Towards a new Paradigm of Board Games

Towards a new Paradigm of Board Games 

Dr. phil Rudolf Kaehr 

ThinkArt Lab Glasgow 

ISSN 2041-4358 

( work in progress, vs. 0.3.5, May/April 2014 ) 

MorphoBoard Games 

Motivations for MorphoBoard Games 

MorphoGames are well motivated by themselves. But there is no reason not to accept another motivation 

too. MoprphoGames offer ideal approaches for a general understanding of morphogrammatics 

and polycontextural logics that had been developed on a much more abstract level in the past. 

MorphoGames offer a playful environment to learn and enjoy the essential structurations and transformation 

rules of morphogrammatics. 

Morphogrammatics is a theory and a system of pre-semiotic patterns and their transformations that 

is fundamental for a study of the morphosphere. 

Semiotic games, like Leibniz games are ‘localized’ on the level of inscription of the semiosphere. The 

basic axiom of the semiosphere is based on perception: What you see is what it is (WYSIWYGapproach). 

MorphoGames of Morphogramatics are ‘localized’ on the level of the morphosphere where the main 

experience is based on cognition: What you see is not how it acts. 

Polycontextural game theory had been developed in the context of an explication and formalization of 

the concepts and stratagems of ‘interactionality’ and ‘reflectionality’, especially on the level of formal 

systems (languages) and MAS (Multi-Agent Systems). 

MorphoBoard Games are not to be confused with Morpho Games from other sources. 

Mathematical Theories of Games 

http : // www.wisdom.weizmann.ac.il/~fraenkel/Papers/GamesHandbk.pdf 

MorphoBoard Game Definition 

Board program 

The parameters of the Game are easily changed. 

The parameters are : Width, Heigth, Patterns (colors) and Randomness. 

The initialization of the Board depends on the chosen values for the parameters. 

Further changes of the rules are obvious. 

This presentation of some very first results of the paradigm of MorphoGames are following closely 

Yves Papegay’s contribution to classical Board Games programmed in Mathematica. 

Yves Papegay, Exploring Board Game Strategies, A Recreational Application of GUIKit 

The Mathematica Journal, Volume 10, Issue 2, 2006 

http : // www.mathematica - journal.com/2006/09/exploring - board - game - strategies/ 

Aim of the paper 

The aim of this paper is three-fold: 

1. Reconstruction of the classical concepts, definitions and programs for Board Games as presented 

by Papegay's Mathematica program. 

2. Elaboration of some dynamics of the parameters of the program to learn the scope of the internal 

possibility of the concept and possible extensions beyond the concept of classical Board Games. 

3. Sketch of the paradigm of Morpho Games and first steps of developing procedures for it in 

Mathematica.

The aim of this paper is three-fold: 

2 MorphoBoardGames.nb 

1. Reconstruction of the classical concepts, definitions and programs for Board Games as presented 

by Papegay's Mathematica program. 

2. Elaboration of some dynamics of the parameters of the program to learn the scope of the internal 

possibility of the concept and possible extensions beyond the concept of classical Board Games. 

3. Sketch of the paradigm of Morpho Games and first steps of developing procedures for it in 

Mathematica. 

Game 

Game definition 

"As quoted, the most important point when designing a board game implementation is to have a 

proper idea of the board itself, which represents the complete status of the game at each play and 

the rules that determine whether or not a play is legal." (Papegay) 

“The game denotes a complete set of interactions between the program—or the physical support of 

the game, whatever it is—and the player. Hence, it is a succession of three phases: 

† an initialization phase when the game starts or restarts 

† a playing phase (i.e., when the user is playing). This is the most common behavior of the game 

and consists in a sequence of successive plays. 

† a termination phase at the end of the game.” 

Board 

“The board represents not only the physical board but, by extension, the complete status of the 

game at a given time. By definition, in a board game, this status is well defined by a mapping 

between a two-dimensional set of locations and additional information (usually qualitative or 

discrete) for each location.” 

Board: Designing, Initializing, Interpretation 

The design of the board follows decisions about its topology. Out of the multitude of possibilities, 

classical Board Games are deciding for a ‘Euclidean’ topology with its Height and Width. 

Interpretation 

The category of interpretation for Board Games is reduced for classical Games to an Interpretation in 

the modus of identity. Because identity is ubiquitous for the classical approach it is not necessary to 

be specially mentioned. It is obvious that the elements of a Game, i.e. the value occupations on the 

Board, are subsumed under the law of identity. 

Definition of a Board 

Here, the parameters of the board are set as follows: 

Width=22; 

Height=15; 

Patterns= {...}. 

In a further development, a menu will allow the user to set the values.

MorphoBoardGames.nb 3 

Width = 22; 

Height = 15; 

Patterns = 

8Item@v, Background Ø Green, Frame Ø TrueD, 

Item@e, Background Ø Red, Frame Ø TrueD, Item@u, Background Ø Blue, Frame Ø TrueD, 

Item@w, Background Ø Yellow, Frame Ø TrueD, Item@s, Background Ø Pink, 

Frame Ø TrueD, Item@z, Background Ø Gray, Frame Ø TrueD Patterns@@xDDD 

s s u s z w z u s u s v v w s s e z w v z s 

w w w z z u e v v u u w z v w z z w z e v v 

w e v e v s u e z e v z v z z u s s e e u w 

w w w u v z u e s z u w u w z e w v u u z w 

w s e w v s w v e z u w w e s z z u u s z s 

z z v v u e v v z u v s s s z w w e v e u u 

w e u z s u w w z z z v w e w s v z w z s z 

w w e e u s u w v w e e z s v e v s s z e s 

u z z s u w v u e z v e v v e w v z e u z u 

z v s e z e w s e z z e s e e s v w e s v w 

z v s e w w z u v w w z e u e e e s w w v s 

w s z w z e z e v s w u v v s u e s v w s v 

s z e v w z e z e z w s s s e w w e e w s v 

u e z v w z u w u s z s u s z w s s e w e e 

v u v v w e e v u w v u z w s v s e e v v u 

Numeric labeling of the colored board


Width = 22; 

Height = 15; 

Patterns = 

8Item@1, Background Ø Green, Frame Ø TrueD, Item@2, Background Ø Red, 

Frame Ø TrueD, Item@3, Background Ø Blue, Frame Ø TrueD, 

Item@4, Background Ø Yellow, Frame Ø TrueD, Item@5, Background Ø Pink, 

Frame Ø TrueD, Item@6, Background Ø Gray, Frame Ø TrueD Patterns@@xDDD 

6 3 4 4 6 3 4 5 2 3 1 1 4 3 5 4 1 6 4 1 3 2 

5 1 4 2 6 5 6 2 5 4 2 2 4 1 3 6 1 3 1 4 1 5 

5 1 4 1 5 2 3 3 4 2 6 5 3 3 5 6 6 5 6 4 6 2 

2 6 1 4 5 2 4 6 1 4 1 6 5 1 1 2 4 3 3 1 6 3 

6 3 6 3 5 3 1 1 3 5 5 6 2 3 3 3 3 4 6 4 6 6 

1 3 2 1 6 1 5 2 5 6 6 3 4 6 3 3 4 2 6 5 5 4 

4 5 3 1 2 1 2 3 1 1 6 1 5 5 3 5 2 1 5 4 6 2 

2 4 6 3 3 4 4 4 6 4 6 2 5 3 2 3 4 4 5 6 1 2 

2 6 5 1 3 6 1 2 4 1 1 5 3 2 5 6 1 3 2 3 1 1 

4 2 4 1 2 4 4 6 6 1 2 1 4 1 3 2 3 1 2 6 6 3 

3 1 2 2 4 4 1 5 5 4 4 2 6 6 4 6 3 1 3 6 5 4 

2 2 5 1 6 6 2 6 2 6 1 4 5 1 1 1 6 4 2 4 5 1 

5 5 6 1 4 6 1 3 1 3 3 4 1 5 2 6 3 1 4 3 3 2 

1 3 5 5 1 3 5 3 5 5 5 4 2 3 5 4 6 3 4 3 6 4 

4 1 2 3 2 6 2 4 1 2 2 5 6 4 3 5 5 1 5 1 2 6 

Board example of the presentation of MorphoGames 

Needs@"GraphUtilities`"D 

Boardinit@D := 

Patterns = 

8Item@v, Background Ø Green, Frame Ø TrueD, 

Item@e, Background Ø Red, Frame Ø TrueD, Item@u, Background Ø Blue, Frame Ø TrueD, 

