[The following was transcribed directly from scans of the published pammplet, regardless of existing typographical errors.] [Cover.jpg] CHESSICAL CUBISM or CHESS IN SPACE By V. R. Parton [Cubic1.jpg] CUBIC CHESS Before the explanation is begun in this booklet of the idea of Space Chess, it will be proper to deal first with the basic types of chessmen. Under my system of classification pieces in chess although having a rather wide variety, can conveniently be divided into four general and basic types which are described in the following paragraphs. A chessman able to move from its position to one adjacent, but having no other sort of move on its board, is the type of man whcih I term Stepper. On the ordinary chessboard, because of two kinds of adjacency for squares, there are two distinctive types of stepping chessmen in consequence (A) The Angular or Diagonal Stepper is one which goes through an angle of its square into the neighboring square. This piece has been called by its Muslim chess name of Fers or Firzan (B) The companion piece, the Lateral or Orthogonal Stepper is one which goes across a side its square into the adjacent square and is called by its Muslim name of Wazir. The Hopper moves across an adjacent position, whether this is vacant or occupied, into the next position beyond in the same direction. The Hopper causes no disturbances to any piece, friend or foe, over which it may chance to hop. On the ordinary chessboard the Angular or Diagonal Hopper passes across an adjacent cornerwise square, whether occupied or vacant, into the square immediately beyond in the same diagonal direction. This piece is known by its Muslim name of Alfil. The piece making its hopping move along ranks and files is the Orthogonal Hopper, and is known by its Muslim name of Dabbaba. The Skewhopper may be regarded as making a change in its direction of hopping as it is passing across the adjacent position. On the ordinary chessboard the Knight or Horseman is of course the skewhopping piece. The move of the Knight can also be regarded in another way, as being formed one step diagonally followed by one step orthogonally, or these two steppings taken in reverse order; such combining of two steps is absolutely one movement. The Runner is a piece which can change its position by running through any number of unoccupied positions in a straight line, but can never pass across occupied positions. On the ordinary board the piece which can run diagonally through empty squares is naturally the Diagonal Runner, or Bishop; and the piece running orthogonally across unoccupied squares is the Orthogonal Runner or Rook. (The general Runner is the Queen) The Flier, in contrast to the Runner just described, can change its position in a straight line only by flying across one or more men in its path. On the ordinary board there can be Diagonal and Orthogonal types of Fliers, The Orthogonal Flier, limited to flying over one piece, friend [Cubic2.jpg] or hostile is called the Cannon. On the ordinary board the King is the general type of Stepper, but general Steppers can of course be just simple pieces, without any role of Kingship. In its essential character geometrically the ordinary chessboard is really a large, basic and enclosing square partitioned into equal sized small squares by lines parallel to the sides of this large enclosing square. The formation of the Cubic Board can be explained in rather similar manner to that of the ordinary chessboard itself; which might be termed the "Squaric Board." In its geometric character, the Cubic Board consists of a large (basic) cube which is partitioned into equal-sized cube-shaped compartments or cells, by means of planes parallel to the faces of that large enclosing cube. This Cubic Board may conveniently be regarded as formed by layers (levels) of cellls. all layers having of course the same arrangements of cells. For a simple method of notation, the layers or levels of cells are denoted by capital letters A,B,C.... upwards, level A being the lowest of the Cubic Board. In each layer "files" of cells are denoted by small letters a,b,c.... left to right from white's point of view, and "ranks" of them are numbered 1,2,3.... forward from white's side of the Cubic Board. The cell specified by"Aa1" in this notation is the lefthand lower corner cell on white's facing side of the Cubic Board, whatever its scale. Cells that are situated directly above the same cell in the lowest layer (A) of the Cubic Board form an upright (straight) set or pile of cells, such a pile is called a "column" A cubic cell has contact with its adjacent cells in one of three distinct ways; this may be contact by face (side), by edge or by corner point (vertex); consequently there are three forms of stepping out of a cubic cell to be considered. An orthogonal step is one made through a face of a cubic cell into the adjacent cell, and a diagonal