Why Maharajah Has Exactly These Pieces

The Answer In One Table

Chess builds its long-range pieces from three moves: the rook’s straight line (R), the bishop’s diagonal (B) and the knight’s jump (N). The queen is simply a rook and a bishop in one piece. Three moves can be put together in exactly seven ways, and here they all are.

Value: the piece’s price in points when you build an army in Maharajah. The prices are set by the game, not derived from the geometry above; the four orthodox pieces keep their usual chess values.

Where seven comes from: each of the three moves is either in a piece or not. That is 2 · 2 · 2 = 8 combinations, and one of them, no move at all, is not a piece, so 8 − 1 = 7. In the picture below every line means “add one more move”.

Why These Three Moves

Turn the board a quarter turn, or look at it in a mirror. A rook still moves like a rook and a knight like a knight: the pieces do not notice which way the board faces. Such a turn or mirror is called a symmetry, and a square has eight of them. The buttons under Fig. 2 let you try three.

The symmetries sort every possible move into families. A straight move stays straight, a diagonal move stays diagonal, and every other move, neither straight nor diagonal, stays crooked. There is no fourth family. The shortest move of each family is one square straight, one square diagonally, and two-and-one: the rook’s step, the bishop’s step and the knight’s jump.

So the rook, the bishop and the knight are not three random inventions. Each is the simplest move of one of the only three families the square has. Mathematicians call these families orbits.

The Queen Began It

About five hundred years ago chess took a step every player now takes for granted: it gave one piece the moves of two. The queen is a rook and a bishop together. Once that is allowed, three more combinations are waiting: rook and knight, bishop and knight, and all three at once. Chess never made them. Maharajah does: the Chancellor, the Archbishop and the Maharajah.

Nothing here is a new kind of move. Every move of every Maharajah piece is a move that some orthodox piece already makes.

The Mathematics You Already Play

A chess player does mathematics on every move without calling it that. Here are the six ideas this article rests on, each as you already know it from the board, with the name mathematicians give it. Part two uses these names.

• A move is a vector. “Two up and one to the right” is the same knight jump from any square. A move is not a pair of squares but a shift: how far across and how far up. Written as a pair of numbers, (1, 2), it is called a vector.
• Moves add up. Two jumps in a row end on the same square in either order. Every move is undone by the same move backwards, and standing still is the move “nothing”. Moves with these rules form what mathematicians call a group.
• Symmetry. You recognise a mate pattern on the other wing as the same pattern, mirrored. Turning or mirroring the board without changing how the pieces move is a symmetry; the square has eight, and they form a group too, called D₄.
• Families of moves. Sorting moves by which ones a turn or a mirror makes into each other gives the families straight, diagonal and crooked. Mathematicians call such families orbits, and the rule “same family” an equivalence.
• Counting combinations. Three moves, each in a piece or not: 2 · 2 · 2 = 8, minus the empty one. All the combinations, each joined to the ones it grows from, make the picture in Fig. 1, which is called a lattice.
• Parity. A bishop never leaves its colour, and a knight changes colour on every jump. That is arithmetic with only two answers, even and odd, called counting modulo 2. It helps to explain why the Archbishop can mate with its king while neither the bishop nor the knight can.

None of this is new to a chess player; it only gets names. With the names, the choice of Maharajah’s pieces stops being a matter of taste and becomes something that can be checked.

Four Surprising Results

We had a computer solve every ending of king and one piece against a bare king. Four results stand out.

1. Bishop and knight in one piece can mate. Neither the bishop nor the knight can force mate with its king. The Archbishop, which is both, mates from every position.
2. Moves add up exactly. On an empty board a bishop has 8.75 moves on average and a knight 5.25. The Archbishop has 14.00, exactly the sum and as many as a rook, because a knight never jumps to a square a bishop could reach.
3. The Maharajah is fast. With its king it mates a bare king in at most 4 moves from any position. The queen needs up to 10.
4. Colour decides. No piece whose every move changes the colour of its square could mate on its own, the knight among them. This is a pattern seen in 18 pieces, not a proof.

