Home

An Essay

Rules

Vinyl Boards

Plastic Pieces

Complete Vinyl Sets

Wood Boards

Wood Pieces

Complete Wood Sets

Software

Ordering

An Essay on Circular Chess

A Modest Proposal
by Alan Mattlage

Modifying Chess strikes most advanced Chess players as about as worthwhile as modifying the Mona Lisa. Why tamper with genius? Nonetheless, below is an outline of a new version of Chess which may prove to be more than a curiosity.

1. An Introductory History

Chess (or perhaps a Proto-Chess) first appeared in India during the sixth century, spreading in subsequent centuries to Persia, China, and the Arab world. During this time, many versions of the game were played. The first European reference to Chess appears in a Spanish nobleman's will, dated 1008. Its primary spread through Europe can be attributed, however, to returning Crusaders. By the 14th century, it had become well-established in Europe.

The rules of Medieval Chess were significantly different from our modern game. Most importantly, medieval Queens and Bishops were far less mobile. They could move exactly two spaces diagonally. This made Medieval Chess slower and simpler than the game we play today.

Modern European players sought a faster and more dynamic game. Hence, a variety of modifications began appearing. In the 15th century, a revolutionary new version appeared. In this version, Bishops and Queens moved in the manner with which we are now familiar. One might say, that the primary difference between Medieval Chess and Modern Chess is the enhanced mobility of these pieces, or perhaps more narrowly, "the invention of the Queen."

Clearly, the new mobility upset all of the strategies of the past. Consequently, players began authoring analyses of the Modern version, e.g., Lucena's Repeticion des Amores e Arte de Axedres and Ruy Lopez de Segura's Libro de la invencion liberal y arte del juego del Axedrea. Other works appeared by Horatio Gianutio della Mantia and Damiano. Slowly, competing rules fell out of popularity, until the Federation Internationale des Echecs (FIDE) adopted roughly our present rules in the 1920's.

The brilliance of Modern Chess can be attributed to its formal simplicity and its substantive complexity. This was a consequence of increasing the mobility of the pieces, without diluting it over a larger board. Modern Chess is played with a small set of pieces on a geometrically simple board, and while its rules can be mastered by children, it quickly becomes mind-bogglingly complex. This makes it as challenging for the grandmaster as it is for the beginner.

FIDE's stabilization of the game did not, of course, eliminate all variations or end attempts to "improve" it, but thus far, modifications have produced only curiosities (if not monstrosities.) Trivial changes don't seem worth bothering about, and significant changes destroy the virtues of the original game. One is tempted to say that Chess has reached a perfection. In any case, any plausible modification must preserve its two crucial characteristics: formal simplicity and substantive complexity. It is with this in mind, that we have developed Circular Chess.

2. Circular Chess

Circular Chess retains all the formal simplicity of Modern Chess, but it enhances the complexity of play. This is accomplished rather in the manner that Modern Chess transformed Medieval Chess: by increasing the mobility of the pieces. Circular Chess uses the same pawns and pieces as Modern Chess, starts them in their familiar positions, and moves them by the same principles as in Modern Chess. Furthermore, the increased mobility of the pieces is not a function of ad hoc changes, e.g., adding Knight moves to the Queen, Bishop moves to the Knight, allowing pawn to move sideways, etc. It is strictly a consequence of the circular board.

Like the traditional board, the circular board is geometrically elegant and elementary. While it appears quite different from the traditional board, the relationships among its squares are really quite similar. By making only slight functional modifications to the board, Circular Chess retains most of the intriguing features of Modern Chess. Circular Chess is Modern Chess, but it is played in a less bounded space, i.e., it is played in a universe where movements return (more or less) to their starting points.

With increased mobility on a board hardly larger than the traditional board, players must rethink all of the standard openings and grapple with a more complex middle game.

2.1 The Board

To understand how the traditional board is transformed into a circular board, imagine the line separating the 4th and 5th ranks of a traditional board shortening to a single point, such that the board takes on a bow tie shape. Each file becomes two isosceles triangles, obliquely arranged and all sharing the board's central point.

