Games and their times

All classifications are arbitrary, but some are more useful than others (to paraphrase George Box).

When thinking about the origins of abstract games, I visualize three broad categories:

• Ancient/Traditional Games. These are games within the culture of a particular people or region. Their exact origins are often unknown; we only estimate their age by their first written references. These games usually lack a fixed rule set (if any), having several regional variations. It is often more accurate to think of them as families of related variants rather than single, fixed games.
• Old Games. These appeared after the commercialization of play. They were either published or patented by their creators. While variants of these games may exist, they generally arise due to the game's popularity and how players engage with their core ludemes.
• Modern Games. Most abstract games published online today fall into this category. Although their creators retain authorship, many have relinquished commercial rights, effectively placing these games in the public commons. Much like open-source software, they embody the spirit of freedom and collaboration that characterized the early internet. 
  

These categories are, of course, not rigid. There are still modern commercial abstract games (e.g., the GIPF project) and older games that were always intended to be public (e.g., Laska). However, I find these distinctions useful for understanding games in their historical context.

Even though the boundaries between these categories are not perfectly defined, it is possible to identify their temporal limits.

The era of old games began around the mid-19th century, when commercialization led to the emergence of the first game publishers. This period also saw the introduction of the patent system, which allowed some types of game rules and physical board designs to be removed from the public commons and be privately owned.

In my opinion, the era of modern games began with the rise of the video games industry, a transformation that took place between 1980 and 1990. This shift is particularly evident in board game magazines such as GAMES and Jeux et Stratégie, as well as in the game's sections of popular science magazines like Science et Vie. During this period, columns devoted to computer games gradually occupied more and more space. One consequence of this shift was the decline — and, in some cases, the disappearance — of these magazines in their original formats, as they struggled to compete with publications specialized in video games and software. Another outcome, in my view, was a decrease in the commercial value of abstract games. This reduced the pressures of commercialization and, coupled with the advent of the internet, opened the era of modern abstract games that continue to this day.

Most abstract game designers align with the era in which they live. Designers from the old era typically created commercial games — for example, Alex Randolph, Sid Sackson, Eric Solomon, and Robert Abbott. Meanwhile, modern designers tend to create games for the public domain, with early examples including Christian Freeling, Wayne Schmittberger, Fred Horn, and Ralph Betza.

However, there are exceptions. Some designers from the old era (approximately 1850–1980) adopted a modern, non-commercial approach to game design. Notable examples include V.R. Parton, Joseph Boyer, Martin Gardner, Sid Sackson (again!), and George Dekle. Conversely, some modern designers still produce commercial abstract games. Perhaps the most famous are Reiner Knizia and Kris Burm.

The commercialization of abstract games appears to be on the decline, even as the global board game market continues to thrive, with its popularity steadily increasing over the past twenty, thirty years. But every category remains alive today. Interestingly, traditional games still appear to be the strongest link. Cultural traditions are very resilient to change, with few dependencies that could threaten their continuity. It seems unlikely that Chess, Go, or Backgammon will lose their public in the medium, long term. In a way, this reflects the Lindy effect at work, with the longevity of traditional games correlating with their cultural significance and resiliency.

In contrast, commercial abstract games appear to be losing relevance, especially when compared to highly competitive industries like video games and social media. And modern games are deeply dependent on the free flow of communication — a privilege we must not take for granted in the decades to come.

Some thoughts about identical development

Some thoughts about identical development
Fred Horn (c) Sep 3, 2012

BUFFALO is a well known game invented by Alex Randolph. 

In his article in The Games Journal Greg Aleknevicus states the game is a further development of a group of games normally referred to as “Fox and Hounds” and in Dutch as ‘Kat en Muis’ (Cat and Mouse).

In fact, it is from another source, because there is also an element of capture involved. It has more to do with ASALTO and all its variants, where two unequal opposite parties try to accomplish different goals.
One wants to capture all of his opponents pieces, the other wants to go somewhere on the Board a) to reach some goal-squares or get off the Board or b) to occupy certain squares.