The full tables are in part two, under What the computations show.

What Is And Is Not Claimed

Claimed: the three new pieces of Maharajah are exactly the combinations of the rook’s, bishop’s and knight’s moves that chess left out.

Not claimed: that they are the only possible new pieces (the board allows more than a trillion symmetric ones), or that geometry decides how strong a piece is in play. Part two gives the proofs and the limits in full.

The Question

Maharajah adds three pieces to the six of chess. The Maharajah moves as a queen and a knight, the Chancellor as a rook and a knight, the Archbishop as a bishop and a knight. The natural question is why these three. Is the choice as arbitrary as in hundreds of fairy chess variants, or is it determined by something?

To answer by proof rather than taste, we describe moves mathematically, as vectors on a lattice organised by the symmetry group of the square. The question “which pieces are missing” then becomes a question about a finite set, and a finite set can be checked exhaustively.

A Move Is A Vector

Let a piece stand on the square (x, y). A move takes it to (x + dx, y + dy). We call the pair (dx, dy) a move vector. A piece can use a vector in two ways:

• leap: one step by the vector, whatever stands in between (the knight);
• ride: repeat the same step while the square is empty (the rook and the bishop).

On an 8×8 board each of dx and dy lies between −7 and 7, so there are 15 × 15 − 1 = 224 non-zero vectors. Those are all the moves the board permits. Figure 3 shows every one of them: the centre is the piece’s square, each cell is one vector. Tap any cell to see its direction and the pieces that use it.

Why 15 and not 8: the vector does not depend on where the piece stands. From the a-file a piece can go at most 7 files to the right, dx = +7 (a → h); from the h-file at most 7 to the left, dx = −7 (h → a). Every value in between is possible too, and so is 0, so dx takes 7 + 1 + 7 = 15 values, and dy the same. That gives a 15 × 15 grid with the piece in its centre; the one vector we drop is (0, 0), staying in place. From any single square only 63 of these vectors stay on the board (one for every other square), but which 63 depends on the square: in a corner they all point one way, in the centre in every direction.

Moves add up: two moves made one after the other are one move, (dx₁, dy₁) + (dx₂, dy₂) = (dx₁ + dx₂, dy₁ + dy₂). With this addition the vectors form the additive group ℤ²: the sum of two vectors is again a vector, the order does not matter (v + w = w + v), (0, 0) is the zero, and every vector v has an opposite −v with v + (−v) = 0. The 15 × 15 grid is a window onto this group, not a group of its own: (7, 0) + (7, 0) = (14, 0) no longer fits on the board.

In plain words: “two up, then one right” and “one right, then two up” end on the same square; standing still is the zero move; and every move is undone by the same move backwards.

The Symmetry Of The Square

Turn the board by 90° or mirror it, and a rook stays a rook and a knight stays a knight. A square has eight such transformations: four rotations and four reflections. Together they form the dihedral group D₄, the symmetry group of the square. A piece whose rules do not depend on how the board is turned (every orthodox piece except the pawn) contains, with each vector, all of its images under D₄.

Each symmetry is an integer matrix acting on move vectors: the quarter turn sends (x, y) to (−y, x), the reflection in the diagonal sends (x, y) to (y, x). So the symmetries respect addition, g(v + w) = g(v) + g(w): turning the board after two moves is the same as turning each move and adding them. In the language of groups, D₄ acts on ℤ² and every symmetry is an automorphism of it. The half turn sends every v to −v, so a symmetric piece always has, with every move, the move back.

Call two vectors equivalent if some symmetry turns one into the other. This is an equivalence relation, and the three group rules are exactly what makes it one: every vector is equivalent to itself (the identity), the relation works both ways (every symmetry can be undone), and it chains (two symmetries in a row are again a symmetry). Its classes split the vectors into families that do not overlap.