Item@w, Background Ø Yellow, Frame Ø TrueD, Item@s, Background Ø Pink, 

Frame Ø TrueD, Item@z, Background Ø Gray, Frame Ø TrueD, 

Item@l, Background Ø Cyan, Frame Ø TrueD, Item@m, Background Ø LightBlue, 

Frame Ø TrueD, Item@n, Background Ø LightRed, Frame Ø TrueD


w e s u s e l z v l l 

s w z m m l w z w m n 

e s w s s l l n m u n 

n m m l e n s l s s z 

v u e z v l l u w n e 

v l w z v w m n l u e 

l n l v v s n m z m s 

v w z m m e u u v v e 

s n l v w u l l m n w 

Interpretation of a Board 

The category of interpretation of a board is not necessarily a specific topic of a classical definition of a 

game. Classical games are conceived as ruled by the identity during a play of its board (width, 

height), modality (randomness, usw), elements (patterns) and rules. 

This is in concordance with the definition of an elementary formal system (EFS) in the sence of Melvin 

Fitting and Raymond Smullyan. 

But there are other approaches to an interpretations of a board and its use available. 

This proposal is distinguishing, at first, between Leibniz, Brownian, Mersennian and Stirling games. 

Classical games are understood as Leibniz games. 

Play 

“By definition, a play is one step of the playing phase. At each play, the player has to select which 

action, among the legal (valid) ones, to perform. “ 

For poly-Games, this decision function of selection is complemented with the election function that 

decides what kind game (Leibniz, Stirling, etc.) shall hold for the next steps. 

Rules 

“The rules are the set of constraints that determine what can be played and how the status of the 

game should be modified by a play.” 

Additional features: Undo, repetition detection 

“How to Undo Remember we said our board representation needed to handle undo operations.There 

are two possible methods : 

(1) Keep a stack in which each stack item holds a whole board representation; to make a move push 

it on the stack and to undo a move pop the stack.Probably this is too slow ... 

(2) Keep a stack storing only the move itself together with enough extra information to undo the 

move and restore all the information in the board position. E.g. in chess you would need to store the 

identity of a captured piece (if any) and enough information to restore castling and en passant capturing 

privileges.” 

http://www.ics.uci.edu/~eppstein/180a/970408.html 

MorphoGame interpretation of a Board 

For a Stirling approach to Board Games, the fact that the concept of patterns,where the ordered 

strings or morphograms of identity-free elements, are crucial, leads to the following elementary rules. 

MorphoGame rules 

Rules in colors 

Rule1. Ê = ‡ 

Rule2. Ê Ê = ‡ ‡ 

Rule3. Ê ‡ = ‡ Ê 

Rule4. Ê Ê Ê ¹≠ Ê Ê ‡ ¹≠ Ê ‡ Ê ¹≠ Ê ‡ ‡ ¹≠ Ê ‡ Á 

Rule5. Ê Ê ¹≠ Ê. 

Rule5 is resolved in metamorphic MorphoGames.


Rule1. Ê = ‡ 

Rule2. Ê Ê = ‡ ‡ 

Rule3. Ê ‡ = ‡ Ê 

Rule4. Ê Ê Ê ¹≠ Ê Ê ‡ ¹≠ Ê ‡ Ê ¹≠ Ê ‡ ‡ ¹≠ Ê ‡ Á 

Rule5. Ê Ê ¹≠ Ê. 

Rule5 is resolved in metamorphic MorphoGames. 

Identification Rules 

In contrast, the rules for identity-based, i.e. classical games, like Leibniz games, are given by the 

following postulates : 

Pos1. Ê ¹≠ ‡, 

with the natural consequences of 

Pos2. Ê Ê ¹≠ ‡ ‡ 

Pos3. Ê ‡ ¹≠ ‡ Ê, and 

Pos4: Ê Ê ¹≠ Ê . 

Wordings of morphogrammatic constellations 

For a Stirlingian game with 3 elements, some typical situations occur. 

1. Ê Ê Ê ª ‡ ‡ ‡ ª Á Á Á, 

Ê Ê ‡ ª ‡ ‡ Ê ª Á Á Ê 

etcetera 

2. Ê Ê ‡ ª rev(Ê ‡ ‡) : reversion 

3. Ê ‡ Ê ª rev( Ê ‡ Ê) : self-symmetry 

Ê Ê Ê ª rev(Ê Ê Ê)| 

Ê ‡ Á ª rev( Ê ‡ Á) 

4. Rules for blanks within the MorphoBoard Game. 

Elimination of blanks: blank|element1|blank|element2 ï element1|element2. 

The rules for morphic patterns are defining the rule-set of the MorphoBoard Games. 

Patterns of Rule4 for Sn (4, 4) 

Ê Ê Ê Ê 

Ê Ê Ê ‡ 

Ê Ê ‡ Ê 

Ê Ê ‡ ‡ 

Ê ‡ Ê Ê 

Ê ‡ Ê ‡ 

Ê ‡ ‡ Ê 

Ê ‡ ‡ ‡ 

Ê ‡ ‡ Á 

Ê ‡ Ê Á 

Ê ‡ ‡ Á 

Ê ‡ Á Ê 

Ê ‡ Á ‡ 

Ê ‡ Á Á 

Ê ‡ Á Ï 

Meta-Rule for morphograms 

Two arbitrary morphograms @mgD i and @mgD j , i, j œ Sn2, of the same complication (length) are morphogrammatically 

equivalent if they don’t belong to the class of morphograms defined by the generalized 

Rule4 with Sn(n,n). 

Range of morphic board constellations 

What ever happens on a MorphoBoard boils down to a system or structuration of morphograms ruled 

numerically by the Stirling numbers of the second kind and their summations by the Bell numbers. 

Therefore, the range of possible constellations is never infinite but restricted by the definition of the 

morphic structurations, counted by the Stirling numbers of the second kind.


What ever happens on a MorphoBoard boils down to a system or structuration of morphograms ruled 

numerically by the Stirling numbers of the second kind and their summations by the Bell numbers. 

Therefore, the range of possible constellations is never infinite but restricted by the definition of the 

morphic structurations, counted by the Stirling numbers of the second kind. 

This fact of finiteness of the morphic constellations enables interesting classifications and reduction 

rules of the range of constellations. 

Hence given a situation with 4 positions, there are by Sn2(4,4), just 15 morphogrammatic constellations 

possible. Therefore, there are just 15x15 = 225 morphic confrontations between two morphograms 

of length=4 possible. 

For the phenotypical realization of a board and its constellations, the range is counted by 

Example 

For m=3, k=2 

m! 

Hm-kL! . 