The extra that was introduced by Alex Randolph were the two different kind of pieces at the conquering side, one Indian with his four Dogs, against a herd of Buffalos. But was this really new?

A long time ago, somewhere in the mid-seventies of the last Century, I bought some printed games from before 1900, which are in Holland known as “Spelplaten” or ‘Platte Spellen’ (“Gameplates” or ‘Flat Games’). One of them was the game De Schapendief, a rather abstract looking game, unlike most others from that period which are mostly Goose-like games. When you examine what is in the Rules at a first look you really cannot believe your eyes. Has Alex Randolph plagiarized an old game?

Of course not! I cannot believe that Alex should have done such a thing (use a game-mechanism) without reference to the original. And this original is only known in Holland, was never ever published after its first printing and the rules are written only in Dutch. So here is a true example of that much talked about phenomenon of separately inventing the same thing. And this has happened with nearly a Century in between!

The Sheepthief

This game is played with two; one is the shepherd, the other the thief.

The shepherd places his 16 sheep on the squares at the bottom-side of the game, next to each other, leaving the middle column open. 

Here he places in his first row the shepherd. The other player places the thief in the middle of the upper row of squares; and places his dogs to his wish anywhere, one in each row on the three upper rows.

Then players move alternately. The shepherd moves first. He may only move one sheep one square forward in a turn. The thief can only move a dog sideways in his own row two squares at the time. 

As soon as the thief can move (slide) a dog to a square occupied by a sheep, this sheep is captured.
Thief and shepherd can move one square in all directions in their turn.

The thief will pay the shepherd one counter for each sheep that reached the grasslands, while the shepherd must pay one counter for each sheep at the end of the game to buy it back.

If the shepherd himself is caught, he must pay five counters ransom. 

§

Greg Aleknevicus' The Games Journal article

To tell you about Buffalo, I first need to tell you about Fox & Hounds (also known as Fox & Geese and at least several other names). For those who do not know this traditional game, it's quite simple: 

Place four "hounds" on the rear-most black squares of a standard 8x8 checkerboard. Place 1 "fox" on any of the front-most black squares. Movement is as in Checkers without the jumping — you may move into any diagonally adjacent square. Hounds may only move forward whereas the fox may move forwards or backwards. Note that there is no capturing. The winner is the player who leaves his opponent unable to make a move.

 

This game is actually quite clever and very useful as a bar trick, I've won a few pints with it. At first glance, it may seem that the hounds have a rather unfair advantage, all they need do is slowly advance each hound in succession and there's no way that the fox can get through. In practice, this proves rather hard to accomplish and it's quite easy for a player to casually move his fox, disrupt the line of hounds and escape. Again, and again and again. Opinion quickly shifts and it seems that the fox is the one with the strong advantage. (Here's where the winning of pints takes place.) You then offer to switch sides at which point you capture the fox every single time. The truth of the matter is that Fox & Hounds is solved and with proper play, the Hounds will always win. The reason it remains interesting is that the correct moves are not at all obvious and figuring them out can be an interesting exercise.

Why mention all this? Well, Buffalo is obviously inspired by Fox & Hounds and I'm almost certain that it too has a solution. The trick is that I have not yet been able to ascertain if this is so. In fact, I can't even say for sure which side is favored.

 

Play is on an 11x7 board with 11 buffaloes placed along one edge (one per column). The Indian and his four dogs start one row in from the other edge. Turns alternate with each player moving one of his pieces. The buffalo player may only move his pieces one space directly forward as long as that space is unoccupied. The Indian player may move his dogs any number of spaces in a straight line (like a Queen in Chess) but may not move onto or through any other piece. The Indian himself may move one space in any direction but not onto a space with a dog. The Indian can move onto a buffalo which is then removed from the game. The buffalo player wins if he can get any one piece to the opposite side of the board and the Indian wins if he can prevent this.