The set of images of one vector is its orbit, and the orbits are these classes. The orbit–stabiliser theorem gives its size: |orbit| = 8 / |stabiliser|, where the stabiliser is the set of symmetries that leave the vector in place. A vector on a line or a diagonal is kept by one reflection, so its orbit has 4 directions. An oblique vector such as the knight’s is kept by no symmetry, so it has 8.

In plain words: take the rook’s step (1, 0). Turning the board by 90° gives (0, 1), (−1, 0) and (0, −1), but flipping the board top to bottom leaves (1, 0) where it was, so the eight symmetries give each of the four directions twice. The knight’s step (2, 1) has no such symmetry: all eight give different vectors, listed below.

Where 7, 7 and 21 come from. An orbit is fixed by its two step lengths taken without signs. A line orbit is (a, 0) with a from 1 to 7: 7 orbits of 4 vectors each, (±a, 0) and (0, ±a). A diagonal orbit is (a, a): again 7 orbits of 4, (±a, ±a).

An oblique orbit is a pair of two different numbers from 1 to 7, and their order does not matter: (2, 1) and (1, 2) lie in the same orbit. There are 7 · 6 / 2 = 21 such pairs, {2, 1}, {3, 1}, {3, 2}, {4, 1} … {7, 6}, and each orbit has 8 vectors: 2 orders times 4 sign combinations. The knight’s orbit, for example, is (2, 1) (1, 2) (−1, 2) (−2, 1) (−2, −1) (−1, −2) (1, −2) (2, −1). Check: 28 + 28 + 168 = 224 vectors in 7 + 7 + 21 = 35 orbits. Put simply, 35 is the number of different moves when direction is ignored: 7 lengths for the rook, 7 for the bishop and 21 knight-like shapes.

The same count comes out without listing a single orbit. Burnside’s lemma says that the number of orbits is the average number of vectors a symmetry leaves in place. Take the whole 15 × 15 grid, zero included: the identity fixes all 225 vectors, each of the three rotations fixes only (0, 0), and each of the four reflections fixes the 15 vectors on its axis. So (225 + 3 · 1 + 4 · 15) / 8 = 288 / 8 = 36 orbits: the 35 directions and the zero vector, alone in its own orbit.

Why exactly three basic moves. A non-zero vector is kept in place either by a reflection in an axis through the sides of the square (the line vectors), or by a reflection in a diagonal (the diagonal vectors), or by no symmetry but the identity (the oblique vectors). So there are three types of direction and no more. The shortest vector of each type is (1, 0), (1, 1) and (2, 1): the rook’s step, the bishop’s step and the knight’s leap. Chess builds its long-range pieces from the smallest member of each type, which is why geometry, not taste, singles out these three.

The Closure Theorem

Write the three long-range moves of chess as R, a ride along a line; B, a ride along a diagonal; N, the knight’s leap. The union X ∨ Y of two pieces is a piece that may move as X or as Y. The queen is the first example: Q = R ∨ B. Union is associative, commutative and idempotent, so the pieces built from these three moves form the Boolean lattice of subsets of {R, B, N}.

In plain words: the order in a union does not matter (R ∨ N is the same piece as N ∨ R), and adding a move a piece already has changes nothing (R ∨ R = R). So a piece built from R, B and N is described completely by which of the three it has.

Union is not a group operation. Moves add up into a group, but a union of pieces has no inverse: a move once added cannot be taken away by another union. That is why the pieces form a lattice, not a group. The group decides what the pieces are built from; the union decides how they combine.

The same count without symbols: each of R, B and N is either in the piece or not, which gives 2 · 2 · 2 = 8 combinations. One of them contains nothing and is not a piece, so 7 remain. Fig. 1 and the table in part one show all seven.

Two corollaries. First, Maharajah introduces no new direction: all its pieces use the same three orbits out of 35 as orthodox chess. Second, the set is minimal. Without any one of the three new pieces it is no longer closed, and any other piece added would bring a new move.