step is one made through an edge of a cell into the adjacent. (These terms are of course intended to correspond properly with the same terms on the ordinary chessboard.) The third distinctive form of stepping is a vertexal steo which is made through a corner point (vertex) of a cubic cell into the adjacent joined by that point. (A vertexal step involves a change of layers for a Cubic piece so moving.) Because a cubic compartment completely surrounded by like cells has 26 neighboring it by face, edge and vertex altogether, the general Stepper in the Cubic Board can step in 26 directions from its cell. [Cubic3.jpg] if interior in this Board. The King in the game of Cubic Chess has the move of the general Cubic Stepper, just as he is the general Stepper on the ordinary chessboard. Under my classification, a Cubic Runner is a piece able to continue moving through empty cells which lie in the same direction or straight line. In the Cubic idea there are naturally three dis- tinctive types of Runners, namely, the Orthogonal Runner which is the Cubic Rook, the Diagonal Runner or Cubic Bishop, and the Vertexal Runner called the Unicorn usually. In the following description the Cubic Board concerned is the 6 x 6 x 6, corresponding in scale with a chessboard of 6 x 6 squares. (a) The Cubic Rook (Orthogonal Runner) moves in a straight line through the faces of the cells, that's moves along ranks, files and columns of cells. For example, situated in the corner cell Aa1, the Cubic Rook moves along the rank Aa1 - Af1, or along the file Aa1 - Aa6, or in the [deletion] column Aa1 - Fa1. From an interior cell this Rook has six directions in which he may move, and four from an edge cell. The Cubib Rook can operate over all the cells of any Cubic Board, as the ordinary Rook over the ordinary chessboard. (b) The Cubic Bishop (Diagonal Runner) moves in a straight line through the edge of cells. For example, situated in cell Aa1, the Cubic Bishop may move along the line Aa1 - Ff6, (up the side facing white); or along the diagonal Aa1 - Fa6, (up the left side of the Board for white), or in the layer A itself along the line Aa1 - Af6, operating over the whole space of this Cubic chessboard. (c) The third distinctive Runner in the Cubic is the Vertexal Runner, to which the name Unicorn is usually given. This Cubic piece moves in a straight line through corner points (vertex) of cells. For example, situated in cell Aa1, this piece may move along the line Aa1 - Ff6. The Unicorn (or Vertexal Runner) can move in eight directions from an interior cell, but only two from an edge cell. In moving the Unicorn is of course changing layers or levels. (d) The general Cubic Runner has naturally the complete power to move along any of the orthogonal diagonal and vertexal directions from its position. The Cubic Queen is given the move of this general Cubic Runner The Cubic Knight or Horse has a skew movement which consists of an orthogonal step followed by a diagonal step; in other words, the Cubic Horse first steps through a face of its cell into one adjacent from which it steps through an edge into the second cell. Even if the intermediate cell is occupied, the move of this Cubib Horse is not obstructed and not prevented. It will be seen that the move of the ordinary chess Knight has simply been transferred into space. [Cubic4.jpg] For example, situated in cell Aa1, the Cubic Horse can go to Ca2 and Cb1, to Ba3 and Bc1, as well as to Ab3 and Ac2 on its own level A. In its actual construction my own Cubic 6 x 6 x 6 Chess Board is formed of six chess boards of 6 x 6 squares, which are placed one above another and held in such position by supporting uprights set at the sides. Each cubic cell is thus represented by some particular square of one of these six (reduced size) chessboards, which correspond to the layers of the Cubic Board. For the ordinary form of my Cubic Chess game, a player has a force of 18 pieces, namely, King and Queen, Bishop and Rook, Horse and Unicorn, with twelve pawns. In the initial arrangement the pieces are placed only one the two lowest layers, A and B. The initial formation in the lowest level, A, is the following (empty squares = O.) ------------ R H U B Q K Black P P P P P P O O O O O O ----------- ----- O O O O O O P P P P P P K Q B U H R White ----------- The player's other six pawns occupy his rear rank of layer B above. The pawn step forward in any of the directions (orthogonal, diagonal and vertexal) towards the opposite side; the pawn also captures in the very same directions as it moves. For example,if a white pawn is in cell " C d 2 ", it can step to the nine cells adjacent in front, namely D c 3, D d 3 and D e 3 with ascending step; to B c 3, B d 3 and B e 3 with descending step; and of course to cells C c 3, and C d 3 and C e 3 on its own level C. The pawn can capture a foe occupying one of thes same squares. In the