FLAT-EARTH DOZER-HEX

Like Hex, but lines of friendly stones may push, abalone-wise, smaller lines of adversary stones by exactly their length difference.

If any stones block the full length of a push, the push stops there. A line may push its own length into empty space. Stones pushed off the edge disappear from the game. No friendly stones may be pushed.  Swap option exists.

Sample Game
  
    OO    XX
 1. p4    p6    
 2. j6    l6    
 3. r6    n6    
 4. q5    o7    
 5. r4    n8    
 6. m9    n8s3  
 7. o5    m7    
 8. m5    s5    
 9. q3    m8    
10. n4    q7    
11. k5    u5
12. q3t6  o7s3  
13. p4v4   u7
14. s7     u5
15. q3     v8
16. p4     k9
17. n4     j10
18. s1     w7
19. n4v4:  s9
20. r8     t6v4:
21. q9     l6v6  
22. Resigns

Final Position:

   abcdefghijklmnopqrstuvwxyzAB
 ___________OO_________  
 \ . . . . . . . . . o \            1
  \ . . . . . . . . o o \           2
   \ . . . . . . . o x . \ X        3
    \ . . . . . . o o o x \ X       4
     \ . . . o o o x o x . \        5
     X\ . . o , , x x x o . \       6
      X\ . . . x . x o x x . \      7
        \ . . x . . o . x . . \     8
         \ . x o . o x . . . . \    9  
          \ x . . . . . . . . . \  10
           ----------------------  
                    OO

This game presents an Abalano-like mutator which seems to provide new games. Notice however that this mutator does not use the Abalone rule, since phalanxes can shift more than one cell (based on the difference between the pushed and pushing phalanx).

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.

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

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?

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…

SLOW PROGRESSIVE GO 9x9

This game uses a progressive mutator in order to increase the dynamics of the original game. However, unrestricted Go drops would kill the game so two restrictions are added: (a) each stone on a drop sequence cannot belong to the same group (b) an atari stops the sequence.

Here's an example at a 9x9 board (white won 41-40). Herein it is used the slow progressive (1222344456667...)

a b c d e f g h i
. . . o x . . . . 1: c7
. . . o x . . . . 2: g3 g6
. . o o x . x x . 3: c3 e8
. . o x . . . . . 4: e2 g8
. . . o x . . . . 5: d2 d5 e7
. . . . o x x . . 6: d4 e3 e5 f7
. . o . o x . . . 7: d1 c4 e6 f9
. . . . o o x . . 8: e1 f6 g9 h3
. . . . . o x . . 9: d3 f8
a b c d e f g h i

Lovely 4 HexGomoku

Gomoku rules on hex board but each player must also drop an enemy stone as
close as possible to his own dropped stone. Wins by making a 4 in-a-row.

Game sample:

  ___oooo________xxxx___
1.  l25 n25     m24 o24
2.  o26 p27     k22 j21
3.  l23 k24     i24 j25
4.  g24 e24     h25 i26
5.  j23 h23     g22 i22
6.  f21 e20     f23 d23
7.  resigns

abcdefghijklmnopqrstuvwxyz

 . . . . . . . . . . . . . 19
. . x . . . . . . . . . .  20
 . . o . o . . . . . . . . 21
. . . x o x . . . . . . .  22
 . o x x o o . . . . . . . 23
. . x o x x x o . . . . .  24
 . . . x o o x . . . . . . 25
. . . . o . . o . . . . .  26
 . . . . . . . x . . . . . 27
. . . . . . . . . . . . .  28
 . . . . . . . . . . . . . 29

abcdefghijklmnopqrstuvwxyz

This variant, played on a hex board, has less directions to win (6 instead of the 8 on a square board) and that provides a better game, since on the square version it's much easier to win.

On the other hand, playing 5 in-a-row with the Love mutator seems very hard to force a win. The game tends to prolong itself without a final outcome.

134* TRICOLEUR HEXXAGON

Rules like Ataxx in a hex board (like Hexxagon, except:

There are three colors: A,B and C. A flips B which flips C which flips A (flips means the enemy piece becomes just like the moved piece, just like Ataxx). Also, a piece captures enemy pieces of the same type (ie, removes them from board).