Next, rotate Black's side of the board 180 degrees, using the line separating the d-file and the e-file as the axis of rotation. The board retains its bow tie shape, but Black's King and Queen sides are now reversed. The oblique relationships between the two halves of each file disappear and each file regains its alignment along straight lines. (See Figure 1.)

At this stage, the board is functionally the same as the traditional board, though the reversal of Black's King and Queen sides and the bow tie shape are somewhat disconcerting. Modern (or Medieval) Chess may be played on this board.

Finally, insert a bow tie shaped, "i-file" between the a-file and the h-file and fan out each wing of the bow tie until all the files are congruent and connected in a single wheel of nine bow tie-shaped files. This creates the Circular Chess board. In the accompanying figures, the center has been cut out to avoid visual confusion. (See Figure 2.) The i-file precipitates all of the new features of the game.

2.2 Three Versions

The circular board can be used to play three versions of Circular Chess: Minimal, Medial, and Maximal Circular Chess. They are composed of 66, 68, and 70 squares respectively. The minimal version employs only i4 and i5. The Medial version employs i3, i4, i5, and i6. The maximal version employs i2, i3, i4, i5, i6, and i7. (See Figures 3, 4, and 5.) The minimal version closely approximates Modern Chess, while the maximal version departs from it most.

3. The Rules

The rules of Circular Chess are identical to Modern Chess, except that movement will extended naturally onto and across the new squares on the i-file.

4. The Opening Game

By extending movement onto and across the i-file, Circular Chess offers players a number of new moves. In Modern Chess, a player's pieces may assume any of 432 different arrangements after the first two moves. Minimal, Medial, and Maximal Circular Chess each permit progressively more possibilities. For example, Medial Circular Chess permits 24 new arrangements. Furthermore, 22 of the old arrangements result in different squares being attacked. Thus, Medial Circular Chess produces 46 functionally new positions for each player after the second move. Minimal Circular Chess permits a lesser increase of new positions, while Maximal Circular Chess permits the greatest increase.

Still, many of the new options created by the i-file appear unadvisable. This underscores the continuities between Modern and Circular Chess. Circular Chess has an oddly familiar quality that can seduce the sophomore player into complacency. On the other hand, the strange new options can also lure the novice into forgetting valuable, Modern Chess fundamentals. One must learn the proper degree of adaptation to the circular board.

4.1 The Knight

Many of the new options appear to be promising opening moves. This is particularly true of the Knight. In Modern Chess, the Knight's natural development is to B3 and secondarily to K2 or Q2. This results in only 16 reasonable Knight configurations. In Circular Chess, the Knight can also be valuably developed to R3. These added avenues of development result in 81 reasonable Knight configurations. Clearly, the number of viable opening positions is substantially increased by the circular board.

4.2 The Bishop

The i-files also allow greater immediate freedom of movement to the Bishop. In Modern Chess, players often find one of their Bishops blocked at K2, K3, Q2, or Q3. This prevents the Bishop from developing to the valuable diagonal running from R2 to N8. In Circular Chess, the Bishop can reach this diagonal despite being blocked in the center. This is done by advancing the Knight pawn and then moving the Bishop through the i-file to N3 on the opposite side of the board.

To further illustrate the effect of the Bishop's initial mobility, consider the mobility of a Bishop on an empty traditional board. From its initial position, the Bishop can immediately attack 7 squares. After its first move, it can attack 10 more squares from two directions and 14 more from one direction. However, in Medial Circular Chess it can immediately attack 14 squares. After its first move, it can attack 16 more squares from two directions and 2 more from one direction:

4.3 The Flanks

The added options of the Knight and Bishop alone suggest that the standard openings will be quickly disrupted by movements (or possible movements) onto and over the i-file. Furthermore, an enormous number of new openings designed to place pressure on the flanks appear viable. The entire circular board is likely to be immediately contested -- from flank to flank and not merely in the center.