variation " Compulsion Cubic Chess", taking is always compulsory, which rule gives the distinquishing name for this var- iation. If two or more ways of taking occur at the same turn, then the player is free to select which he shall make. The capture of the enemy King gives the victory. This Cubic Board of 6 x 6 x 6 is used for my ideas of Cubic Draughts and Cubic Halna. "Scacia" is the variety of chess game in which the capture of all the enemy pieces is the objective of play, kings being only regarded as ordinary pieces. Taking in Scacia is compulsory. Should [Cubic5.jpg] two or more ways of taking occur at the same turn ofplay, however, freedom to choose between these alternatives of capturing is allowed to the player. One piece only can be takenat a time. TAMERLANE VARIATION OF CUBIC CHESS The game of Tamerlane " Great Chess " is one which I have specially translated into Cubic form for playing on the 6 x 6 x 6 Board. Of enlarged kinds of chess played during the Middle Ages, by far the most well-known and written about is Tamerlane's Chess, for which over thirty chessmen each side are required and a chessboard of 112 squares. The name is due to its being associated with the great Mogul (Tartar) emperor Timur, nicknamed Leak the Lame, living in the 14 century (1337-1405) who was a devoted player of this particular enlargement of chess. For my Cubic idea, the pieces in the player's force retained from the " Great Chess " are the following: two Rooks and two Horsemen (Knights) ; two Dabbaba, two Alfil and two Zurafa, with King and Fers, and pawns. The characteristic piece, Zurafa, in the Tamerlane game has a move which is a cornerwise step followed by and orthogonal (Rook) move of several squares. Quite unlike the Horse, Dabbaba and Alfil, the Zurafa cannot pass across any occupied positions. In the new forms, the Dabbaba is the Cubic Orthogonal Hopper and the Alfil the Cubic Diagonel Hopper; the fers is the Cubic Diagonal Stepper. The pieces are initially arranged on the two lowest levels A and B. The arrangement in the player's rear rank of layer A is left to right ) R - Z - F - K - Z - R with pawns in rank in front; the arrange- ment in the player's rear rank of layer B is D- A - H- H - A- D with pawns in rank before. [Sphinx1.jpg] SPHINXIAN: CHESS Probably at first thoguht any idea of chess played in four dimensions must appear to be nothing more than pure fantasy, or at most just something fanciful of a truly theoretical and abstract nature. The game of Sphinx Chess is my idea of chess in four dimen- sions, being a design for such chess game in a rather simplified form and of quite practical type, fairly easy to understand as regards its basis and it rules of play. The nature of Sphinx Chess is really determined by the characteristics that "define" the Sphinx Board on which it is played. The basis of the Sphinx Board here concerned is a system of nine identical small chess boards arranged in the simple pattern of three-by-three. The "reduced" form of this Board consists of four small boards which are arranged in two-by-two formation, the simplest pattern possible. The principle of Sphinx type Boards is essentially the idea of a board of small boards, which raises up to the next degree as it were, the old customary idea of a board as being formed of squares, and under which the term "small board" becomes the substitute for that word "square" of the definition. The nine component boards which form the Sphinx Board are miniature chessboards of 4 x 4 squares in size, that is quarter chess boards. For a notation easy to understand and apply, the nine small boards are regarded as lettered a - i; and the squares of each of these as numbered 1 - 16 in a straightforward manner, so that each square or position on this Sphinx Board is properly specified by means of such letter-number method of notation. For example, the square termed "g7" will clearly denote square 7 on the small board lettered "g". ---------------------------------//--------------------------------------- [Note from transcriber: For clarification, the following diagrams has been added.] Numbering system of the squares [ 1][ 2][ 3][ 4] [ 5][ 6][ 7][ 8] [ 9][10][11][12] [13][14][15][16] Lettering system of the boards [a][b][c] [d][e][f] [g][h][i] ---------------------------------//--------------------------------------- Those squares which bear the same number are termed quite naturally " corresponding positions or squares" for each in such group has exactly the same placing on its respective small board, as square 16 for example denotes, the right hand lower corner square on any of the nine miniature chess boards. It is through such "corresponding squares" that basic connections between small boards are logically and practically established on the Sphinx type chess board. The following paragraphs deal with the explanation of my interpretation and development of the