Initial setup:

abcdefghijklmnopqrstuvwxyzABCDEFG
        A . . . b . . . c           1
       . . . . . . . . . .          2  
      . . . . . . . . . . .         3
     . . . . . . . . . . . .        4
    B . . . C . . . a . . . A       5
   . . . . . . . . . . . . . .      6
  . . . . . . . . . . . . . . .     7
 . . . . . . . ___ . . . . . . .    8
c . . . b . . /   \ . . B . . . C   9
 . . . . . . .\___/. . . . . . .   10
  . . . . . . . . . . . . . . .    11
   . . . . . . . . . . . . . .     12
    a . . . A . . . c . . . b      13
     . . . . . . . . . . . .       14
      . . . . . . . . . . .        15
       . . . . . . . . . .         16
        C . . . B . . . a          17
abcdefghijklmnopqrstuvwxyzABCDEFG

A player wins the game by eliminating one enemy color.

---

Game sample (abc even turns, ABC odd turns):

 1. k13                           abc ABC
 2. u13q13  w17     e13i13      
 3. j16     k13     q17m17 k1       9-12
 4. q1t2    j8      v6     D12     12-12
 5. m13k11  k15     m5p6   G9D8    12-13
 6. i9k7    j8l6    v6v8   s3      13-13
 7. l10     l14     B6     p6s5    12-17
 8. t2p2    t4      C11    w9      16-14
 9. l14p14  D8z8    l10l8  f4      13-17
10. t4x4    p2l2    w17u15 C11A9   16-15
11. b6b8    j6      n16    g3      13-21
12. x4A5    j2      y9w11  y9      17-17
13. B8x8    g3k3    l6o5   o15     12-19
14. s3q5    c9      A5y7   z16     16-18
15. o15s15  l2n4    k13o13 e5e7    14-20
16. C13z14  v12     z16B14 x2      16-20
17. u15x16  o5r6    o13s13 e7b8    12-24
18. x8v10   w3      v12x10 z14z16  15-22
19. s13u11  q13s11  r6u7   k11n12  14-23
20. x10t10  v10v12  x4     D12A11  17-20
21. p14s13  n12p14  u7u9   k7o7    16-20
22. v12t14  u11r12  w11s11 y5      17-20
23. t12     u9x8    o17    o7s7    14-24
24. y5w7    r14     x16t16 y17u17  16-21
25. s13s15  n16-n14 u7     v12     11-26
26. u17u15  y15     C5z6   C11     13-24
27. u7u9    t12v14  v12x14 w7y5    10-25
28. z16w15  z6z4    D10    z14     14-21
29. u9y9    r14v14  o17r16 s5      13-22
30. w3t4    B14B12  x14    x6      16-20
31. y9B10   r16t14  s5     s11u9   11-24
32. resigns

Final Position:

abcdefghijklmnopqrstuvwxyzABCDEFG
        c c . . . . . . b          1
       . B . . . . . . b .         2
      . . B . . . . . . . .        3
     B . . . B . . : . c c .       4
    . . . . . . B B . . c . .      5
   . . . A . . . . . . c . . .     6
  . . . . . . . . A . . . . . .    7
 B . . . . A . ___ . . . . . . .   8
B B . . . . . /   \ B . . . . . .  9
 . . . . . . .\___/. . . . A A .  10
  . . . . . . . . . . . . A A .   11
   . . . . . . . B . . . . a .    12
    . . . . . . . . . . . . .     13
     . . . . C . . C C a a .      14
      . . B . . . . . . . .       15
       . B . . . . . . . .        16
        B . C . . . . . .         17
abcdefghijklmnopqrstuvwxyzABCDEFG

This is a remarkable game, with lots of strategic planning in order to achieve local victories based on the deployment of its different units. An army configuration can be strong in one sector and vulnarable in another.

VERY SLOWLY PROGRESSIVE Y

Progressive Y (with the drop restriction of just one piece per friendly group per turn) with the following move sequence:1 2 2 2 3 4 4 4 5 6 6 6 7 8 8 8...