For the case of just 2 elements involved in a constellation of 3 positions, the abstract morphogram, 

[Êıı], has a representation of 6 concrete realizations, i.e. the set {[abb], [acc], [baa], [bcc], [caa], 

[cbb]}, all representing the morphogram [Êıı]. 

Hence, the number of confrontations between the phenotypical representations of [Êıı] and [Êıı] 

on a MorphoBoard is 6x6=36. 

Equivalence classes of games 

The distinctions of different phenotypical representations of genotypical constellations, morphograms, 

enable to define a theory of equivalence classes of MorphoGames. 

MorphoGames that appear phenotypically as different may still be morphogrammatically equivalent. 

This not to confuse with the trivial statement that what we can play in red we can also play in green. 

MorphoGame strategies 

At first, there are two simple strategies to consider: 

1. Strategy: Elimination 

Morphic sameness is eliminating the morphograms. 

a.) vertical and horizontal 

u w m n 

z s v z 

ï [ ] 

b.) horizontal 

n l e w 

v e m w ï [ ] 

c.) vertical 

n l e 

v e m ï [ ] 

2. Strategy: Reduction 

Morphic sameness is reducing the morphograms up to one morphogram. 

Rules have to specify which morphic representation of the reduction owith the second strategy 

survives. 

a.) vertical and horizontal 

u w m n 

z s v z 

ï 

m 

v 

n 

z 

b.) horizontal 

n l e w 

v e m w ï v e m w 

c.) vertical 

n l e 

v e m ï e m


n l e 

v e m ï e m 

Explanation of the elinination rules 

Rule1: Ê = ‡ 

If there are just two elements on the board as neighbors available, then they get eliminated independently 

of being the same or different. Also the blanks between the elements are eliminated. 

Ñ Ñ Ñ Ñ Ñ Ñ 

Ñ v Ñ Ñ l Ñ 

ï É 

v Ñ Ñ l 

Ñ Ñ Ñ Ñ 

ï v l ï [ ] : horizontal 

v 

l 

ï@D : vertical (for final steps) 

Rule2: Ê Ê = ‡ ‡ 

The same as for Rule1 with two or more elements. 

v Ñ l 

v Ñ l 

ï@D : vertical 

v Ñ v 

l Ñ l 

ï@D : horizontal 

Rule3: Ê ‡ = ‡ Ê 

m 

n 

n 

m 

ï [ ] : horizontal+vertical 

Iteration of Rule3 

l 

m 

n 

u 

n 

m 


n 

w 

n 

l 

z 

l 


Rule4: Ê Ê ‡ ¹≠ Ê ‡ Ê ¹≠ Ê ‡ ‡ ¹≠ Ê ‡ Á 

This rule marks the difference between patterns of elements. It hold in both directions: vertically and 

horizontally. 

Following Rule4: Ê ‡ Á ¹≠ Ê ‡ Ê , the two patterns 

therefore not eliminable. 

MorphoPalindromes 

v 

w 

e 

and 

l 

s 

l 

are morphogrammatically different and 

A higher level of sophistication is achieved with the strategy to detect not just morphic equivalences 

but morphic palindromes.


Morphic palindromes are a subclass of morphograms. The detection of Morphic palindromes is demanding 

for a new abstraction of detection and separation. 

Therefore, palindromic games are of a higher gaming level than the basic morphogrammatic games. 

Two arbitrary morphograms @mgD i and @mgD j , i, j œ Sn2, of the same complication (length) are morphogrammatically 

equivalent if they are morpho-palindromic. 

There are three main categories to distingush for a detection of morphic palindromes. 

Case one: Head and body are morphogrammaticall equivalent 

The MorphoGame related strategy to detect and eliminat patterns is based on the presumption that 

the head and the body, defined by repetition, reversion and accretion, are of the same length. 

Odd palindromes, i.e. morphogrames of length 2n+1, entail a medium pattern between the two 

parts, head and body. For even palindromes, the medium part is empty. 

Even palindromes 

Therefore, the head and the body of the palindrome shall be ‘parallelized’ horizontally in the game. 

If two patterns are building together a morphic palindrome then they shall be eliminated. 

1. Repetition 

2. Reversion 

3. Accretion 

Iterability scheme for @1, 2, 2, 3D ; 14 

head of palindrome 

@1, 2, 2, 3D 

á ¯ ä 

reversion accretion repetition 

@2, 3, 3, 1D 

@3, 2, 2, 1D 

@3, 4, 4, 1D 

@4, 2, 2, 1D 

@4, 5, 5, 1D 

@2, 3, 3, 4D 

@3, 1, 1, 2D 

@3, 2, 2, 4D 

@3, 4, 4, 5D 

@4, 1, 1, 2D 

@4, 2, 2, 5D 

@4, 5, 5, 6D 

@1, 2, 2, 3D 

@1, 4, 4, 3D 

- [1,2,2,3] = tnf[3,4,4,1]; 

val it = true : bool 

- [1,2,2,3,1,4,4,3] = [1,2,2,3,2,3,3,1]; 

val it = false : bool 

Example 

The two neighbor patterns [ Ê ‡ Ê Á ‡ Á ] and [ Á ‡ Á Ê ‡ Ê ] are morphic palindromes, hence they 

are eliminable. 

In this case, the two patterns are also just morphogrammatically equivalent. 

[1,2,1,3,2,3] = mg [3,2,3,1,2,1] 

[1,2,2,3,3,4,4,1] = mg [1,4,4,3,3,2,2,1] 

-ispalindrome@1, 2, 2, 3, 3, 4, 4, 1D; 


Odd palindromes 

Serial case


n 

n 

z 

e 

e 

, 

l 

l 

n 

w 

w 

, 

l 

s 

l 

w 

l 

s 

l 

: vertical 

e s u s e : horizontal 

If @mgD is a vertical or horizontal palindrome it is eliminable : 

@mgD œ PAl : @mgD ï @D 

Counter example 

m 

z 

z 

z 

l 

l 

v 

w 

ï not-[] 

Examples 

-ispalindrome[1, 2, 1, 3, 1, 2, 1]; 


-ispalindrome[1, 2, 3, 2, 1]; 


-ispalindrome[1, 1, 2, 3, 3]; 


Case two: Head and body are morphogrammatically different 

A more intriguing case is given if the body and the head of two even palindromes are morphogrammatically 

different. 

Trivially, the morphogram [1,1,1,1] is a palindrome. The same holds for the morphogram [1,2,1,2]. 

But both are morphogrammatically different, [1,1,1,1] mg ¹≠ [1,2,1,2]. 

Hence, the new abstraction is ruled by the property of being a palindrome. 

Both morphograms of the example are palindromes of the same length, hence they can be eliminated. 

In contrast, the morphogram [1,1,1,2] is not a palindrome. Hence, the comparison of [1,1,1,2] and 

[1,1,1,1] is not eliminative. 