You can't get much simpler than this, but it has proved to be an interesting challenge. The game has flip-flopped back and forth as players try a variety of different "openings". First one approach will be attempted, effective counter-moves will be discovered and it will seem that one side must surely be the "winning" one. Then another approach will be tried, and it will seem that the other side has the guaranteed win. At this point I haven't explored Buffalo enough to make any definitive statements but while it does seem that the buffalo player has the easier time, I have the sneaky feeling that the Indian has the guaranteed win. Even though the game will lose all interest once (if) I'm ever able to figure it out completely, it has been an enjoyable exercise working through the various approaches and responses.

Of course, you could easily just ignore all this nonsense about "solving" the game and play it as a very quick and simple contest but I'm not sure that it's all that engaging as a game rather than an interactive puzzle.

On Natural & Artificial Games

Can we use the terms natural/artificial for games, in general?

One way to look at this is to relate naturalness to simplicity. Simple games like Hex, Tic Tac Toe or, perhaps, Go, seem almost like discoveries, rather than inventions. But the fact that Hex was only "discovered" in the 1940s does give us pause to ponder. Simplicity is culturally dependent, just as are more obvious or trivial things - e.g. the positional numerical system, or the moral statement "slavery is wrong" -- which were not so in the past.

Another way to look at this is using History. Games, or game concepts, that seem to be invented independently are, at least, cognitively attractive to humans. Perhaps they are cognitively attractive to conscious beings in general, and so, they may even exist in alien cultures. Some examples of those are race games, or Tic Tac Toe/Gomoku variants. Possibly, seed games, Mancalas, also belong here. So, a natural game would be an instance of one of these archetypal game concepts. Of course, as usual, the frontier is blurred. Bao is a very complicated Mancala game. Overly baroque variants can hardly be seen as natural. Shall we include Checkers/Alquerque or Chess games? And how many add-ons can a game include and still be considered a natural one?

A third way to try to make sense of this separation between natural and artificial, is to look into the game's history.

Games like Chess, Go, Mancala and Checkers have evolved through centuries, absorbing gaming experience into their progressive adaptable rules. As in biological natural selection, these games are more like species, with their life trees,
their historical compromises, their multiple branches (cultural instead of biological). So, in this case, every game started in one or more human minds, in some raw artificial state, and was tested in a social environment. Like most species, most games must have become extinct quite quickly. But a few were able to adapt to the cultural intricacies of the memetic landscape of its inventors. And then, when society changed, games also changed, like any adaptable, flexible population of organisms. Of course, the analogy only goes so far. Unlike fossils, if they are rediscovered, games can be brought back and enjoy a new life eventually under new clothes. (The Game of Ur, and Senet, with their reconstructed rules, are famous examples).

Just as for the concept of species, we might differ our final judgement about our recent games. Perhaps Hex will have a biography (it already has thriving children, like Y). Game inventors, nowadays, have an immense set of game ideas and, like alchemists, they try to mix them, mould them into new games. Most of those, as in the past, will become extinct and never get beyond being artificial concepts. Others, because they say something to generations of players, will be carried along with our evolving society and will start to have a biography. They will, slowly and in unexpected ways, become natural concepts.

[also posted in rec.games.combinatorial]

Board Games Studies 2011

This year edition of BGS happened in Brugge, Belgium. The meeting was held at the KHBO-Spellenarchief. The game archive has several thousand board games and has a database (in Dutch) which is being slowly uploaded with game information and, hopefully, game rules. Here are some pictures from their public section:

And here is a panoramic view (as usual, click to enlarge):

Lots of thousands of games were donated by the Dutch game collector and inventor, Fred Horn, who still collaborates with the Museum. Here is a picture of him (the middle guy, talking math & games with Jorge Nuno and Carlos Santos, two Portuguese friends):

Irving Finkel presented "On to Square Two – the question of dissemination" talking about how games travel between cultures. This travel is useful for game survival but has also the problem of game contamination. Some games may even replace and destroy old games. For a game Historian the modern waves of European migration (like colonists or missionaries) to the rest of the world meant terrible news.

In the colloquium a participant (I don't recall his name...) presented a rediscovered picture from an old book, picturing Adam & Eve in the garden of Eden. Some say they are playing a board game (I saw just them picking and eating grapes or something like that...). Anyway, the picture is pretty:

Michel Boutin made a nice presentation called "The structure of games" were he made an historical perspective of how games were classified in the past.