Game:

1.  k7 (X started)
2.  n8 k9
3.  j8 g9
4.  k5 n6
5.  m5 m7 f10
6.  m9 q9 h8  l6
7.  h6 o7 f8  o9
8.  h4 p6 p8  g7
9.  i3 i5 i9  l4  i11
10  i7 q7 h10 b10 m11 e11
11  j4 c9 j10 n10 t10 q11
12  m3 j2 o11 s11  j6 e9
13  l2 f6 l10 r10 c11
14  e6 d10 ...and wins next move.

Final position:

abcdefghijklmnopqrs
          .            1
         o x           2
        x . o          3
       o x x .         4
      . x o x .        5
     x x o o o o       6
    o o o x x x o      7
   . x o x . o o .     8
  x o x x o o x o .    9
 o o x o x x x . x x  10
. x o . x . o o x o . 11
abcdefghijklmnopqrstu 12

In most games (Go, Hex, Y, Moku...) the progressive variant without any restriction does not provide a propoer strategic background. In our experience, a progressive mutator sequence (and there are several, as we already saw during this blog) does have a very good partner, the group restriction mutator. Using both, each multiple drop within a turn is balanced so that no group of connected pieces is extended by more than one piece (except, of course, when two groups merge by placing one stone inbetween).

Y using the Lovely mutator

LOVELY modifier: along with one's own stone one must play an opponent
===============  stone as (euclidean) close as possible to one's own

                             ooo      xxx
                            ---------------
          .             1.  k7,j6    l6,k5
         . .            2.  n8,o9    i7,h6
        . . .           3.  j10,i11  g7,f6
       . . . .          4.  g9,f10   h10,g11
      . . o . .         5.  m9,n10   m11,o11
     o o x x . .        6.  k11,l10  k9,l8
    . x x o . . .       7.  resign
   . . . . o o . .      8.
  . . o . x o x . .     9.
 . . x x o x x . . .   10.
. . . o x o x o . . .  11.
abcdefghijklmnopqrstu

This mutator is based on Vincent Everaert's In Love Gomoku. One problem, imho, with In-Love mutator is that it permits tabu cells, ie, isolated cells where it's not legal to drop pieces. This tabu cells can produce much more drawish games (which seems the case with Gomoku). Using a closer cell to place the enemy stone, tabu cells are eliminated while still allowing all the tactical possibilities of the original mutator.

1-11 REVERSI

The scoring and moves of the game are exactly as at standard reversi,
with the four initial moves in the four centre cells.
From the second turn onwards, each player plays a move for himself, one for his opponent, and another for himself, to constitute his turn.

Game sample:

  =xx==oo=  
1. d4  e4
2. e5  d5
3. e3  d3
4. c4  f3
5. d6  d7
6. e2  b4
7. c5  d1
8. e1  f1
9. g2  h3
10 h1  f4
11 g5  g3
12 a4  g1
13 c1  e6
14 f5  h6
15 h2  b5
16 h4  a3
17 f6  a5
18 d8  g4
19 g6  c8
20 c3  f2
21 b8  g7
22 h8  h7
23 b6  h5
24 d2  g8
25 f8  c2
26 f7  e8
27 b3  b2
28 a1  a8
29 b1  e7
30 c6  a2
31 a6  c7
32 b7  a7

1-0 (35-29)

Final Position:

a b c d e f g h

x x x x x x x x  1
x x x x x x x x  2
x o x x x x x x  3
x o o x x o o x  4
x x o x o o o o  5
x o x o o o o o  6
o o o o o o o o  7
o o o o o x x x  8

Corners are much easier to grab, since players may drop an opponent stone on any valid empty cell. There are much more traps on each turn, and the possibilities for each move are quite large.

134* Amazons - Some Notes

Remembering the rules of 134* AMAZONS:

As Amazons but
* Players move 1,3,4,4... friendly Amazons per turn
* Different pieces to move within a turn.
* If 4 different pieces cannot be moved, the player loses.
----------------

Some notes:

It's difficult to know how safe it is to leave holes in the walls of one's territory.

