . ./   o~o~x~~~

For actually playing the game, naturally, as always, the first player has an enormous advantage; a sure win, in fact, by the usual strategy stealing argument.  But beyond the very smallest boards it is very hard to find.

This advantage can be left as is, giving the weaker player first move; or (say for more formal games), one of the usual equalizing methods can be used.  Probably the simplest is the "cut-and-choose" method of 3-move equalization (mentioned on another thread recently), whereby one player plays 3 opening moves, black-white-black, then the other player chooses which colour to be.  It is also conceivable that even 2-move or 1-move equalization would be suitable, as e.g. the corner cells are not quite so valuable as the central ones, so an opening move there might well be a losing one, but only just, making 1-move
equalization a viable option.

Trabsact Sagme Diaries