@mgD mg 1 ¹≠ @mgD 2 Ô @mgD 1 ,@mgD 2 œ PAl : @mgD 1 || @mgD 2 ï [] 

Example 

horizontal palindromes: 

l n l 

v w z 

s n l 

ï l n l œ Pal, v w z 

s n l œ Pal, 

l n l ¹≠ mg s n l , v w z ï [] 

Program to detect morphic palindromes 

Bisymmetric e/v-version 

BiSymTest[x] = 

TabView[ 

Grid /@ 

{x , 

"Palindrome" -> SameQ[ 

Grid[ 

Reverse[First[Grid[Reverse /@ 

x]]]] , 

Grid[ 

Reverse /@ First[Grid[Reverse[ 

x]]]], 

Grid[Transpose[x]]], 

Clear[x]

BiSymTest[x] = 

TabView[ 

Grid /@ 

{x , 

"Palindrome" -> SameQ[ 

Grid[ 

Reverse[First[Grid[Reverse /@ 

x]]]] , 

Grid[ 

Reverse /@ First[Grid[Reverse[ 

x]]]], 

Grid[Transpose[x]]], 

Clear[x] 

}] 


Wee MorphoPalindrome Checker Palin (4, 2) 

« Å▸ ¡ 

v 

e v 

e v e 

List Version 

PalindromeQ[i_String] := StringReverse[i] == i 

MorphoPalindromeQ[i_String] := ReLabel[StringReverse[i]] == i 

Example for lists 

"Palindromes 4to5"@MenuViewD 

@1,2,3,1D 

@1, 1, 2, 3, 1, 1D rule1 @1, 1, 2, 3, 1, 1D 

@4, 1, 2, 3, 1, 4D rule3 @1, 2, 3, 4, 2, 1D 

@4, 1, 2, 3, 1, 5D rule4 @1, 2, 3, 4, 2, 5D 

@2, 1, 2, 3, 1, 3D rule1 @1, 2, 1, 3, 2, 3D 

@3, 1, 2, 3, 1, 2D rule2 @1, 2, 3, 1, 2, 3D 

The MorphoBoard configuration 

The MorphoBoard configuration is defined by the board measures, the number of elements and a 

random function that maps the elements onto the Board. 

The MorphoTransition rules 

The Transition mappings are considering different modi of parallelism of the reduction rules. 

Termination and Evaluation of the MorphoPlay 

The scores of the play are determined by the numbers of steps and the number of not resolved 

patterns of the reduction game. 

Examples for a manual play 

The examples show the strategy how to achieve the results of: 

First play: 18 steps, with 2 patterns left. 

Second play: 15 steps, with 3 patterns left.


Different logics for different paradigms 

Classical games 

Obviously, classical board games are ruled by classical two-valued logic. This is abbreviated by the 

identity rules of the classical game: Pos1: Ê ¹≠ ‡. An element of a game has a unique value: it is or it 

is not on the board. Logically speaking, it is true (false) that an element occupies a place on the 

board. 

For the game, what counts are sequences of the identical and unique elements. Therefore, the classical 

two-valued logic is ruling the rationality of the game. A blank on a classical board has no logical 

significance. It just marks the elimination of a value. 

Hence, the rules of a classical game are concerning the status of the sequences of connected identical 

elements on the board. 

Classical sequences of the same elements might be ordered horizontally and vertically, but there are 

also possibilities to connect the elements diagonally. 

u 

v 

A diagonal connection is illustrated by the example 

m 

s 

u 

u 

. The diagonal rule reduces the blue diago- 

n 

e 

nal sequence 

u 

Ñ 

Ñ 

v 

u 

u 

to the figure 

Ò 

m 

s 

n 

v 

Ò 

„ 

e 

. 

All three modi of connection, horizontal, vertical and diagonal, are gap-free. 

This reflection on the logic of games shouldn’t be confused with the genre of Logic Games. 

MorphoGames 

Things are different for MorphoGames. What counts for the game are not the elements and their 

values as such but the differences between the elements that are localized on the board. Because of 

this differential definition of the states of a MorphoGame, not only the involved and localized elements 

are of importance but the gaps, represented by the blanks, too. 

To put it in a abbreviation, the logic of MorphoGames is not two-valued but a combination of at least 

two 2-valued logics and the logics between the two 2-valued logics covering the logical status of the 

gaps inscribed as blanks. 

Hence, the rules of MorphoGames are concerning the status of the sequences of horizontally and 

vertically connected complex differences and their gaps on the board. 

For example: 

w Ñ blanks Ñ m 

w Ñ blanks Ñ m 

ï 

w w 

m 

m 

ï Ñ Ñ 

: vertical + horizontal move and 

elimination. 

Desiderata 

The MorphoBoard Games are not yet programmed. 

This note gives just the general concept with its rules and the program for the Board constellations. 

The rules have to be applied by a human player. A programmed version will hopefully follow soon. 

Board configuration one 

n z l v l s z s l z n 

m e z m w e n m e z w 

n l n u u l u l s s l


n u z l n l n v n w v 

v w u w m n m z z z m 

m w z s v z m n s e e 

w l e l u e l u v z m 

w s v l u v z m u w l 

m n w n v z n s u v z 

n l e w l v m n e w v 

v e m w s l l w v l m 

Decompositions of board one 

One possible solution of the Board is given by the following 18 steps of the Game with 2 patterns left 

(omitting any moves of the squares). 

This demonstration of the MorphoGame with ‘board one’ is quite static and relies only on the existing 

configurations of the board. There are also no ‘diagonal’ interpretations involved. 

Therefore, the action of moving elements (squares) on the board to produce new constellations is not 

yet included in this description. 

The rules are horizontally and vertically applied to the patterns. 

1 Ñ Ñ Ñ Ñ Ñ Ñ Ñ 1 6 6 

1 Ñ Ñ Ñ Ñ Ñ Ñ Ñ 1 6 6 

17 Ñ Ñ Ñ Ñ 17 5 5 13 13 l 

17 Ñ Ñ Ñ Ñ 17 5 5 13 13 v 

12 12 2 Ñ Ñ 2 18 8 Ñ 8 m 

12 Ñ 2 Ñ Ñ 2 18 8 Ñ 8 e 

18 10 10 11 11 15 Ñ 15 v 9 9 

18 10 10 11 11 15 Ñ 15 u Ñ Ñ 

4 Ñ Ñ 4 14 Ñ 14 s u 9 9 

Ñ Ñ Ñ Ñ Ñ Ñ Ñ 3 Ñ Ñ 3 

4 Ñ Ñ 4 14 Ñ 14 3 Ñ Ñ 3 

The number at the corners of a deleted pattern (configuration) indicates the number of the steps of 

the play and its empty frame. 

Vertical application of Rule3 

1 = n z l v l s z s l 

m e z m w e n m e 

ï Ñ Ñ , 

with vertical application of Rule3 : Ê ‡ = ‡ Ê 

2 = u w m n 

z s v z 

ï Ñ Ñ 

, with Rule3 : Ê ‡ = ‡ Ê, 

3 = n e w w 

w v l l 

: 

n 

w = e v ï w l 

= w l 

ï Ñ Ñ : vertical, 

with Rule3 : Ê ‡ = ‡ Ê for 

n 

w = e v , 

and Rule3 as Identity Rule for 

w 

l 

= w l 

,


w 

l 

w 

l 

ï Ñ Ñ 

: vertical + horizontal. 

4 = 

m n w n 

n l e w 

v e m w 

: horizontal is not valid becaus of double "n", 

and not valid for vertical because of double "w" : 

m n w n ¹≠ n l e w 

v e m w : vertical, 

Hence, what holds is : 

m n w 

n l e 

v e m 

vertically + horizontally and n l e w 

v e m w horizontally. 

5 = u l 

n v 

ï@D : vertical + horizontal. 

6 = z n 

z w 

ï@D : horizontal. 

7 = l n 

u z 

ï@D : vertical + horizontal. 