Players should always leave some loose cannons scattered to prevent enemy blitzkrieg construction. Too much friendly pieces together are prone to quick isolation, too few to direct attacks.

There is an interesting strategic tension between going edgewards, where it is easier to make territory but also risks being shut in, or going central where one is safe but has little chance to make territory.  This feature is largely absent from the original game.  It is remarkably similar to Go, though in dynamic form rather than static form.

Initial moves of an actual game (first player is X):

1   a1-b2/m2
2   a5-c5/a3    e1-c3/b3    m1-l2/c2
3   i1-k3/m3    m13-l12/l3  b2-c1/k1    e5-e8/k2
4   c2-d2/d1    c5-e5/b2    a13-c11/l11 i13-j12/k11
5   k3-g3/k3    m5-j8/j11   a9-d6/i11   e13-i13/i12
6   e9-h12/h13  j12-k13/j12 m9-k9/m11   d2-g5/d2
7   i9-h9/h11   j8-g8/g13   e8-e10/g12  d6-c7/g11
8   k9-j8/j9    e5-f6/j10   g5-h5/j7    i5-j5/j6
9   g3-f4/j4    h9-i8/i4    g8-g4/h4    c7-c5/e5
10  j5-k4/j5    j8-h6/j8    f6-c6/g2    h5-d9/i9
11  f4-e3/j3    i8-g6/g5    e10-g7/h7   g4-e6/g5
12  h6-i7/h8    c11-f8/g8   d9-d7/e8    c6-b6/c6

a b c d e f g h i j k l m
. . x # . . . . . . # . .   1
. # # # . . # . . . # o #   2
# # . . x . . . . # # # #   3
. . . . . . . # # # o . .   4
. . x # # . # . . # . . .   5
. o # . x . x . . # . . .   6
. . . o . . x # o # . . .   7
. . . . # o # # . # . . .   8
. . . . . . . . # # . . .   9
. . . . . . . . . # . . .  10
. . . . . . # # # # # # #  11
. . . . . . # o # # . x .  12
. . . . . . # # x . o . .  13
a b c d e f g h i j k l m

Two uncommon progressive variants (2/2)

KIWI PROGRESSIVE CHESS
======================
Same as Scottish Progressive Chess except that mate may be delivered only in a move series with no captures.

Some games:

1. e4
2. d5 Kd7
3. e:d5 Nf3 Bb5+
4. Kd6 Bg4 B:f3 B:d1
5. c4 Nc3 d4 Ne4 #

r n . q . b n r
p p p . p p p p
. . . k . . . .
. B . O . . . .
. . O O N . . .
. . . . . . . .
O O . . . O O O
R . B b K . . R

 1. f3
 2. e5 h5
 3. Bh5 Be6 B:f6+
 4. K:f6 Qh4 Q:h2 Q:h1
 5. d4 d:e Qd5 Q:h1 Qf3 +
 6. Ke6 b5 Bb7 B:f3 Nh6 Bb4 +
 7. c3 c:b4 N:f3 B:h6 B:g7 B:h8 Nc3
 8. a5 a:b4 b:c3 c2 Kd5 Kc4 R:a2 c1Q +
 9. R:c1 +
10. Kd5 R:b2 c6 c5 c4 c3 h4 Rd2 c2 Rd1 +
11. R:d1 +
12. Kc4 Kc3 c1N Nb3 Nd4 Na6 Nc7 Nd5 Nf4 Ng2 #

. . . . . . . .
. . . p . . . .
. . . . . . . .
. p . . O . . .
. . . . . . . p
. . k . . N O .
. . . n O O n .
. . . R K . . .

 1. e4
 2. e5 Ke7
 3. d4 d:e5 Bg5+
 4. f6 e5 Bg4 B:d1
 5. Nc3 R:d1 R:d5 R:d8 R:f8
 6. Ke6 f:g5 Nh6 R:f8 R:f2 R:f1 +
 7. K:f1 h4 h:g5 g:h6 h:g7 Rh5 g8Q +
 8. Ke7 Na6 R:Q:g:b Rh2 R:R
 9. Ne2 Ng3 N:h5 Nb5 N:c7 N:a6 Nc5 N:b7 Kg1
10. K:h4[3] K:e5:e4[4] K:b7[3]
11. K:h7[6] Kc2[5]
12. K:a2...d7[7] a5 a4 a3 a2 a1Q
13. resign

. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. k . . . . . .
. . . . . . . .
. . K . . . . .
q . . . . . . .