ordinary chessmen (rook, bishop, queen, pawn, king) into their forms logical to suit the characteristics of the Sphinx Board, and practical for play on such board. (1) The Sphinx rook can move from one small board to another if both are situated vertically or horizontally in the same line of small boards. This rook must go, when so changing small boards, to that square or position on his new board which corresponds with, that is, which has the same number in its notation as the square from which he has quitted on his former board. For example, a Sphinx Rook on square a1 can move to b1 or c1 horizontally, or to d1 or g1 vertically. If the new square is occupied by some enemy piece, this rook can of course, capture the [Sphinx2.jpg] particular foe on that square. Because the Sphinx rook gets from a1 to c1 horizontally by way of square b1, then occupation of that intermediate square (b1) will obstruct and consequently prevent such movement of the rook horizontally from a1 to c1. In like manner, square d1 is situated "between" squares a1 and g1 vertically in the Sphinx idea, so that if it should be occupied, then the rook's path from a1 to g1 would be clearly obstructed by whatever piece might be there. It will be understood that a piece on b2 or b3 would not prevent the rook on square a1 going to c1. (The square b2 is between a2 and c2; and the square b3 between a3 and c3.) ---------------------------------//--------------------------------------- [Note from transcriber: The phrase 'and c3.)' in the original document appeared to read 'and a3.)', or possibly 'and o3.)', because of the typing correction that was made. Through logical deduction, it was determined that the author intended the phrase 'and c3.)' and this was used to minimize confusion.] ---------------------------------//--------------------------------------- On any small board the rook will have his ordinary move. The idea of playing chess on the Sphinx basis means that such a rook must still maintain his horizontal and vertical types of moving when he goes from one small board to another. Such is only strict Sphinx analogy with that orthogonal movement which the rook has on the ordinary chessboard itself. In general that Sphinx idea is also true for any kind of chess- man having a suitably defined movement on the normal chessboard. As far as it is practical the normal move of each particular chessman is simply widened in order to meet the major characteristics of the Sphinx Board. (2) The Sphinx bishop can move from one small board to another that is situated in the same diagonal direction. Such a bishop, like the Sphinx Rook already described, go to that square or position on his new small board which corresponds with, or has the same notation number as his square of departure on his former small board. For example, a Sphinx bishop on square a4 can move to e4 or to i4 diagonally, and can even capture a foe occupying such square. Because this Sphinx bishop gets from a4 to i4 by way of e4 then occupation of that intermediate square or position (e4) would obstruct and therefore prevent the bishop moving from a4 to i4 diagonally. In like manner, a Sphinx bishop situated at c5 can get diagonally to square g5 across the position e5, which is "between" c5 and g5 in the Sphinx idea, but that intermediate square (e5) must not be occupied. On any small board, the bishop will have its customary chess move. It will be noted that a Sphinx bishop operated either on the five small boards a,c,e,g,i, or else on the four middle boards b,d,f,h. In order to fully cover the Sphinx network, a player required four bishops. (3) From what has already been explained about the rook and bishops for chess played on the Sphinx Board, the character of the Sphinx Queen will naturally follow, as she will clearly enjoy the powers of both those Sphinx pieces, so as to conform by strict analogy with the ordinary chess queen, theoretically considered as the piece which is a compound or amalgam of rook + bishop. [Sphinx3.jpg] Like the Sphinx rook, this queen can operate over all squares in the Sphinx network. On any small board the queen will have her normal chess movement. (4) The idea of the Sphinx pawn raises no special question regarding its powers to move and take. First, a pawn advances "one step" with vertical step straightforward, whether on the same small board or to that board directly in front of the one where this pawn is standing. For example, a white pawn situated on square g 13 can advance a step to g 9 on the same small board or to d13 on the board directly before "g". Secondly, a pawn will take with a cornerwise step forward left or right. For example, from square e14 a white pawn could capture some foe either on a14 or on c14, which positions are corresponding squares on the small boards "a" and "c", cornerwise in front of the board "e". On any small board a Sphinx pawn will make