Orthodox chess accepts union as a principle (the queen) but applies it to only one of the three pairs. Maharajah applies the same principle to every pair and to the triple.

How Many Pieces Are Possible

A symmetric piece is fixed by what it does with each of the 35 orbits: ignore it, leap along it, or ride along it. Riding differs from leaping only when at least two steps fit on the board, that is when 2·max(|dx|, |dy|) ≤ 7. That holds for exactly 9 orbits, those of length up to 3. The other 26 have only two options.

Why riding needs 2·max(|dx|, |dy|) ≤ 7: a rider makes the same step again and again, so its second step lands twice as far away. Along (3, 0) a piece on the a-file reaches d, then g: two steps fit. Along (4, 0) the second step would need 8 files, but the board has only 7 to spare, so “ride along (4, 0)” and “leap along (4, 0)” are the same piece. The length column counts orbits by their longer coordinate: length L has L + 1 orbits, (L, 0), (L, 1) … (L, L). So lengths 1 to 3 give 2 + 3 + 4 = 9 orbits, and lengths 4 to 7 give 5 + 6 + 7 + 8 = 26.

The choices are independent, so they multiply. Two orbits with three options each already give 3 · 3 = 9 combinations. For all 35: 3⁹ = 19,683 for the short orbits times 2²⁶ = 67,108,864 for the long ones. One combination ignores every orbit: a “piece” that cannot move at all. That is why 1 is subtracted.

So the 8×8 board admits exactly 3⁹ · 2²⁶ − 1 = 1,320,903,770,111 symmetric pieces, more than 1.3 trillion. That count leaves out pieces that move only forward (the pawn, the shogi pieces), pieces that jump over others (the grasshopper, the xiangqi cannon) and pieces that can be blocked (the xiangqi horse). The conclusion is simple: no set of pieces can be “complete” among all possible ones. Completeness only makes sense relative to a chosen base.

A Wider Base: 17 Pieces

What if the base includes short steps? Along lines a piece may have nothing, a step W or a ride R; along diagonals nothing, a step F or a ride B; plus the knight or not. That makes 3 · 3 · 2 − 1 = 17 pieces. Orthodox chess has 5 of them (the king only as a royal piece), Maharajah has 8. The other nine are known from other traditions: the shatranj ferz, the shogi dragons, fairy pieces.

The count works as in “How Many Pieces Are Possible”: 3 choices along lines times 3 along diagonals times 2 for the knight is 18, and the one combination with nothing at all is not a piece.

An Atlas Of Known Pieces

Below are 31 pieces from chess, Maharajah, shatranj (the old Persian game chess descends from), xiangqi (Chinese chess), shogi (Japanese chess) and fairy chess. Each thumbnail shows the moves from d4 on an empty board. Tap a piece to open it on the large board. There you can move it by tapping a square and switch on a heat map of how many moves it has from every square.

What The Computations Show

For each piece we computed its mobility on an empty board and solved the king-and-piece-versus-bare-king ending by enumerating every position (about 262 thousand for each side to move). The solver is checked against known results: the longest mate with a queen comes out at 10 moves and with a rook at 16, as in the endgame tablebases.

Where 262 thousand comes from: the white king, the black king and the white piece can each stand on any of the 64 squares, and 64 · 64 · 64 = 262,144. Placements where two of them share a square or the kings stand side by side are illegal and are skipped.

Average: mean number of moves over the 64 squares. Corner and Centre: the fewest and the most moves from one square. Diameter: the most moves needed between two squares. Forced win: share of positions (White to move) where mate is forced. Longest mate: in moves, with best play by both sides. Maharajah’s pieces are highlighted.

Betza is the standard short notation for fairy pieces. W is one step along a line, F one step along a diagonal, N the knight’s leap, K = W + F the king’s step; R, B and Q ride, and a doubled letter rides too (NN, the nightrider). Letters combine: RN moves as a rook and as a knight. In the pawn’s mfWcfF, m means move only, c capture only, f forward.