This variant is slower than the Scottish one. The mate restriction adds more strategy to the move sequence, since mates are harder positions to get and so, both players may try different approaches without concerning about annoying Fool's mates.

Two uncommon progressive variants (1/2)

PORTUGUESE PROGRESSIVE CHESS
============================
Same as Scottish Progressive Chess except that all moves within a turn are with different pieces, (castling counts both).

Game example:

 1. e4    
 2. e5  Nf6
 3. Nc3 Nf3 Bc4
 4. Bb4 O-O Nc6 d6
 5. O-O d4  a4 Bd2 Qe2
 6. B:c3 Nd4 c5 b5 a6 h5
 7. N:d4 b:c3 Bb3 Bc1 Rb1 Qf3
 8. c:d4 a5 Bg4 Rc8 Qb6 g6 Kg7
 9. Bb2  c:d4 Rfc1 Kh1 Qe2 f3
10. e:d4 Be6 b4 h4 Rh8 g5 Nh5 Kg6 Rc6 Qc7
11. a:b4 B:e6 B:d5 Ra1 c3 Rb1 Qd2
12. a:b4 d5 R:e6 f5 Nf6
13. B:f6 d4 c:b4 Ra2 Qd3 e:f5+
14. K:f6 Re8 Re3 Qf4
15. b5   Rd1 Rb2 Qf1
16. K:f5 Rb8 Re5 h3
17. b6 Rb5 Rdb1 g3 Kg1 Qd3+
18. Kf6 Q:f3 g4
19. b7 R:d5 Rb2 Q:f3+
20. g:f3 Re2 R:b7
21. R:b7 Re5
22. resign

Final Position

. . . . . . . .
. R . . . . . .
. . . . . k . .
. . . . R . . .
. . . O . . . .
. . . . . p O p
. . . . r . . O
. . . . . . K .

This variant provides more 'natural' games that the typical progressive rules. It is more similar to FIDE chess and, in a sense, captures more closely the army metaphor behind the original chess concept (at a given time, every soldier should be able to move).

Neutral Mutator (part IV)

As noted with NeuY, neu-games can be somewhat smaller than their standard counterparts, and maintain the same level of interest.  This is partly because with many more options per turn, (roughly the square of the number!), the game need not be so lengthy for the same total number of options; and partly because the extra move options mean somewhat less space is required to execute the same strategic plan.

For example, we have found that 9-a-side NeuY is fully as interesting as regular 13-a-side Y.

NeuGonnect: Gonnect is already a very intriguing variant of Go, so this game may almost appear to be gilding the lily!  However, as always, the neutral-stone transformer adds intriguing new ideas to the game, while keeping the essential ideas of the parent game largely intact. In Go-like games, we have found the best option is to count neutral stones as liberties for both opponents.  Also, if two replacements are made on a turn, each must be separately legal in the order played; so that living groups in normal Go are still alive. All the other earlier remarks apply here as well. So as explained, Gonnect might also be played on a smaller board. However, as the 13x13 size already seems an ideal balance between the connection and the Go aspects in the parent game, we suggest that Neu-Gonnect not be reduced beyond 11x11, preferably 12x12.

Rules:
"""""
1) Placement styles as for Neu-games generally.
2) If 2 neutrals are flipped, each must be separately legal in the order played.
3) Captures are as in Go, but neutral stones count as liberties for both players.

Play is compulsory; suicide illegal.  The winner is whoever completes an orthgonally connected chain of stones between two opposite sides.