its ordinary move or take. A pawn cannot advance two squares consequently there is no taking of pawns "enpassant" in Sphinx Chess. Thirdly, when a white pawn reaches the last rank in a, b, or c, and likewise when a black pawn reaches the rear of g, h, or i, then such pawn advanced will naturally be promoted into a queen or other piece (5) As regards the character of the Sphinx king, naturally on a small board the king will have his usual chess move of stepping one square in any direction around his position. By analogy in Sphinx play, this king will therefore move with a step whether diagonally, vertically or horizontally in direction, from his own small board to any adjacent to it. When the Sphinx king changes his small board, he must go to that square on the new board corresponding with the square which he has just quited on his former board. For example, if the white Sphinx king stands on square h7, then he is able to step to g7, d7, e7, f7, and i7, all which positions are corresponding with his square h7. He can of course, capture a hostile piece standing on such a corresponding square. With the five types of pieces which have now been described, the game of chess can be played in Sphinx form. The only matter really involving some preliminary work is now the actual construction of the Sphinx Board itself on which to play this game. The geometrical character of the Sphinx Board is essentially a field or space of four dimensions. The proof of this may be outlined as follows: The positions of the squares on a small board involve two dimensions or co-ordinates, one for the rank and the other for the file (or column) in which any particular square is situated on its small board. The arrangement of the nine components is itself involved with the remaining two dimensions or co-ordinates, one denoting that column and the other that horizontal line of small boards in which a particular small board is placed in the Sphinx. The idea of Sphinx Boards is really the principle of associating two [Sphinx4.jpg] patterns, where one patternwhich is the form of the identical small boards is embedded as it were in another pattern, namely, the system in which these component boards are themselves arranged to form the Sphinx Board. The basis of the Sphinx idea creates a flat system of flat boards. This particular type of Board is easy to see with the eyes and to under- stand with the imagination, and thus it offers a good practical advantage as regards the very important matter of moving or manoeuvre of pieces on such a board. Moreover, the Sphinx Board is not difficult to construct, and may even be drawn in roughish outline on a suitable piece of stiff paper or cardboard. In the game of Sphinx Chess, the ordinary knight piece is used generally to represent a special sort of chessman which is "hybrid" in its Sphinx character. This particular Sphinx piece is given the name of "centaura"; on any small board the centaura moves and take exactly as the ordinary chess knight, but in order to change positions from one small board to another, the centaura must move or take like the Sphinx Queen. By such Sphinx modification, chess knights are transformed or converted into suitable and practical pieces for Sphinx Chess. This term "centaura" shows the links with the ordinary name "knight or horsemen". For playing on the Sphinx Board, a player has the normal force of sixteen chess pieces, knights representing centauras. The player places his king, queen and two bishops in his central rear small board, a rook and a centaura with two pawns in each of the corner small boards, and four pawns in the central small board "e". Initially the black pieces are arranged as follows: king on square b3 and queen on b2; bishops on b1 and b4; rooks on a 2 and c3; centauras on a3, c2, with pawns on a1 and a4, on c1 and c4, and e1 - e4 ( central small board). Initially the white pieces are arranged: king on square h15 and queen of h14; bishops on h13 and h16; rooks on g14 and i15; centrauras on g15 on a3, c2, with pawns on g13 and g16, on i13 and i16, and on e13 - e16 ---------------------------------//--------------------------------------- [Note from transcriber: For clarification, the following diagrams has been added.] Black Black Black [P][R][C][P] [B][Q][K][B] [P][C][K][P] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] Black [ ][ ][ ][ ] [P][P][P][P] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [P][P][P][P] [ ][ ][ ][ ] White [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ] [P][R][C][P] [B][Q][K][B] [P][C][R][P] White White White B=Bishop C=Centaura K=King P=Pawn Q=Queen R=Rook ---------------------------------//--------------------------------------- For practical purposes the white king may be confined to moving within "h" and the black within "b". The "reduced" Sphinx Board is as already stated, formed of a system of four small boards arranged