Four conclusions that follow from geometry alone:

1. Mobility adds up exactly. The knight’s vector is not a multiple of any line or diagonal vector, so a knight’s squares never coincide with a rider’s: A = B + N = 8.75 + 5.25 = 14.00, the same as the rook.
2. A union can do what neither part can. Neither the bishop nor the knight can mate alone; the Archbishop mates from every position. The reason is square colour (Fig. 6).
3. The Maharajah mates in at most 4 moves from any position; the queen needs up to 10.
4. Parity. No piece whose every move changes square colour (W, N, W+N, the nightrider) can mate alone. This pattern holds across 18 pieces; it is not a proof.

In plain words: number the files and ranks from 1 to 8 and add the two numbers. Dark squares have an even sum, light squares an odd one. A bishop’s move changes the sum by an even number, so the bishop never changes colour. A knight’s move changes it by 1 or 3, an odd number, so the knight changes colour on every move. The Archbishop has both moves and can reach any square, which is why it mates while the bishop and the knight alone cannot.

The Limits Of The Claim

So that the conclusion is not read more broadly than it is proved:

• Not every possible piece. Maharajah holds 7 of more than 1.3 trillion symmetric pieces. It is complete under union on the base {R, B, N}, that is, with respect to chess’s own long-range moves.
• Riding the knight. For lines and diagonals chess has both a step (the king) and a ride (rook, bishop); for the knight it has only the leap. A nightrider, which rides the knight’s vector, exists in neither chess nor Maharajah.
• Other kinds of piece, those that move only forward, jump over other pieces or can be blocked, lie outside this model.
• Strength and value. Geometry does not decide what a piece is worth in play. That is a question for engine measurement, not for this article.

History Of The Idea

The set of chess pieces has always been extended. In shatranj the queen’s place belonged to the ferz, a single diagonal step, and the bishop’s to the alfil, a leap to (2, 2). In Europe at the end of the fifteenth century they were replaced by the modern queen (R ∨ B) and bishop, and that is when the union of moves entered chess.

Combining the rook and bishop with the knight has been proposed many times: Pietro Carrera (1617) on a 10 × 8 board, Henry Bird (1874), José Raúl Capablanca (1920s) with the Chancellor and Archbishop, Christian Freeling’s Grand Chess (1984) on 10 × 10, Yasser Seirawan and Bruce Harper’s Seirawan Chess (2007) on 8 × 8. The queen plus knight (the amazon) appears in regional and fairy chess. The historical dates follow commonly cited sources and were not checked against primary sources.

What is new in Maharajah is not the pieces themselves but that all seven unions are collected on the ordinary 8 × 8 board, with players building their armies from the same budget.

Methods And Reproducibility

Every number here was computed, not taken from the literature. kxk.c (C, about 3 seconds) computes mobility, colour binding and diameter, and solves the K + X versus K ending by retrograde analysis: backwards from the mate positions, half a move at a time, until the set of won positions stops growing. Legal positions have the kings apart and the piece on its own square; stalemate is a draw; the black king may take an undefended piece. The thumbnails on this page are drawn from the same move definitions. The source of kxk.c opens in the browser from the link above: compile it with cc -O2 kxk.c -o kxk and run ./kxk.

Conclusion

Orthodox chess builds its long-range pieces from three moves, the line, the diagonal and the knight’s leap, and uses four of their seven unions. The Maharajah, Chancellor and Archbishop are exactly the missing three. The Maharajah set is therefore the closure of the chess set under the union of its own moves. Nothing new is added; what the queen began is completed.

And the mathematics was on the board all along. Every player who knows that a bishop keeps its colour, that a knight jump can be undone, or that a pattern works the same on the other wing, has been using vectors, symmetry and parity without the names.

You can build an army with these pieces in the army builder or see how each one moves in Learn.