This is particularly true in that the i-file opens up new avenues of attack on the a, b, g, and h-file pawns (and on the Rook in Maximal Circular Chess.) On the other hand, the i-file opens up new avenues of defense as well. Players will need to make early decisions regarding the security of their flanks and the advisability of castling, which no longer seems unambiguously good. This poses new problems. Players may find that an alternative to castling is needed, depending upon how the opening unfolds.

Players must quickly but carefully assess the possible lines of attack and defense before choosing to castle. In fact, it appears that castling may often be unwise. Deprived of the castled position, the King will frequently remain in the heat of the battle during the middle game. A possible alternative to castling might involve moving the King to the second rank and thereby linking the Rooks without castling and then opening up files on the flanks for Rook development.

In any case, the difficulties of the middle game are clearly complicated by subtle options for a checkmate in the middle game. The contest in Circular Chess is, therefore, less like battles between modern rulers and more like the battles between Alexander the Great and Darius III where both occupied the field of battle and capturing the King was always a tactical goal.

5. The Middle Game

The changes that occur in the middle game result from the increased mobility of the all of the pieces. To measure this increase, one can assess the "a priori mobility" of each piece. To arrive at a value for a priori mobility, one must first count the number of squares to which a piece can move, and do this for each square that the piece can legally occupy. The a priori mobility for that piece is the sum of these totals. For example, on a traditional board, the Rook can legally occupy any of the 64 square. From each of those squares, it can move to 14 squares (when the board is empty, of course.) Since, 64 x 14 is 896, we can assign this value as the a priori mobility of the Rook on the traditional board. Each version of Circular Chess results in different a priori mobility values for each piece. These values increase as one shifts from Modern Chess to Minimal, Medial, and Maximal Circular Chess.

Increased piece mobility means, of course, an increase in the inter-relationships between the pieces. This is especially true in that the increased mobility is not diluted over a proportionally larger game board. The circular board multiplies possible combinations just as increasing the principles of movement for the Bishop and Queen multiplied them in the 16th century. Below is a table showing the a prior mobility of each piece for the various versions of chess.

A Priori Mobility of Pieces

6. The End Game

The end games of Modern Chess feature wide-open play and/or attempts to promote pawns. These remain characteristics of Circular Chess. However, Modern Chess entails special strategic considerations regarding pawns on the a-file and the h-file. These are complicated by the i-file. Furthermore, since pawn promotion requires moving the pawn six spaces from its initial position, pawn promotion on the i-file is not possible. Players must, therefore, be careful when moving pawns onto the i-file, but normally the value of capturing a piece on the i-file will out-weigh the reduction of one's options to promote.

7. Patterns of Mobility and the Values of Squares

Thus far I have discussed the enhanced mobility of the pieces and its effects on the opening, middle, and end games. I have mentioned only in passing, however, its effects on the value of the center. Using the figures describing the a priori mobility of pieces, one can get a clearer idea of the patterns of mobility for each piece and the relative value of the squares on the board.

7.1 Patterns of Mobility

On a traditional board, Knights, Bishops, and Queens become more mobile as they move to the center of the board. In the center, they enjoy both the full extent of their mobility and they have easier access to all parts of the board. The mobility of Rook, however, remains constant where ever it is placed. Of course using Rooks to control the center is preferred, but this is not because of their increased mobility, but because of their threat to other pieces that seek to occupy the center.

In Circular Chess, these patterns of mobility change. Both Knights and Bishops (especially) are more uniformly mobile across the whole of the board. The mobility of Rooks, on the other hand, is increased when they are moved to one of the undivided circles in the middle of the board. As one would expect, the Queen's pattern of mobility reflects a combination of the Bishop's and the Rook's. This has the consequence of converting Bishops into something like weak Rooks, and converting Rooks into something like powerful Bishops, but it does not reduce the power of the Rook. Indeed, its overall power is increased. Only its pattern of mobility is more Bishop-like.