two-by-two. This may be regarded as the ordinary chessboard partitioned into four (separate) quarterboards of 4 x 4 squares. With the proper Sphinx adjustments mentally made, ob- viously the Sphinxidea can be played on the ordinary board, with the chessmen initially arranged in the ordinary manner. For example the squares QR1, QR5, K1 and K5 are "corresponding positions" as are QB2, QB6, K, Kt2 and K Kt6 for either player. ---------------------------------//--------------------------------------- [Note from transcriber: The last paragraph contains the phrase 'K, Kt2', this was most likely meant to read simply 'K Kt2' or the second rank of the King's Knight file.] ---------------------------------//--------------------------------------- In Sphinx Chess the perpetual check may be claimed as victory. [Sphinx5.jpg] THE COMPULSION SPHINX CHESS VARIATIONS. From the explanations given so far about the character of my Sphinx idea, the reader will have gained an insight into various fundamentalfeatures of this Sphinx Chess. Even with such a satis- factory grasp of the concepts and rules here involved, the beginner has still two new fields of Sphinx play to face,their questions to solve. One problem is how he should open and develop his campaign against the enemy force: this is possibly a case of trials and errors on the grand scale. The essential problem which the novice must face is naturally how to manoeuvre and force checkmate finally against the hostile king in that mysterious, tortuous and cruel labyrinth of the Sphinx Board. The act of combining moves together to create some satisfactory form of logical sequence in play, is obviously harder in Sphinx Chess than in the ordinary game. The radical difficulty to be mastered is how to associate, by Sphinx interelations, plays on nine individual small boards quickly and wisely as one complete and logical scheme of play. Because checkmate in Sphinx involves some awkward new types of planning, it seems preferable that the novice should approach the proper game by first practising his abilities on the two following var- iations. The first variant to which I have given the name Compulsion Sphinx simplifies matter in two ways. It introduces a special rule with regard to capture of enemy pieces. Instead of the customary rule that the player can freely choose between taking a foe and dec- lining to take it, the rule of compulsion is now applied: the player is obliged to capture an emeny piece when he can. If two or more foes should be liable for capture at the same time, however, the player is permitted to make which take he prefers. A further simplification is that in Compulsion Sphinx the idea of checkmate is completely omitted as an element in its basis. Kings hold no special role in this game. Losing his king has no serious con- sequence for a player, who may sacrifice his monarch as readily as he might one of his pawns ot the enemy. The aim in Compulsion Sphinx Chess is to capture all the enemy pieces. An advantage of this variant for the novice is that the power of the hostile queen is partly curbed in play. By reason of being compe- lled to take, thus that enemy queen may be forced into some position for her own destruction perhaps by a mere pawn! Losing Sphinx Chess also applies the rule of compulsion in taking, and similarly treats kings on the same level as other pieces, capturing a king at any time in play being of no special merit. The purpose in this game is to force the opponent to capture all the chess- men in one's own army. The player who cna get rid of his pieces is the winner, not the loser! [Ecila1.jpg] ECILA CHESS. The idea which this chapter describes under the name of Ecila occurred to me a little while after I had designed the games of Sphinx Draughts and Chess. It resembles the Sphinxian idea to some extent bec- ause it applies to ordinary space a principle very similar to that which is applied in the flat ( or plane) for the Sphinxian boards, and as the basis of the movement of chessmen. The general principle of the Ecila idea is that a number of space boards are arranged specially in a simple pattern in ordinary space. As they involve three dimensions and the pattern in which they are arranged is also three-dimensional, the Ecila idea is consequently one involving a space of the sixth dimension. Though a variety of "Ecila Boards" can be designed, my own interest was the playing of Ecila Chess and Ecila Scacia in a form and manner both reasonable and practical. The simple type of Ecila board was therefore taken as the basis for such games. The simplest form of ordinary space chessboa rd is the Cube par- titioned into eight cubic cells, or into 27 such cells if a larger scale of board os required. In its basis the Ecila board (of the type inevented fb r the palying of ches and Scacia) is a system of eight Cubes arranged in the corners of a large "enclosing