AbGamers

When we talk about abstract board gamers there is a relevant distinction. On one hand, there are people that don't mind giving lots of time and attention to abgames and, on the other, people that try an abgame or two from time to time. The first ones are hard to find and the net, like in r.g.a, is a perfect way for these guys (well, we) to meet and discuss and play abgames. It's difficult, if not impossible, to replace the net in this.

The second ones are much more common. There are thousands and thousands that like to play abgames. I live in Portugal, a country with no social history of abgames except for Chess and Checkers. From my experience, me and some local friends were able to promote abgames events alongside with hundreds of schools, where the finals alone gathered 500+ young people (we are preparing the 4th edition). We also made a set of 10 small books, discussing historical puzzles and games, that sold thousands for each copy. And Portugal only has a population of 10 million. I guess that from all these tens of thousands, some got interested and started reading and playing a bit more than before (I know that that happened at least in some cases). This is another way to contribute to the abgames future community.

[this blog has been too quiet, just like the main website, the World of Abstract Games. In these last months I didn't had much time to process new games (but I'm still collecting them) so be patient and check or recheck the 500+ pure abstract games already posted]

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.)

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.

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.

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 . . . .
. . . . . . . .
. . . . . . . .

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] -- Dec 11, 2001 post

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.

Clues for a Conceptual Toolbox for the Game Designer

Abstract Games are very curious mathematical objects. They tend to be are deprived of any significant cultural context to present what – in principle – really matters to another curious thing called a game player. This act of cultural removal is rather subjective, but anybody can sense it, when we see several people around the world play the same game and enjoying it the same way. Many games, like Go or Chess are played on several dozens countries at this very moment. Not even conjectural prohibitions based on Religion, Law or by political reasons, manage to destroy the memory of the rules that define those persistent and rather illusionary things…

Here in, an abstract game is: (a) A game for 2 players; (b) with no luck element and no hidden information; (c) where each game turn (except perhaps the first and the last one) consists of a move by the 1st player and another by the 2nd player; and (d) the set of rules agreed by both players do not leave any space to ambiguity.

This, in a way, restricts the field of study, but leaves an endless Ocean of possibilities and magical wonders! We can say, with no hesitation, that Humanity only scratched the surface of such great Sea of Games. Herein, the goal is to present a set of conceptual tools for those who like to design and test new abstract games, to inform people how to avoid common errors and pitfalls and so, in conclusion, to create better games.

There are two concepts that I find most relevant! The first one is Depth, or strategic complexity, meaning the capability of a certain game to have a certain number of degrees of skillfulness. The exact number of different skills is never exactly known, even because no one reached an precise definition of what is a degree of skill. A possible way is to say that a player is one skill above the another if he wins 2 out of 3 games. Even saying that, skill levels of different players are not disjoint, since the concept is not transitive. Different ways of playing work better on certain players than others. Just like Soccer! The more skill levels a game has the better, it means that a person can continuously learn about the game for a long time (even more than a single life or an entire civilization). A deep game also gives the players more chances to recover from crises (i.e., bad moves), or stating differently, if a player on a relatively bad position is replaced by a better player, this one may still be able to balance the game.

The other condition is Clarity. Clarity means that most board positions should be as close as possible to the way the Human mind sees things. This concept will surely be very different for a Martian, but since no ET was found yet, let’s stick to this human-centered vision. Tic-Tac-Toe and Hex are exceptionally clear. There is not, however, a direct correlation between clarity and the description of the rule set. Some simple games to describe tend to mess a lot when an actual game is played, but a difficult game to describe will possibly have some clarity problems as well. For instance, many Chess Variant designers find on clarity a merciless judge. People are used to the way a Knight moves, but some similar fairy pieces (with the same right to existence as any other) create a problem of lack of clarity that prevents players to be interested on such opaque game.