8 = z z z 

n s e : vertical 

z 

m 

9 = 

w 

l 

: vertical + horizontal 

v 

z 

10 = l e 

s v 


11 = l u 

l u 


12 = v w 

m w : horizontal 

13 = s s 

n w : vertical 

14 = 

v z n 

l v m 

s l l 

ï@D : vertical 

15 = e l u 

v z m 

ï@D : vertical + horizontal 

16 = u u 

l n : vertical 

17 = 

n 7 7 16 16 l 

n 7 7 16 16 l 

: horizontal move, neglecting blanks


n 7 7 16 16 l 

n 7 7 16 16 l 

ï n l 

n l 

ï @D : horizontal move, removing blanks 

18 = 

12 12 2 Ñ Ñ 2 m 

12 Ñ 2 Ñ Ñ 2 m 

w 10 10 11 15 Ñ Ñ 

w 10 10 11 15 Ñ Ñ 

4 Ñ Ñ 4 14 Ñ 14 

Ñ Ñ Ñ Ñ Ñ Ñ Ñ 

4 Ñ Ñ 4 14 Ñ 14 

ï w Ñ Ñ Ñ Ñ Ñ m 

w Ñ Ñ 4 14 Ñ m ï w w 

m 

m 

ï @D : vertical + horizontal move 

According to the rules, this final constellation cannot be further resolved . 

Hence the play ends after 18 steps with 2 patterns unsolved. The score is (18, 2). 

5 13 13 l 

5 13 l 

5 13 13 v 

8 Ñ 8 m 

8 Ñ 8 e 

15 v 9 9 

15 u Ñ Ñ 

ï 

5 13 v 

8 Ñ m 

8 Ñ e 

15 v Ñ 

15 u Ñ 

ï 

Ñ 

Ñ 

s 

v 

u 

u 

l 

v 

m 

e 

. 

s u 9 9 

s u Ñ 

Test of (4.) with ReLabel 

m n w n 

n l e w 

v e m w 

88m, n, w, n


Ñ 

Ñ 

s 

v 

u 

u 

l 

v 

m 

e 

ï 

Ñ 

Ñ 

s 

v 

Ñ 

Ñ 

l 

v 

m 

e 

. Following this path, further reductions are possible: 

Ñ 

Ñ 

s 

v 

Ñ 

Ñ 

l 

v 

m 

e 

ï Ò, with the figure 

Ñ 

Ñ 

s 

v 

Ñ 

Ñ 

plus the reduction of the blanks and finally the equivalence 

of s and v by Rule1 we get the terminal state Ò. 

l 

Ñ 

And by a self-application of Rule1 onto the figure 

v 

m 

it reduces it via 

Ñ 

Ñ 

to the terminal state Ò. 

e 

Ñ 

As a result we have the situation that by applying the morpho-rules consequently what includes its 

self-application, all MorphoGames are re-deducible to a terminal state Ò or ‡. That is, all MorphoGames 

terminate in the final state Ò or ‡. 

But this approach makes sense only for the final steps of the difference-theoretic concept of the 

MorphoGame. 

A second run 

A second approach to the previous configuration of the Board for the run one is given by the following 

15 steps with 3 patterns left as shown by the resulting constellation of the board by run two. 



n l n u u l u l s s l 









15 7 Ñ Ñ Ñ Ñ Ñ Ñ 7 14 15 

15 7 Ñ Ñ Ñ Ñ Ñ Ñ 7 14 15 

9 Ñ Ñ Ñ Ñ 9 u 11 11 6 6 

9 Ñ Ñ Ñ Ñ 9 n Ñ Ñ Ñ Ñ 

13 12 10 Ñ Ñ 10 12 11 11 Ñ Ñ 

13 12 10 Ñ Ñ 10 12 n s Ñ Ñ 

4 Ñ Ñ Ñ 4 5 Ñ 5 v Ñ Ñ 

4 Ñ Ñ Ñ 4 Ñ Ñ Ñ 14 Ñ Ñ 

13 3 3 13 v 5 Ñ 5 14 6 6 

1 Ñ 1 Ñ 2 Ñ Ñ Ñ Ñ Ñ 2 

1 Ñ 1 13 2 Ñ Ñ Ñ Ñ Ñ 2 

1. = n l e 

v e m 

ï @D : vertical

2. 

l v m n e w v 

s l l w v l m 

ï @D : vertical 

In contrast, the horizontal interpretation of H2.L doesn' t hold. 

88l, v, m, n, e, w, v


v 

m 

n 

v 

n 

13. = 

4 

4 

w 

w 

ï 

m 

m 

w 

w 

ï @D 

m 

14. = z z 

u 

u 

ï @D 

15. = n m 

n 

w 

ï @D 

This final constellation cannot be resolved according the rules. 

Hence, the score is (15,3). 

Ñ 9 u 11 11 

Ñ 9 n Ñ Ñ 

Ñ 10 12 11 11 

Ñ 10 12 n s 

4 5 Ñ 5 v 

4 Ñ Ñ Ñ 14 

ï 

v 

u 

n 

n 

Ñ 

s 

v 

. 

v 5 Ñ 5 14 

2 Ñ Ñ Ñ Ñ 

2 Ñ Ñ Ñ Ñ 

Self-application of rules 

A further reduction is possible with the idea of a self-application of the difference rules. Again, this 

step makes sense only after the difference-oriented run is exhausted. Applied from the beginning 

would ruin the game. 

v 

u 

n 

n 

Ñ 

s 

v 

ï 

v 

n 

Ñ 

s 

v 

ï v n ï 

Strategies for morphograms 

The two examples show clearly the strategy how to detect and separate morphogrmmatically similar 

patterns and how to eliminate them. 

First, a pattern has a clear frame which is defined by its environment. 

Hence two or more patterns that are similar must have the same horizontal and vertical frame. The 

frames are defined by their local Width and Height. 

A frame is separated, horizontally and vertically, from its environment by different other not overlapping 

patterns. 

board 

1. 

z z z m 

n l e 

v e m 

w 

w 

ï 

z z z m 

n l e 

v e m 

w 

w 

: part of the board


2. 

z z z m 

n l e 

v e m 

w 

w 

horizontal z z z m ëvertical 

w 

w 

environments of the pattern 

n l e 

v e m . 

3. 

n l e 

v e m : 

horizontally separated pattern by environments w w and z z z m . 

4. 

n l e 

v e m ï n v = l e = e m ¹≠ w w 

: vertically, reduction by Rule3, with 

environments w w and z z z m . 

n 

v = l e = e m ï @D. 

5. n l e w = v e m w : horizontal reduction by Rule3, including w . 

6. Horizontal strategy with additional separation criteria 

board 





board 



n l e w 

v e m w 

l v m n e w v 

s l l w v l m 

n l e w 

v e m w ; l 

s ; v m n e w v 

l l w v l m 

horizontal separation of 

n l e w 

v e m w by l s . 

Program-assisted recognition of patterns with ReLabel 

Program-assisted recognition of patterns 

The cognitive training necessary to play MorphoGames might be supported by some simple but 

helpful programs. They might be implemented as tools into the game.


ReLabel@L_ListD := L ê. 

Map@Ò@@1DD Ø Ò@@2DD &, Transpose@8DeleteDuplicates@LD, Range@Length@Union@LDDD


n l e w = v e m w ï @D 

ReLabel[{n, l, e, w, l, v, m}] ¹≠ ReLabel[{v, e, m, w, s, l, l}] 

{1, 2, 3, 4, 2, 5, 6} ¹≠ {1, 2, 3, 4, 5, 6, 6}, 

Therefore these patterns cannot be eliminated by the existing rules. 

n l e w l v m ¹≠ 

v e m w s l l 

A further separation beyond n l e w = v e m w implying 

n l e w l v m ¹≠ v e m w s l l 

or more elements, stops with the double occurrence of the element "l". 

Also the identity of the elements doesn't count, their order is of relevance. 