cube, " the primary or master cube of the system as it might be termed. The four upper Cubes of this Ecila board as seen from above are specially lettered ( A B ) ( C D ); and the four lower Cubes ( E F ) ( G H ) Each of the eight Cubes is itself partitioned into eight cubic cells, the upper group being specially numbered (1 2) and the lower (3 4) group (5 6) (7 8) Should the Cubes be divided into 27 cells numbering of these cells follow the same method; the group of nine cells being numbered 1- 9, the middle group 10 - 18, the lowest group 19 - 27. All cells are thus definitely fixed in position in the Ecila board by means of this notation of letter and numbers, each cell having its wn letter and number "address." This particular Ecila Chessboard or structure, represents the sort of cube in the sixth dimension which corresponds to the cube 2 x 2 x 2 in ordinary space. If the cubes of this Ecila board are each partiti ned into 27 cells however, it represents something in the sixth dimension which may be regarded as intermediate between the "Super cubes" 2 and 3 in that space. [Ecila2.jpg] For practical purposes it becomes necessary that the cells of the eight Cubes are simply represented by squares. Consequently any hor- izontal group of four (or nine) cells, becomes a "miniature chessboard" of four ( or nine ) playing squares. The structure of the Ecila Chessboard is formed of four levels or platforms in the case of the minimum size. As these levels are essen- tially related in pairs, the distance between the second and the third platform is made greater than the distance between the top pair, or that between the lower pair of levels. A platform, square in shape, has a large square section removed from its centre in order to help vision. It also helps in the stress ing of this basic feature, Cubes situated in the corners of the "enclos ing" or "master" cube. At each of the corners of a platform there is marked out a "miniature" chessboard of four squares. Two pairs of colours are used for colouring the squares, white and black, red and green. The white and black are used in the case of the squares representing cells of the Cubes A,D,F,G.; the red and green for those representing cells of the Cubes B,C,E,H. The platforms are supported by the four uprights so placed as to hold the platforms at the middle points of their sides. This allow the "miniature chessboards" at the corners of the platforms to be clear of any sort of obstruction to vision and handling of chessmen situated there. Having now explained the general nature of the Ecila board I next turn to the character of the pieces which I used for the playing of Ecila Chess and Ecila Scacia. One is not likely to use every sort of Ecila chessmen at the same time. ( Some of the pieces are more easy to understand and use than others.) In this idea the three pieces regarded as having the simple basic moves are Rook, Bishop, Unicorn. The moves of other Ecila pieces can be described or defined in terms of these three moves. (1) The Ecila Rook moves parallel to the edges of the "enclosing or master cube" that is, parallel to the edges of the Ecila board generally. A rook situated at A1 can move in his own Cube A to A2, or A3 or to A5 on the platform below. When the Rook changes his Cube he must go to that square or cell in his new Cube which corresponds to ( has the same number as) his cell of departure in his former cube. For example: the Rook at A1 can go to B1 or C1 on the same platform, or to E1 on the third platform. (2) The Ecila Bishop makes an oblique move parallel to the diagonals of the faces of the 'enclosing cube" that is, parallel to the diagonals of the top, base or sides generally of the Ecila structure. A Bishop situated at A1 can go to his own A to A4, or to A6 or A7 below. When the Bishop changes his Cube, he must go to the cell (in his new Cube) corresponding to his cell of departure in his former Cube. For example, the same Bishop at A1 can go to D1 on the same platform, or to F1 or G1 on the third platform. [Ecila3.jpg] (3) The Ecila Unicorn moves parallel to lines joining diametrically opposite vertices of the "enclosing Cube". For example: a Uni- corn at A2 can go to A7 in his own Cube, or to the corresponding cell H2 in the Cube H, which is diametrically opposite to Cub A. Again, a Unicorn at B1 can move to B8 (of his own Cube) or to the corresponding cell G1 in the diametrically opposite Cube G. Comparison of these three Ecila pieces shows that from his cell (or square) the Rook guards six cells, as likewise does the Bishop from his position, but the Unicorn guards only two cells. However, the Rook can reach any cell (or square) of the Ecila board in one or more moves, but the Bishop is limited to operating about on only one quarter of the total number of these cells. the four colours already mentioned are used to distinguish the fields of operation of