The joint combination of this two features give a glimpse on the quality of the game. Tic Tac Toe suffers due to limited strategic possibilities, Go scores highly but is penalized by subtle rules, Hex scores very highly (I dare to say that 19x19 Hex would approach Go in depth, while retaining a much better clarity). A computational approach of this subject would say that given a game tree with N nodes, complexity would be the total nodes required to formulate sensible strategies, and clarity would be the ability to search deeply inside the game tree in order to achieve them.

Well, but this are just general concerns that you should have in mind. What about the real and objective game´s ruleset? The next section will talk about some basic game mutators. That is, modular concepts that can be applied to almost any game, creating new games (not necessarily better ones).

Basic Mutators

As we all found by experience, there are many good ideas out there hidden on obscure games. The next list is not intended to be complete, but it tries to dissect as many basic concepts as possible, in order to provoke people to mix them in some strange, new and hopefully also in skillful ways.

Let’s state some extra principles to those already stated on the initial Abstract Game definition. The game may have a board, consisting of cells linked together in some specific ways (a square tiling, an hexagonal tiling, a rhombus, …). Each player has, at least, one set of stones of a certain color (let’s say, black stones for the first player, white stones for the second), where the possible playing options are stated by the rules defining the game.

Each mutator must be seen as a rule, or part of a rule, that needs some preconditions before execution (i.e., may only works on certain game states/positions), and may create, when executed, events that enables other mutators to work (even on the same player´s turn). On the following list, some mutators have glimpses of possible preconditions and events.

Pass A player does not affect the game state. It is the null mutator.

Drop This is, probably, the most applied rule on abstract games. A piece may be dropped into a certain board cell. The drop restrictions can be various. The most common is that the cell must be empty. Other options would restrict it to a certain area (e.g., must enter into the players initial zone), on local conditions (e.g., must be near a friendly stone), or global ones (e.g., the board must not have more than x stones).

Move The move mutator is also very widely used. A stone already on the board, can move from a cell A to a cell B. This movement may be subject to certain restrictions, like intrinsic ones (e.g., it can just move to an adjacent cell, to orthogonal/diagonal cells, …), contextual ones (e.g., it can move to an empty cell, a cell not attacked by the opponent, only moves if it has x adjacent friends, …), or global ones (e.g., the total number of stones define how each stone can move).

Capture A set of stones, either friendly and/or unfriendly, are removed from the board and those cells become empty. This usually is an action caused by the execution of another mutator (most cases, this is a consequence of moving). Capturing can be a consequence of a certain pattern, like custodian capture (like Hasami Shogi), simple jumping (Checkers), cannon capturing (like in Xiang-Qi), bombing (all adjacent enemy stones are captured). Capturing may provoke several lateral effects, like Suicide (the captured piece is destroyed like in The Way of Go), or Protean capturing (the piece inherits the captured stones abilities, like in Cannibal Chess).

Jump A jump is simply using another stone to move to another cell not in range otherwise. This not include the Chess Knight, since it does not need another stone or piece to make its move.

Merge Two or more stones occupying the same cell are transformed into a different piece. Bashke, Laska and Focus use this concept in the Checkers game world.

Pivot The pivot mutator is a generalization of the Jump mutator. Usually a jump uses the intermediate stone as the pivot to move on a straight line. General pivot moves have much more liberty. Other kinds of pivot moving are scaling (check Scalus for use of that concept), and rotation (check Kefren or Twirls of Action) also known as Twirls, named by Claude Chaunier. These are just two possible ways to explore Pivot moves.

Swap A stone (the swapper) can swap position with another stone (the swapped). Possibly, the swapper will be on moving range from the swapped.

Shift Shift also means push a set of stones into a specific direction (e.g., check Epaminondas or Abalone). This shifting may produce other events, like single or group capturing.

Pile Piling inserts an extra dimension to bidimensional boards. There are several ways to pile, namely Staking (the new stone is placed on the top) and Queuing (is placed on the bottom).
This may provoke a change event, meaning that the new stone merged the piled piece. This also implies possibly a splitting mechanism.