Some moves of a neu-gonnect game:

   --x----?------o----?--
1.  g7   j10    j7   h8
2.  i9   g9    g9h8  j7
3.  i8   g4     h7   i10
4.  i7   i4     h6   h10
5.  i6   k4    g4i4  g9
6. k4i10 g7     j5   j6
7.  k6   g8    j6j7  h8
8.  k7   l9     k8   k9
9.  j8   l8    l8l9  h7

   a b c d e f g h i j k l m  
1  . . . . . . . . . . . . .  1
2  . . . . . . . . . . . . .  2
3  . . . . . . . . . . . . .  3
4  . . . . . . o . o . x . .  4
5  . . . . . . . . . o . . .  5
6  . . . . . . . o x o x . .  6
7  . . . . . . ? ? x o x . .  7
8  . . . . . . ? ? x x o o .  8
9  . . . . . . ? . x . ? o .  9
10 . . . . . . . ? x ? . . . 10
11 . . . . . . . . . . . . . 11
12 . . . . . . . . . . . . . 12
13 . . . . . . . . . . . . . 13
   a b c d e f g h i j k l m

'o' just created a local deadlock using two neutral stones.
'x' must prevent that the upper enemy group reaches the right edge.
Also, a neutral battle is brooding at the board centre.

We end this article with a great position on a NEUMOKU game (make 5 in-a-row):

i j k l m n o p q r s
. . . . . . . . . . .  -2    (declined 3rd swap)
. . . . . . . . . . .  -1  
. . . . . . . x . . .   0  -x----?------o----?--
. . . . . . . o ? . .   1.  n3  m5     o2   n4
. . . . o x x x x o .   2.  l6  q2     o3   p2
. . . . o x o ? o ? .   3.  o4  p5     m2   q1
. . . x . o x o . . .   4.  n2  n6    n4p2  o2
. . ? . ? . . x o . .   5. q1o2 n2     p4   q5
. . . x . ? . . o . .   6. n2q5 n3     p3   m3
. . . . . . . . . . .   7. n3p5 q5     p1   q3
. . . . . . . . . . .   8.  p0  q6    q5q6  p3
                        9.  l4  k5    m3q3  p2
                       10. p2q2 q1     r2   r3

Neutral Mutator (part III)

Another game that fits nicely with this mutator is Y. The complexity of this game on even quite small boards is amazing:

     abcdefghi    --x---?----o---?--
5       ?          e1  f4   e4  h2
4      x ?         c1  c3   f2  e5
3     x o ?        g1  d4   a1  i1
2    x . o o       b2  g3  h2i1 a1
1   ? x x ? o     c3d4 g1  resigns

'o' must swap (as there is only one empty cell), and though momentarily creating a winning path, must immediately destroy it by exchanging one of his vital stones. But in doing that, 'o' provides a winning path for 'x'.

Neu-Reversi can be played in two ways.

(a) NeuVersi: a neutral stone can be dropped anywhere on the board;
(b) AdNeuVersi: it must be adjacent to some stone(s) of some player(s).

In both games, the playing of a genuine stone must follow reversi rules, and neutral stones can be turned over to the player's colour as if they were opponent stones; and if two neutrals are flipped, both must be legal reversi moves in the order played.  As always, if a player cannot make a legal move he is obliged to pass, and if both are thus obliged to pass the game ends and the count is done.  Note, there is no need for the extra game-end placement default-option rules in these games.

The discoverers have tried both versions, and found them both to have their own intriguing characters, which are noticeably (though not overwhelmingly) different from the parent game.  In general, as with all Neu-games, there tend to appear more strategic plans and tactical resources than with parent games, for the same-sized boards.

Here is an example for NeuVersi :

    --x---?-----o---?--
1.  e3  g7    f3  c4
2.  f4  f6    b3  h8
3.  c5  b4    a4  g4
4.  a2  h4

. . . . . . . .  1
x . . . . . . .  2
. x . . x o . .  3
o o x x x x ? ?  4
. . x x x . . .  5
. . . . . ? . .  6
. . . . . . ? .  7
. . . . . . . ?  8
a b c d e f g h

'x' tries to get the h8 corner in the next turn. However, 'o' moves 4... h4 a8, with the following result:

. . . . . . . .  1
x . . . . . . .  2
. x . . x o . .  3
o o o o o o o o  4
. . x x x . . .  5
. . . . . ? . .  6
. . . . . . ? .  7
? . . . . . . ?  8
a b c d e f g h

if 'x' swaps h8, there will be only one neutral left (a8) which provides no captures. So, the move is illegal and 'x' cannot take the corner. [cont.]