Numeric presentation of a MorphoBoard 

2 5 4 2 1 5 3 6 5 3 3 

2 4 6 3 1 5 2 1 2 5 5 

4 1 2 6 2 5 5 2 6 6 6 

1 3 6 5 3 4 6 6 6 3 1 

2 4 6 2 3 4 4 2 3 6 6 

2 2 5 4 6 1 5 4 4 6 6 

4 1 6 5 2 5 5 6 5 5 3 

2 4 5 2 6 6 2 1 5 1 1 

6 2 1 2 3 2 5 3 5 2 2 

1 6 3 4 4 4 6 6 4 1 6 

5 3 6 1 5 3 1 2 1 3 2 

1 1 

1 

2 2 is not accepted because it has a prolongation of 1 in 2, 

1 6 

1 

1 

hence is not separated or the same as the neighbor 2 

6 

. 

ReLabel@82, 5, 4, 2, 1, 5, 3, 6, 5, 3, 3


Definition of the environments of patterns 

Classical situation 

Positions of colored tiles are given by a succession of calls to the pseudorandom number generator. 

(Papegay ) 

Visualizing the Board 

We are now able to initialize the game given the vlues for its Heights, Width and Patterns (colors). 

Choosing some colors, we can define the function View for a nicer display of the board. (ibd) 

Transition function 

“To deal with corner and boundary situations, we define the BoardValue function to access the values 

of the board. It returns -1 if the arguments for location are outside the bounds of the board. This 

allows us to ignore the boundaries of the board when considering the neighbors of a location.” (ibd) 

Full Mathematica Program for the Board Game MHaki by Yves Papegay 

Needs@"GraphUtilities`"D 

Needs@"GUIKit`"D 

NewGame@D ê; Not@NeedRandomnessD := HBoard = InitBoard@D; InitPlay@D;L 

NewGame@s_: SeedD := HSeed = s; SeedRandom@sD; Board = InitBoard@D; InitPlay@D;L 

NewGame@s_: SeedD := HSeed = 7; SeedRandom@7D; Board = InitBoard@D; InitPlay@D;L 

NewGame@"new"D := HSeedRandom@D; Seed = Random@Integer, 31 991D; 

SeedRandom@SeedD; Board = InitBoard@D; InitPlay@D;L 

InitBoard@D := Table@InitPosition@i, jD, 8i, Height


Function@x, BoardValue@xD ã BoardValue@First@yDDDD &, yDDDD, 8p


NBPlayGame@"new"D 

With a Parameter Interface (to do) 

H* Frame for Board *L 

Width = 22; 

Height = 11; 

Patterns = Range@6D; 

NeedRandomness = True 

H* Colors *L 

HMakiColors = 8Green, Blue, Red, Yellow, White, Pink< 

H* HMakiColors@@x+vDD *L 

View@D := Show@Graphics@Raster@Transpose@Map@Reverse, Transpose@BoardDDDD ê. 

x_Integer :> HMakiColors@@x + 2DDDD 

H* FaceNeighbours@p_D *L 

FaceNeighbours@p_D := 

Map@Plus@p, ÒD &, 880, 1


Width = 20; Height = 15; Patterns = Range@5D; NeedRandomness = True; PixelSize = 12; 

Clear@ScoreD 

InitScore@D := Score = 80, 0, Width Height< 

UpdateScore@D := Module@8n = Width Height - Length@Select@Flatten@BoardD, Ò == 0 &DD


In the simplest case of a MorphoGame it starts with a parallelism by the tuples of similar neighboring 

values. 

BoardValue@posD = 888i, j


That is: 

a = mg 

Hence, in ML: 

b iff ReLabel(a) == ReLabel(b). 

fun teq a b = (tnf a = tnf b); 

And more in the sense of differences of E=equal and N=nonequal: 

fun teq a b = (ENstructure a) = (ENstructure b); 

Difference ε/n-notation of morphograms 

The fact that the presentation of the morphograms by specific elements is arbitrary has to be considered 

as crucial. Therefore, not the elements are determining the morphic patterns but the differences 

between the elements. 

This is well depicted for the example [Ê ‡ Ê]. 

Ê ‡ Ê : morphogram 

\ê \ê 

1. n n 3. : ε ê n - structure 

\ê 

2. ε 

A useful notation is given with the matrix of the ε/n-structures. 

Ê Ê Ê Ê Ê ‡ Ê ‡ Ê Ê ‡ ‡ Ê ‡ Á 

ε - 

ε 

ε 

ε - 

n 

n 

n - 

ε 

n 

n - 

n 

ε 

n - 

n 

n 

Example for the ε/n-structures of 

z 

w 

v 

m 

l 

z 

z 

w 

v 

= n - 

n n , m 

l 

z 

= n - 

z 

n n , hence w 

v 

mg 

= 

m l 

z 

. 

With ReLabel: 

ReLabel@L_ListD := L ê. 

Map@Ò@@1DD Ø Ò@@2DD &, Transpose@8DeleteDuplicates@LD, Range@Length@Union@LDDD


iê j 1 2 3 4 5 6 7 

1 e z m w e n m 

2 l n u u l u l 

3 u z 8i, j< 8i, j + 1< l n v 

4 w u 8i + 1, j< 8i + 1, j + 1< n m z 

5 w z s v z m n 

6 l e l u e l u 

7 s v l u v z m 

iê j 1 2 3 4 5 6 

1 e z m blank w m 

2 l n u Ñ u l 

3 u z 8i, j< blank 8i, j + n< v 

4 w u 8i + 1, j< blank 8i + 1, j + n< z 

5 w z s Ñ v n 

6 l e l Ñ u u 

7 s v l blank u m 

BoardValueMorpho@8l1_, c1_, l2_, c2_


Bval@BoardValueMorpho@Hi, j - 1L, Hi + 1, j - 1LDD 

mg 

= 

val@BoardValueMorpho@Hi, j L, Hi + 1, j LDD, 

mg 

= 

val@BoardValueMorpho@Hi, j + 1L, Hi + 1, j + 1LDDF 

mg 

¹≠ 

val@BoardValueMorpho@Hi, j + 2L, Hi + 1, j + 2LDD 

short : 

8i, j + 1< 

8i + 1, j + 1< 

8i, j + 2< 

8i + 1, j + 2< 


mg 

¹≠ 


Horizontal environment, upper 

8i - 1, j - 1< 8i - 1, j< 8i - 1, j + 1< 

8i, j -1< 8i, j< 8i, j + 1< 

val@BoardValueMorpho@Hi, j - 1L, Hi , jL, Hi, j + 1LDD 

¹≠ mg 

val@BoardValueMorpho@Hi - 1, j - 1L, Hi - 1, jL, Hi - 1, j + 1LDD 

MorphoBoard Games in the Framework of Graphematics 

The distinction between classical and morphogrammatic board interpretations and rules motivates to 

involve additionally to the Stirling games of MorphoGames a group of other graphematic systems 

(structurations). 

Different paradigms 

An elementary group of not mixed approaches or paradigms is listed as: Stirling, Mersenne, Brown 

and Leibniz systems. 

Stirling structurations are the domain of morphogrammatics and therefore of MorphoGames. 

Leibniz structurations are the domain of identity based systems of abstract logical, arithmetic and 

semiotic calculi, therefore of classical board games. 

Brownian and Mersennian structurations are two non-orthodox systems that are not genuinely morphogrammatic. 

The additional structurations are becoming relevant for game theory and games if they are set into a 

interactional context that involves parallelism. 

Brownian games are commutative, while Mersennian games are iteration invariant. 

Leibniz games are special cases of such a parallel setting: they collapse with their neighbor systems.