Change This means changing the stone status. After the application of such mutator, a stone acts and reacts differently to the same conditions. Some examples include: stone promotion (increase its power) and demotion (decrease it), freezing (cannot move), stoning (cannot move or be captured), make royal/unroyal, …

Local Interactions After a move is done, the actual cell interacts it some local neighbors (the adjacent stones, the nearest orthogonal neighbors, …) and affects them. For instance, there are gravity forces (attracts all by one or more cells), and magnetic forces (attracts opposite color, repels equal ones). Some games where this is applied are Magnetic Go and Magnetic Chess.

Momentum A momentum mutator creates multi move games. It works like this: A previous moved stone will repeat its behavior on the following turns while it’s valid. Until now, from our knowledge, this was only used on Chess Variants.

Progressive This mutator affects the way turns are defined. The typical progressive mutator adds an extra movement for each player’s new move (one move for Black, two for White, three for Black, …). Other progressions are possible, softer ones (1, 2, 2, 3, 3, 4, 4, …) and wilder ones (1, 3, 5, 7, …). This obviously reduces the game length, and for some games it is a nice way to play a fast variant (give it a try with 9x9 Go). The set of movements could be sequential or simultaneous, it depends on the context where it is applied.

Save It’s a kind of active passing. The player gives the turn to the other player, but it saves the move for later use, i.e., next turn it can move twice in a row. This is a very strong mutator, and should be used with extra restrictions, in order to keep the game interest.

The produced events can activate more than one mutator. For example, a multiple move/capture is an application of a certain capture mutator within itself.

Improving the Spark

There is no magical formula for making an abstract game with depth and clarity. That implies a little of luck, insight or something else that creates the ‘spark’. I will speak of the something else, and also about the fact that the spark, if not treated right, may be lost, transformed into a poor game that lacked the basic care of any newborn.

Let’s start on the initial setup. First, the game designer should decide what shape will define the board and if it begins empty or not, if there is still the possibility to drop stones afterwards. A related point is to decide the total number of stones. A good rule is to see how stones are capable of moving (if they move at all). Board density (i.e., the average number of stones per cell) is relevant on this decision. A game with moving stones and growing density may face ‘traffic’ problems on the endgames. Usually if stone mobility is high, then density should be low, and vice versa.

If the designer chooses an initial setup, he must see if that setup does not go through another global pattern before the game really begins (i.e., both players found that to attack or defend, they should position their stones into a certain tactical pattern). In those cases, the designer should change the initial setup to that intermediate one. It will speed the initial phase, without decreasing its depth (this of course, may be risky, if the designer or the game testers are not able to see other potential good openings, on those cases, the game depth will suffer).

The number of moves of a typical game is an essential thing to note. Very short games are not very interesting, except for children, very long games take much time to be played and tend to be rather tedious. Perhaps if Go was presented today, it would suffer from this fate, many people would not be interested because it takes too long to finish a game, and they would miss its remarkable depth. Ralf Gering marks 20 turns (i.e., 20 moves for each player) has a minimal mark for a average game score, in order to have some interest. Of course, this also depends on the number of moving options, but too many options reduce clarity! This is a tight business! For maximal turns, an original game that takes more than 100–120 turns will need a good marketing!

An important subject is to avoid mirror tactics. This happens when one player can mimic the other, in order to achieve a draw (like in Halma) or even victory (like in Hip on even square boards). This can be done by using odd boards (that is with a center cell, usually called Tengen), allowing captures, or asymmetrical positions (i.e., that after a certain move, the other player cannot mimic it).

Two more things about the initial phase, Handicaps and Equalizers. Handicaps are always a good way for two players with different skills still manage to get some fun paying (ups, I mean playing) the game. This is done by creating a better position for the weaker player, by giving him some extra moves, extra material or easier winning goals.

Equalizing means to balance first (or even second) player’s advantage. This can be done, using an Handicap system; or by using the N-move equalizer: After N moves, the player on disadvantage may choose which side to play. When N is 2, this rule is also known has the PIE rule (i.e., you cut, I choose). There are other ways, like giving two moves per player, except for the first game move, but these are less general and may not work everywhere.

