Plus Equals #5

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Marcel Duchamp once said that “while all artists are not chess players, all chess players are artists.” Whether or not you agree with that, there’s no denying that chess enjoys a distinguished reputation, not only as a game capable of prodding the deepest recesses of the intellect, but also as an egalitarian pastime whose appeal has endured for centuries across cultures and social strata. Easy to learn and hard to master, chess’s simple rules belie a vast ocean of possibility, and after hundreds of years of exploration, we still haven’t gotten to the bottom of it.

To excel at chess is to envision staggeringly complex branching sequences of moves, to live simultaneously in the present and many possible futures. In this light, it might be likened to the Cubist paintings of the early 20th century, which often merge multiple movements, moments, and perspectives into expressive hybrid compositions. And indeed, Duchamp’s fascination with chess spilled into some of his Cubist works (Fig. 1).

A Cubist painting of two people playing chess
Fig. 1: Portrait of Chess Players (Marcel Duchamp, 1911, oil on canvas, 39⁵⁄₈ × 39⁹⁄₁₆″)

So I think it’s fair to say that at the very least, chess enables art. And chess’s seemingly boundless framework for intricate strategic and tactical invention—derived from a deceptively simple playing field—is enticing for an artist interested in combinatorics, as I happen to be.

In an influential 1950 paper entitled Programming a Computer for Playing Chess, the mathematician Claude Shannon estimated the number of ways a chess game could possibly be played out: 10120, which is larger than the number of atoms in the observable universe. If anyone ever manages a full accounting of those possibilities, my hat is off to them, but it’s just a bit too ambitious for me.

Instead, I skipped ahead about 50 years to a computer game called Battle Chess (Fig. 2), one of the many beneficiaries of Shannon’s pioneering paper. Battle Chess plays like any other computer chess game, but its name comes from its distinctive visuals. When one piece attacks another, the attack is literal: a knight decapitates a bishop, a rook devours a queen, a king blows up a knight, and so on. These animations are amusing, if unsubtle, but I’m less interested in their particulars than their logistics. In all, 36 separate animations (Fig. 3) needed to be created to cover every possible attack, which is much more workable as a starting point for a combinatorial exploration of chess than the mind-melting Shannon number.

A Battle Chess screenshot, with two rows of cartoony, pixelated chess pieces on a chess board
Fig. 2: Battle Chess (Interplay Productions, 1988)
A diagram showing each of the six chess pieces—pawn, bishop, knight, queen, king—with each connected to six branching sequences of pairsBBPBNBBBRBQBKRRPRNRBRRRQRKQQPQNQBQRQQQKKKPKNKBKRKQKKPPPPNPBPRPQPKNNPNNNBNRNQNK
Fig. 3: Diagram of the 36 unique attack animations in Battle Chess. Each piece is capable of attacking any other piece, including a piece of its own kind.

A square on a chess board can’t be occupied by more than one piece at a time, and in that sense, those 36 attacks are about territory: one piece displaces the other. But what if the chess board’s territory weren’t so exclusive? A square simultaneously occupied by multiple pieces introduces new avenues of thought about those pieces and their formal, functional, and symbolic traits. What possibilities do merging these pieces suggest?

The idea of merging chess pieces recalls not only the aforementioned Cubist paintings of Duchamp and others, but also 0 through 9, a series of paintings and prints Jasper Johns produced beginning in 1960. Each work in the series superimposes the numerals zero through nine on top of each other (Fig. 4). The process of navigating the images’ visual morass to discern the individual numerals forces the viewer to focus on the numerals’ form, to truly observe something that’s usually merely seen. As a typography enthusiast, one aspect I find especially interesting is how the numerals interact with each other so chaotically despite coming from the same typeface. If I were to merge chess pieces in a similar way, how might the outcome be affected if the pieces’ designs anticipated their merger?

A lithograph of the numerals zero through nine, hand-drawn and superimposed on top of each other
Fig. 4: 0 through 9 (Jasper Johns, 1960, lithograph, 18¹⁵⁄₁₆ × 24½″)

To find out, I made my own two-dimensional chess set, with the design of its six pieces paying careful attention to how they overlay each other (Fig. 5). The simpler the design, the less chaotic the overlays, so I made a point of establishing some fairly tight constraints, settling on a 50×50 grid as a shared basis for the six drawings. And to keep everything as recognizable as possible, the set is modeled on the Staunton style, the standard for international tournament play, familiar to beginners and grandmasters alike.

Six 50×50 grids, each of which has all six chess pieces drawn on it semitransparently and superimposed on top of each other. On each grid, a different piece is made opaque and given focus.
Fig. 5: This chess set’s six pieces are all designed on a 50×50 grid and optimized to overlay each other.