To define a reasonable board game in the framework of Brownian, Mersennian and Stirling structurations, 

a simple parallelism of the path (steps) of the game is of necessity. 

Tabularity vs. linearity 

With this, the emphasis on the tabularity of the game, with its board and its planar rules for the 

distinction of patterns, environments and successions, a new approach, compared to the previous 

studies of graphematic calculi, is promoted. 

First, common to all games is that they need a workspace (board) on which parts are selected for 

manipulation by the rules typical for the chosen rationality (kind) of the game. The kind of the game 

defines the characteristics of the elements (operands) for the applied rules (operators). 

Second, a reasonable game has a beginning and an end. Hence it is ruled by the initial and final 

conditions of the game. 

The mentioned categories are stable. There is no interplay between the categories. Between board 

and part, elements and rules their is no interplay. A board is a board and a part of a board is not the 

board. 

Metamorphic games where the basic categories are involved in complex interplays are possible only 

as a multitude of discontextural games. The shall be called poly-Games. 

General frameworks 

Systems 

Leibniz 

a a b b 

a b a b 

Stirling turn 

Mersenne 

Stirling 

á ä 

a a b 

a b a 

ä á 

a a 

a b 

a a b 

a b b 

Brown 

Pascal 

á ä 

Brown ¬ Stirling Ø Mersenne 

ä ¯ á 

Leibniz 

types\values aa ab ba bb combinatorics 

Leibniz aa ab ba bb m n 

Mersenne aa ab ba - 2 n - 1 

Brown aa ab - bb J n + m- 1 

N 

Stirling aa ab - - ⁄ M 

k=1 

S Hn, kL 

n 

http://memristors.memristics.com/Handouts/Kindergarten%20and%20Differences-Handouts.html 

http://memristors.memristics.com/Kindergarten%20and%20Differences/Kindergarten%20and%20Diff 

erences.html 

Leibniz Games (Identification) 

Graaphematic identity rules 

‡ ‡ ¹≠ ‡ 

‡ ¹≠ Á 

Wording 

Two elements are not equal one element. 

Different elements are different and not equal.


Board frames 

" i, j œ Board HHeigth i, Width jL : 

val@BoardValue@Hi, jLDD id 

= 

H* vertical *L 

val@BoardValue@Hi, jLDD = id 

val@BoardValue@Hi + 1, jLDD, 

valBBoardValue@Hi, j + 1LD, H* horizontal *L 

val@BoardValue@Hi, jLDD = id 

valBBoardValue@Hi, j + 1L, Hi + 1, jL, Hi + 1, j + 1LD, 

with = id œ Pos1 

Leibnizian game rules 

r1: 

n 

n 

n 

fl [] : vertical 

r2: n n n fl [] : horizontal 

r1.2: 

Ñ n Ñ 

n n n 

Ñ n Ñ 

fl [] : mixed 

Grid@ 

88Item@" ", Background -> White, Frame -> TrueD, 

Item@n, Background -> Blue, Frame -> TrueD, 

Item@" ", Background -> White, Frame -> TrueD Blue, Frame -> TrueD, 


Item@n, Background -> Blue, Frame -> TrueD White, Frame -> TrueD, 


Item@" ", Background -> White, Frame -> TrueD











Mersenne Games (Differentiation) 

The basic graphematic rules for the Mersenne differentiation calculus 

Á ‡ ¹≠ ‡ Á 

Á Á = ‡ ‡ 

The basic rules of the calculus of differentiations 

Rule 1. () () = Ø 

Rule 2. (()) = () 

3. Substitution rules 

Wording 

Rule1: A differentiation between 2 differentiations is an absence of a differentiation. 

Rule2: A differentiation of a differentiation is a differentiation. 

In colors 

Rule1. ‡ ‡ = Ø 

Rule2. ‡ = ‡ 



val@BoardValueMersenne@HHi, jL, Hi + 1, jLLDD 

Mers 

= 

val@BoardValueMersenne@HHi, j + 1L, Hi + 1, j + 1LLDD, 

with Mers = œ Rule1, Rule2 

Mersennian game rules 

Rule1 : Vertical HserialL 

z 

z 

ï @D : Rule1, 

Applications of Rule1 

l 

u 

l u fl l l 

fl @D 

Rules2: Horizontal (parallel) 

z z ï z : Rule2 

Applications of Rule2 

z z z ï z : Rule2


e z w 

e z w 

s s l 

ï 

Ñ s l 

: Rule2, environment 

n w v 

n w v 

First steps of a run for a Mersennian Game 












n z l v l s z s l Ñ n 

m e z m w e n m e Ñ w 

Ñ l n u u Ñ u l s s l 

Ñ u z l n Ñ n v n w v 

v Ñ u w m n Ñ z z z m 

m Ñ z s v z Ñ n s e Ñ 

Ñ l e Ñ Ñ e l u v z m 

Ñ s v Ñ Ñ v z m Ñ w l 

m n w n v z n s Ñ v z 

n l e Ñ l v m n e w v 

v e m Ñ s l l w v l m 

Vertical 

Rule1 : 

l 

l 

, 

u n 

u , n 

fl @D 

Rule1 : 

m 

m , 

w z 

w , z 

fl @D 

Horizontal, Rule2 

e e fl e , 

s s , l l , u u , 

z z z 

Brownian Games (Distinction) 

The basic graphematic rules for the Brownian distinction calculus 

Á ‡ = ‡ Á 

Á Á ¹≠ ‡ ‡ 

Basic rules for the Brownian distinction calculus based on the graphematic rules. 

Rule 1. () () = () 

Rule 2. (()) = Ø 

3. Substitution rules


Rule 1. () () = () 

Rule 2. (()) = Ø 

3. Substitution rules 

Wording 

Rule1: A distinction of 2 distinctions is a distinction. 

Rule2: A distinction of a distinction is no distinction. 

In colors 

Rule1. ‡ ‡ = ‡ 

Rule2. 

‡ = Ø 



val@BoardValueBrown@HHi, jL, Hi + 1, jLLDD 

Brown 

= 

val@BoardValueBrown@HHi, j + 1L, Hi + 1, j + 1LLDD, 

Brown 

with = œ Rule1, Rule2 

Examples 

The genuine Brownian rules might be translated into the two rules: 

Brownian rules : 

Rule1 : 

w 

w 

ï w : vertical, serial 

Rule2 : w w ï @D : horizontal, parallel 

Graphematic rules : 

w 

w 

w 

v 

w 

w 

v 

w 

ï w w ï@D : vertical, horizontal 

ï @D : vertical 

l 

u 

l u ï u u 

l 

l 

: vertical 

l 

u 

l u 

ï @D : horizontal 

With horizontal chain and environment 

e z w 

s s l 

n w v 

ï 

e z w 

Ñ Ñ l 

n w v 

Direct Brownian game rules 

Rule1 

Vertical (linear) setting. Rule1 is sufficient to deal with uni-linear events selected from the board. 

z 

z 

ï 

z 

Rule2 

Has no corespondence in a uni - linear setting. HOverlapping is excludedL 

Horizontal (parallel) setting. 

Rule2 demands for a planar (tabular) definition of the workspace. This is realized by a parallel setting 

of the event chains.


z z z ï z 

s s l ï l 

First steps of a run for a Brownian Game 












n z l v l s z s l Ñ n 


Ñ l n u u Ñ u l s s l 


v Ñ u w m n Ñ z z z m 


Ñ l e Ñ Ñ e l u v z m 

w s v l u v z m Ñ w l 


n l e Ñ l v m n e w v 


Comparison of Brownian and Mersennian games 

Games under Mersenne and Brown rules are complementary. There are also dual games for each 

type of games. 