7.2 The Values of the Squares

One can also calculate a rough measure of the value of each square with regard to the mobility it affords the pieces. For each square, by summing the number of squares to which each piece can move (were it to occupy that square,) one can derive an a priori value for each square. For example, in Modern Chess, four Knights, two Bishops, four Rooks, and two Queens can occupy each square. Combining their mobility on each square yields a value of 84 for the four center squares, 80 for the adjacent twelve squares, etc. This figure declines to 60 for the corner squares.

The progression is less regular in Circular Chess, due to the absence of two or more i-file squares. Still, a rough progression can be identified. For example, in Medial Circular Chess, the values of squares in the outer circle (ranks 1 and 8) average in the 70s. Squares in the inner circle (ranks 4 and 5) average in the low 100s. The weakest squares are, of course, the "corner" squares: a1, h1, a8, and h8. They are valued at 62. Curiously, the strongest squares are on the third circle: c3, f3, c6, and f6. These are valued at 111.

Overall, the inner, undivided circles are substantially higher in value than the outer, divided circles, due to the mobility of Rooks and Queens on these circles. However, these are "a priori" values. When two or more pawns are placed on the inner circles, mobility on these circles is reduced, whereas Bishops can frequently find ways to penetrate pawn defenses from the outer circles. These considerations indicate that players will need to attend to the whole of the board, and not merely concentrate their attention on the center squares (or even just the undivided circles.)

8. Psychological Considerations

Much of the initial difficulty in playing the game will stem from adjusting to normal chess relations as they are played out in a new spatial configuration. Players will have to learn to comprehend arcs and spirals as easily as they now comprehend straight lines. This may tax new faculties of the brain, making some strong Modern Chess players weaker, and some weak Modern Chess players stronger. That is, these amorphous cognitive abilities for spatial understanding (as opposed to the more crisp skills of logical analysis) may play a more significant role in establishing a player's strength than we currently might think.

Because there are three versions of Circular Chess, players can become progressively acclimated to the game by first playing Modern Chess on a circular board, i.e., they might play on a circular board in which none of the squares on the i-file are active. This way, they will experience the need to see relationships in arcs and spirals, but they will not be required to entertain any new logical relationships between the pieces.

Once becoming comfortable with the spatial configurations of the circular board, players can progress to Minimal Circular Chess, where only i4 and i5 are active squares. In this version, the new logical relationships are relatively few. As playing Minimal Circular Chess becomes natural, they can progress to Medial and/or Maximal Circular Chess. In any case, different players may find a preference for one or another of these versions. Each version presents different problems for play, progressively adding to the yet ever-present relationships of Modern Chess.

9. General Conclusions

Circular Chess appears to hold a good deal more promise for interesting play than any other variation on Modern Chess. This is because it retains the simple rules of Modern Chess, while employing a simple, but strikingly different board. Furthermore, the circular board is functionally less different from the traditional board than its appearance would suggest. However, the differences in the configuration of the board have profound effects on the specific strategies that one must employ. Furthermore, these new strategies must be formulated for a game that becomes rapidly more complex than Modern Chess and poses unique opportunities and dangers for swift and unexpected victories and defeats. Finally, it will likely require different cognitive abilities.

Whether or not Circular Chess is an "improvement" on Modern Chess is impossible to say. I suspect, however, that since the complexity of Modern Chess outstrips the human understanding, any vaunted improvement is meaningless. Modern Chess has indeed reached a degree of transcendental perfection. Consequently,
Circular Chess can at best offer three new versions of that perfection. Their participation in that perfection will be due entirely to all that they share with the Modern game.

Still, Circular Chess offers a new field of play for a familiar game. That field is for now a blank slate upon which new openings must be written, or more accurately, Circular Chess represents three New Worlds for Modern Chess. Each New World awaits exploration by its Ruy Lopez. Only then will we know if Circular Chess deserves a respectable place beside its venerable parent.