Besides the beginning, there is also the end! How the game should stop? What will be the winning goal? There are several classical concepts:

1. Territorial – wins the player with more controlled cells
2. Pattern – wins the player that first achieves a certain pattern of stones or cells: n-in-a-row, n-in-a-group, n-enclosed
3. Connecting – link two or more edges, link two or more special cells, link all friendly stones
4. Capturing – capture x enemy stones, capture some key stones (i.e., royal stones)
5. Reaching – reach a set of key cells, surround a certain stone or cell area

The designer should take special attention on one thing. On a typical endgame, the winning player has enough power to win? Is he able to decide the final outcome of the game? In Chess, we know that King + Bishop vs. King is a draw. If almost all Chess games would end on this position, then another winning rule would be needed (e.g., the Bare King rule – a player looses all other pieces are captured).

After the rule set is defined, the designer should look into each single rule and ask some questions: Is this rule necessary? Why is it so? Is it a logical consequence of some other rule(s)? If so, it should be placed on the notes section, not among the rules! Does the rule interacts with the other rules to create some more tactical possibilities? Or is it totally independent? If so, and if the rule decreases clarity without giving some to the game as an whole, then the designer should rethink about keeping the rule.

Combinatorial Game Theory talks about game temperature. A hot game state is one where the player has the advantage to move. Otherwise a cold game is one where the player does not want to move (in that sense, a game with a pass rule is never cold, since players may pass their turns). Some samples: Hex is hot and gets hotter. An extra move never hurts the player and usually puts them in a winning position, more so towards the end of the game. Go starts medium hot then cools down to lukewarm. Towards the end of the game moves become less effective until they are not worth making. Gonnect starts hot then suddenly turns freezing cold at the end. An extra move is good during the early and middle games, but can become a game-loser in the endgame. This does not give the designer a way to determine a level of quality, but can give him insights about how his own game reacts from the opening until the endgame.

Final Words

A really nice thought is to imagine that maybe some games invented in this century will be played in the year 3002 (perhaps one of your own games), where Hollywood, Microsoft, Intel, the Computer Game Industry, and so many other powerful businesses would already entered into oblivion…

A new way to play computer-Go?

Computers have started to outperform humans in games they used to lose [full text here]

[...] Deep Blue and its successors beat Mr Kasparov using the “brute force” technique. Rather than search for the best move in a given position, as humans do, the computer considers all white's moves—even bad ones—and all black's possible replies, and all white's replies to those replies, and so on for, say, a dozen turns. The resulting map of possible moves has millions of branches. The computer combs through the possible outcomes and plays the one move that would give its opponent the fewest chances of winning.

Unfortunately, brute force will not work in Go. First, the game has many more possible positions than chess does. Second, the number of possible moves from a typical position in Go is about 200, compared with about a dozen in chess. Finally, evaluating a Go position is fiendishly difficult. The fastest programs can assess just 50 positions a second, compared with 500,000 in chess. Clearly, some sort of finesse is required.

In the past two decades researchers have explored several alternative strategies, from neural networks to general rules based on advice from expert players, with indifferent results. Now, however, programmers are making impressive gains with a technique known as the Monte Carlo method. This form of statistical sampling is hardly new: it was originally developed in the Manhattan project to build the first nuclear bombs in the 1940s. But it is proving effective. Given a position, a program using a Monte Carlo algorithm contemplates every move and plays a large number of random games to see what happens. If it wins in 80% of those games, the move is probably good. Otherwise, it keeps looking.

This may sound like a lot of effort but generating random games is the sort of thing computers excel at. In fact, Monte Carlo techniques are much faster than brute force. Moreover, two Hungarian computer scientists have recently added an elegant twist that allows the algorithm to focus on the most promising moves without sacrificing speed.

The result is a new generation of fast programs that play particularly well on small versions of the Go board. In the past few months Monte Carlo-based programs have dominated computer tournaments on nine- and 13-line grids. MoGo, a program developed by researchers from the University of Paris, has even beaten a couple of strong human players on the smaller of these boards—unthinkable a year ago. It is ranked 2,323rd in the world and in Europe's top 300. Although MoGo is still some way from competing on the full-size Go grid, humanity may ultimately have to accept defeat on yet another front.

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