the four Ecila Bishops, each of these moving on squares of only one colour. The Unicorn in Ecila is very limited in his operation. Of the pieces in the ordinary chess set, the pawns are used for Unicorns in actual play. (Only four Unicorns at most should be used. ) If the Cubes of the Ecila board are 27 celled, this chessmen has more range and value. In explaining the nature of the Ecila Queen two definitions can be given. First the "conventional" description of a Queen is that this piece moves and takes like both Rook and Bishop. Secondly, the Ecila Queen can be defined as the piece moving like any of the three basic Ecila pieces: R,B,U. Another composite Ecila piece is usefully created in the Narwhal (Sea Unicorn) who can move and take like both Rook and Unicorn. This chessman is best represented by an inverted Rook which naturally suggests this power of a Rook as one of his components. The Narwhal Provides a Piece inter- mediate in strength between the Rook and Queen, for the Rook attacks six, the Narwhal eight and the Queen (conventional) twelve cells or squares from a position. From these types of Ecila pieces I will now turn to the sorts which may be considered to be the "leapers" in my idea, the Horse or Knights of Ecila. The piece which first moves with a Rook step to a new Cube and follows this immediately by a Bishop step in the new Cube is what I call the "Ecila Horse". For example: a Horse at A1 can go to B1 and then move to cell 4,6 or 7 in this new cube B. Any occupation of the intermediate cell B1 has no obstruction on the Horse, because it is a "leaping" move with regard to B1. The Ecila Horse can attack or guard nine cells from his position; he can operate on half cells of the board. The idea of combining a Rook step with a Unicorn step is the Ecila chessman which I have name the "Hippogriff." This piece first moves to a new Cube with a Rook step and follow it immediately with a Unicorn step in that new Cube. For example: from A1 the Hippogriff can reach B8 ( via the intermediate cell B1)and C8 (via C1) and E8 (via E1.) From his position he can attack three cells alone. The move of the Hippogriff is really a "stepping move" which involves four dimensions. [Ecila4.jpg] The combination of a Bishop step to a new Cube, followed by a Unicorn step in that the new Cube, gives the Ecila "Wyvern". For example: a wyvern at A1 can get to D8 (via the intermediate cell D1) or to F8 (viaF1) or to G8 (via G1). The Wyvern has really a "stepping" move in which five dimensions are involved. The three pieces just described are not easy to understand and to use as the Rook and Bishop: they are in a certain sense the elaboration of the main idea. The Ecila distinction between this trio ( Horse Hippo- griff, Wyvern) and the rest is, that the Horse, Hippogriff and Wyvern have no movements atall in a single cube. In general, when one wishes to use the Knights of the ordinary set of chessmen to represent something in Ecila, then they can be used for Ecila Horses, Hippogriffs or Wyverns. The idea of commencing a Ecila game (Chess or Scacia) from a standard ized arrangement of the chessmen is not very satisfactory of practical. The following method, being flexible, is more appropriate. The players alternately place their pieces one at a time on vacant squares (cells) of the Ecila board. The order in which they place their pieces should be quite free for the Ecila games of Scacia and Schlagschach. The chessmen all placed, White makes the first move, making a take if he must. Black then plays and so on. In Scacia, the player who captures all of his opponent's pieces is, of course the winner, if the game is played to the strict rule of Scacia as described in chapter The inverse idea of Scacia, the game Schlagshach is won by the player first losing all of his pieces to the opponent. When the Ecila Chess game itself is played, then the placing of the pieces at the commencement is not so free as for Scacia and Schlag- shach. For Chess, the method of placing is modified: A player must first of all place his King. (It can be in any vacant position he chooses.) After that the pieces are placed one at a time in vacant spaces of the Ecila board but with the obviously necessary restriction that no piece may be placed so as to give check directly to the King. Kings in Ecila can follow the "conventional" definition of a monarch by having the power to move with a Rook step or a Bishop step. In the case of the minimum size of Ecila board ( the cubes having eight cells) it will be seen that such a monarch is indentical in power with the "conventional" Queen. Consequently, rather strong forces may be required for the defeat of that Monarch. For practical purposes the Ecila K move should be limited to the Rook step, or even more drastic to the Bishop step. Alternately, the King can be restricted to moving within that cube where he is initially placed.