With the design established, the combinatorial intention is to find every possible merger of two, three, four, five, and six pieces (Fig. 6).

We first find all the pairs, beginning with the pawn and merging it with each of the other five pieces, in ascending order of value: pawn/bishop, pawn/knight, pawn/rook, pawn/queen, pawn/king. Next, we do the same with the bishop. Since we already have a pawn/bishop pair, we can safely ignore bishop/pawn and skip ahead to bishop/knight, followed by bishop/rook, bishop/queen, and bishop/king. Like the bishop’s redundant first pair, the knight’s first two pairs—knight/pawn and knight/bishop—are also already accounted for, as are the rook’s first three pairs, and the queen’s first four pairs, all of which can be ignored. By the time we get to the king, every pair that could possibly include it has already been produced. In all, the six pieces make 15 unique pairs.

Now the process is repeated for each pair. Pawn/bishop yields the trios pawn/bishop/knight, pawn/bishop/rook, pawn/bishop/queen, and pawn/bishop/king. The next set of trios are based on pawn/knight. Since we already have a pawn/bishop/knight trio, we can ignore the redundant pawn/knight/bishop and skip ahead to pawn/knight/rook, followed by pawn/knight/queen and pawn/knight/king. A clear pattern begins to take shape: when adding a piece to a merger, its value can’t be less than that of the preceding piece, because the merger that would result has already been produced. Once we’ve used this formula to find all the sets of three, we can repeat the process to find all the sets of four, which will in turn yield all the sets of five.

A diagram of chess piece mergers. Pairs branch from single pieces, trios branch from pairs, quartets branch from trios, quintets branch from quartets, and a single sextet branches from the first quintet.PPBPNPRPQPKPBNPBRPBQPBKPNRPNQPNKPRQPRKPQKPRQKPNQPNRQPNRKKPNRQKPBQPBNRPBNQPBNKPBRQPBRKKPBRQKPBNQPBNRQPBNRKKPBNRQKBBNBNRBNQBNKBNQKBNRQBNRKBNRQKBRBQBNRQKQKKNRNQRQRKNKNRQNRKNQKRQKNRQKBRQBRKBQKBRQK
Fig. 6: Diagram of the 57 ways the six chess pieces can be merged

When the dust settles, the six pieces can be merged 57 different ways. Including the original six individual pieces brings the tally to 63, which is one less than the number of squares on a chess board. That final square contains all that remains: the empty set, a merger involving no pieces at all.

Of the many ways to think about these mergers and their formal, functional, and symbolic traits, the recurring themes for me are accord, conflict, and the inconvenient reality that those notions aren’t mutually exclusive.

Functionally, the mergers are force multipliers, since they usually have more mobility than any of their constituent parts would have on their own. A pawn/rook can move as far as it wants horizontally or vertically, and it can attack diagonally. Adding a rook to a bishop effectively turns it into a queen. And even the queen, the most powerful piece on the board, benefits from merging with a knight (Fig. 7).

Three chess boards. The first shows eight spaces a knight can move to, the second shows 27 spaces a queen can move to, and the third shows 35 spaces a hybrid queen/knight could move to.QNNQ
Fig. 7: Adding a knight’s L-shaped movements to the queen’s ability to strike in any direction makes for a powerful combination.

But as much as we might appreciate what this says about the virtues of collaboration, we also see that the more complex mergers noticeably struggle to maintain their formal integrity. Even though the pieces have all been designed to work together, the mergers can’t help but get jumbled as more pieces are thrown into the mix, raising questions about where we draw the line between an effective team and an inefficient bureaucracy (Fig. 8).

Chess’s preexisting symbolism offers the mergers even more specific allegories. Those of us who firmly believe in the separation of church and state might see something disquieting in how fluidly the bishop merges with the king (Fig. 9). Critics of the military industrial complex may find it significant that the knight’s lack of symmetry makes its presence consistently conspicuous (Fig. 10).

Three images: The first is all six chess pieces superimposed on top of each other, making a complex hybrid image. The second is a bishop/king hybrid. The third is a bishop/knight/king hybrid, with the knight sticking out from the other two symmetrical pieces.
Fig. 8–10: Is the combination of all six pieces the ultimate merger or a muddled mess (left)? The fluid merger of the bishop and the king is essentially a theocracy (center). The knight’s conspicuous asymmetry could be seen as questioning the military’s role in civil society (right).

I’ve been playing chess for most of my life, and I’m not bothered by the fact that its inherent beauty is something I’ll never be able to adequately articulate. But getting even a tiny step closer with an exploration like this is gratifying. The encodings attached to pawns, bishops, knights, rooks, queens, and kings—the way they look, behave, and reflect our understanding of each other—have a richness that extends far beyond the bounds of the chess board. Those six little chess pieces contain multitudes.

Rob Weychert

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