Stirling Games (Difference) 

Stirling games have a first representation by MorphoGames. 



val@BoardValueMorpho@HHi, jL, Hi + 1, jLLDD 

mg 

= 

val@BoardValueMorpho@HHi, j + 1L, Hi + 1, j + 1LLDD, 

with = mg œ Rule1 - Rule4 

Elementary rules for morphoGames: Frame rules 

Very first elementary rules for morphoGames are given by the rules for frames only. This approach is 

abstracting from the specific rules (or logic) of the constellations (patterns, morphograms), and is 

just taking the abstract frames, independent of their internal structure or coloring, defined by their 

width and heights only and separated by their (empty) environment into account. 

Frame rule1


Frames of the same size which are separated, vertically or horizontally or both, by an empty environment 

can be eliminated. This applies for single patterns too. 

This might happen automatically by a first run. And then by pattern construction by the frame rule2. 

Frame rule2 

The composition of constellations (patterns) can be manipulated in all directions: the horizontal, 

vertical and the diagonal. 

Frame rule3 

The minimum of adjacent (adjacency) of the squares have to be chosen to define the game. 

The minimum adjaceny of the neigborhod of frames is obviously 1. 

Adjacent by Emily Lanie 

Example 

First steps for a morphFrame game 

Example of the first step of a morphoFrame Game with an adjency of 1. 

ï 

First steps of a run for a Stirling Game 


m e z m w e n m e z w











vertical, first steps 

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ z n 

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ z w 

n Ñ Ñ Ñ Ñ l Ñ Ñ Ñ Ñ Ñ 

n Ñ Ñ Ñ Ñ l Ñ Ñ Ñ Ñ Ñ 

v w Ñ Ñ Ñ Ñ m Ñ Ñ Ñ Ñ 

m w Ñ Ñ Ñ Ñ m Ñ Ñ Ñ Ñ 

w Ñ Ñ Ñ Ñ Ñ Ñ Ñ v Ñ Ñ 

w Ñ Ñ Ñ Ñ Ñ Ñ Ñ u Ñ Ñ 

Ñ Ñ Ñ n Ñ Ñ Ñ Ñ u Ñ Ñ 

Ñ Ñ Ñ w Ñ Ñ Ñ Ñ e Ñ Ñ 

v e m w Ñ Ñ Ñ Ñ v Ñ Ñ 

vertical, final first steps 

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ n 

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ w 

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ 


v Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ 

m Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ 

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ v Ñ Ñ 


Ñ Ñ Ñ n Ñ Ñ Ñ Ñ Ñ Ñ Ñ 

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ e Ñ Ñ 

v e m Ñ Ñ Ñ Ñ Ñ v Ñ Ñ 

Horizontal, first steps 












Test with ReLabel 

EqualBReLabelB: n z l v l s z s l z n >F, 

ReLabelB: m e z m w e n m e z w >FF 

ReLabel@8n, z, l, v, l, s, z, s, l, z, n


ReLabel@8n, z, l


6 3 4 4 6 3 4 5 2 3 1 1 4 3 5 4 1 6 4 1 3 2 

5 1 4 2 6 5 6 2 5 4 2 2 4 1 3 6 1 3 1 4 1 5 

5 1 4 1 5 2 3 3 4 2 6 5 3 3 5 6 6 5 6 4 6 2 

2 6 1 4 5 2 4 6 1 4 1 6 5 1 1 2 4 3 3 1 6 3 

6 3 6 3 5 3 1 1 3 5 5 6 2 3 3 3 3 4 6 4 6 6 

1 3 2 1 6 1 5 2 5 6 6 3 4 6 3 3 4 2 6 5 5 4 

4 5 3 1 2 1 2 3 1 1 6 1 5 5 3 5 2 1 5 4 6 2 

2 4 6 3 3 4 4 4 6 4 6 2 5 3 2 3 4 4 5 6 1 2 

2 6 5 1 3 6 1 2 4 1 1 5 3 2 5 6 1 3 2 3 1 1 

4 2 4 1 2 4 4 6 6 1 2 1 4 1 3 2 3 1 2 6 6 3 

3 1 2 2 4 4 1 5 5 4 4 2 6 6 4 6 3 1 3 6 5 4 

2 2 5 1 6 6 2 6 2 6 1 4 5 1 1 1 6 4 2 4 5 1 

5 5 6 1 4 6 1 3 1 3 3 4 1 5 2 6 3 1 4 3 3 2 

1 3 5 5 1 3 5 3 5 5 5 4 2 3 5 4 6 3 4 3 6 4 

4 1 2 3 2 6 2 4 1 2 2 5 6 4 3 5 5 1 5 1 2 6 

Vertical 

3 

6 

6 

5 

5 

2 

2 

3 

= palin 

3 

1 

1 

3 

1 

1 

2 

1 

1 

5 

¹≠ palin, 

1 

1 

3 

1 

1 

2 

1 

1 

= palin, 

1 

6 

2 

3 

6 

5 

, 

3 1 

3 1 

6 4 

3 1 

¹≠ palin, but morpho - equivalent 

Horizontal 

4 4 4 

6 1 2 

= palin1 »» palin2, palin1 ¹≠ mg palin2 

3 1 1 4 

4 2 2 4 

= palin »» palin 

4 4 1 5 5 4 4 2 6 6 = palin 

1 5 5 4 4 2 = palin 

2 4 4 6 6 1 = palin 

Palindrome tests


-ispalindrome Htnf@3, 6, 6, 5, 5, 4, 4, 3DL; 


-ispalindrome Htnf@3, 1, 1, 3, 1, 1, 2, 1, 1, 5DL; 

val it = false : bool 

-ispalindrome Htnf@1, 1, 3, 1, 1, 2, 1, 1DL; 


-ispalindrome@1, 2, 3, 1D; 


-ispalindrome Htnf@1, 6, 2, 3, 6, 5DL; 


-ispalindrome Htnf@3, 1, 1, 4DL; 


-ispalindrome Htnf@4, 2, 2, 4DL; 


-ispalindrome Htnf@6, 6, 1, 5, 5, 4, 4, 2, 6, 6DL; 


- ispalindrome Htnf@1, 5, 5, 4, 4, 2DL; 


- ispalindrome Htnf@2, 4, 4, 6, 6, 1DL; 


Augmenting games with new elements 

Up to now, the scope of the patterns, i.e. the number of elements on the board had been, once set, 

constant. 

From a classical, identity-oriented point of view, an addition of elements is straight forwards and 

simple. 

Morphogrammatics is, again, not based on identical elements, but on patterns, i.e. morphograms. 

Hence, an extension of the scope of the patterns has to take the specific laws (properties, characteristics) 

of morphogrammatics into account. 

This leads to a morphogrammatic concept of addition (coalition). And in more complex situation to a 

morphogrammatic approach to multiplication (cooperation). 

Coalition 

Coalition in morphogrammatics is a retro-grade recursive and super - additive operation. 

Hence, a morphogrammatically adequate extension of the range of elements is not abstract but retrograde 

defined by the ‘elements’ already in the game. 

Steps towards a program for MorphoBoard Games 

FaceNeighboursMorpho@p_, q_D := 

Map@Plus@p, q, ÒD &, 

880, 1

SpotMorpho@p_, q_D := FixedPoint@ 

Function@y1, Union@y1, Apply@Union, Map@Select@FaceNeighboursMorpho@ÒD, Function@ 

x1, BoardValueMorpho@x1D ã BoardValueMorpho@First@y1DDDD &, y1DDDD, 8p

Towards a new Paradigm of Board Games

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