Neutral Mutator (part II)

In these neu-games, the instances of two game plans being executed quite independently of each other, is much more common than in regular games. The neutral structures serve to undermine enemy positions and to create optional paths to the game's goal. It is a very flexible tool.

I realized that this mutator could be used in many more games. But before continuing, as Bill Taylor noted, the rules would not be totally specified unless we say what happens when the board is almost full:

1) When there is 1 space and no neutrals left: player to move fills the space;
2) when there is 1 neutral and no spaces: mover converts the neutral to his own;
3) when 1 of each: mover fills the space, then opponent converts neutral.

As a default option, these end-game rules always apply.

A second experiment was Gomoku. We were surprised to see what a remarkable game was discovered. The complexity and balance of Neumoku seems extraordinary.

NEUMOKU: On a unlimited square board, each player may:
       * Drop a friendly stone plus a neutral stone
       * Flip two neutral stones into friendly stones and then flip another friendly stone into a neutral stone
       The goal is like Gomoku.
       PIE RULE: After the third move, the second player may swap sides.

A sample game:

   --x----?-----o----?--
1.  n3  m5     o2   n4
2.  l6  q2     o3   p2
3.  o4  p5     m2   q1
4.  n2  n6    n4p2  o2
5. q1o2 n2     p4   q5
6. n2q5 n3     p3   m3
7. n3p5 q5     p1   q3
8.  p0  q6    q5q6  p3
9.  l4  k5    m3q3  p2
10. p2q2 q1     r2   r3

Actual Board:

j k l m n o p q r s
. . . . . . . . . .  -1
. . . . . . x . . .   0
. . . . . . o ? . .   1
. . . o x x x x o .   2
. . . o x o ? o ? .   3
. . x . o x o . . .   4
. ? . ? . . x o . .   5
. . x . ? . . o . .   6
. . . . . . . . . .   7

Here, 'o' has a winning sequence:

   --x---?-----o---?--
11.  o5  l5    r7  s1
12.  s7  l3    q4  t0

j k l m n o p q r s t u
. . . . . . . . . . . . -1
. . . . . . x . . . ? .  0
. . . . . . o ? . ? . .  1
. . ? o x x x x o . . .  2
. . . o x o ? o ? . . .  3
. . x . o x o o . . . .  4
. ? ? ? . x x o . . . .  5
. . x . ? . . o o . . .  6
. . . . . . . . . x . .  7
. . . . . . . . . . . .  8

Next turn, 'o' wins at column 'q' or the diagonal from p4 to t0. [cont.]

Neutral Mutator (part I)

Joao Neto has found a most promising new game mutator which seems applicable to a number of games. It is most applicable to games which begin with an empty board and continue with players adding a piece per turn. Obvious examples are: almost all connection games, Go, Go-moku, Reversi. Initially, this idea was meant to create a Hex variant, called Nex (or Neux):

NEX: On a Hex board, at each turn, the player must do one of:
       * Drop a friendly stone plus a neutral stone;
       * Flip two neutral stones into friendly stones and then
          flip a different friendly stone into a neutral stone.
The goal is as for Hex, with a connecting path including no neutrals.  

Here is a sample game:

    Vertical   Horizontal                      
  --v-----?-----h------?--                          
 1. b11   f6    d7     b8
 2. e7    g5    h3     f4
 3. i3    g3    g6     e6
 4. f6g5  a11   d10    h4
 5. f9    d9    g3h4   d7
 6. e4    f3    f4e6   d10
 7. i4    d6    h5     e3
 8. d6d7  g5    e3f3   f4
 9. i5    c4    h7     f8
10. h6    d3    g7     c10
11. d3c4  f6    f8d9   g3
12. i7    b10   i6     b9
13. b9b10 h6    c10b8  f3
14. c8    k5    d5     b11
15. e5    e9    a11b11 b8
16. e9k5  b9    e8     k2
17. j7    k7    j6     j2
18. k6    g4    j5     i2
19. k4    k1    k3     j3
20. j4    g8    i2j2   d5
21. Resign

Final Position:

  a b c d e f g h i j k
1  . . . . . . . . . . ?
2   . . . . . . . . h h ?
3    . . . v h ? ? h v ? h
4     . . v . v ? ? h v v v
5      . . . ? v . ? h v h v
6       . . . v h ? h ? h h v
7        . . . v v . h h v v ?
8         . ? v . h h ? . . . .
9          . ? . h v v . . . . .
10          . v h ? . . . . . . .
11           h h . . . . . . . . .
              a b c d e f g h i j k

These games are full of tactical subtleties. An interesting feature is that no piece is totally useless, because it can always be used to swap two neutrals. Swap battles tend to occur after some critical mass of neutrals is achieved. Forcing moves, (i.e. where the player forces the opponent to drop a piece of their own colour) are a key to success in this game,   as that is the only sure way to stop him flipping two '?'s next turn.

[cont.]

13(4) progressive mutator

Amazons is another example of a game using the progressive 13(4) mutator. So, the first player makes one move, then the second player makes three moves, and then, on every remaining turn, both players make four moves. There is a restriction (we should always attach restrictions to progressive mutators): a piece can only move once per turn. Here is a board after 7 moves:

a b c d e f g h i j k l m n o p q r s
. . . # y # . . . . . y . . . o . . . 1
. . . . # # . # . . . . . . . y . . . 2
y . . . # . # . . . . . . . . . . . . 3
. # # # # . # . . . . . . . . . . . . 4
. # . . # . . . . o . . . . . . . . . 5
# # . . . . . . . . . . . . . . . . . 6
# . . . . o # . . . . . . . . . . . . 7
. . . # . . . . . . . . . . . . . . o 8
. . . y . . . . . . . . . . . . . . . 9
. . . . . . . . . . . . . . . . . . . 10
. . . . # o . . . . . o . . . . . . . 11
. . . . y . . . . . . . . . . . . . . 12
. . . y . . . . . . . . . . . . . . . 13
. . . . # . . . . . . . . . . . . . . 14
. . . # . . . . . . . . . . . . . . . 15
. . . y o . . . . . . . . . . . . . o 16
. . . # . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . 18
. . . . . . . . # . . . . . . . . . . 19
a b c d e f g h i j k l m n o p q r s

Moves: Y vs O

1. s4d4/b4
1... a7d7/g4 d19g16/g7 l19l16/i19
2. a8a5/b5 d1d2/d1 d4d3/d4 h19c14/c4
2... h1g2/e2 d7e6/e3 l16l11/e4 a12b11/b6
3. a5a9/e5 d3a3/a7 d2e1/g3 p19p2/h2
3... g2j5/f1 e6f7/f2 b11f11/a6 g16e16/e14
4. a16d16/d17 c14d13/d15 a9d9/d8 s12e12/e11


A progressive game of Y

Here follows a Y game with a 1222 progressive mutator and the following restriction: both drops must be on non-adjacent groups.

   1          .
   2         . .
   3        . . .
   4       . . . .
   5      . . . . .
   6     . . . . . .
   7    . . . . . . .
   8   . . . . . . . .
   9  . . . . . . . . .
     /       /       /
    a b c d e f g h i

         X       O
    1. c5      c7 e8
    2. d7 f8   c6 e7
    3. d6 f6   d8 g8
    4. f7 d4   f9 e5
    5. e6 h8   g9 i9
    6. a5 b7   b5 a6
    7. b6 a4   g7 a4
    8. h9 a3   b6 b3
    9. resigns

Final position:

   1          .
   2         . .
   3        O X .
   4       X . . X
   5      X . X . O
   6     O X O X X X
   7    . X O X O X X
   8   . . . O O X O X
   9  . . . . . O O O O
     /       /       /
    a b c d e f g h i

'O' cannot play both b4 and b5 (the two drops would belong to the same group). So 'X' make occupy one of those cells plus another below to connect the bottom edge.