# Cluemeister's Corner

### Results for December

In December, the Cluemeister posed the following question: How many different possible notations are there for a chess move? This is a complicated question, and as such the Cluemeister didn't expect anyone to answer it completely (but was prepared to be pleasantly surprised). However, he did invite readers to take guesses, the closest of which would win honors and a prize at Minicon 46. He was therefore surprised that, despite a murmuring of interest, no guesses were forthcoming for the entire month, except for one pity guess tendered on Facebook to soothe his lonely heart.
One might attempt a ballpark figure by multiplying the number of types of chess pieces (6) by the number of squares on the board (64) to arrive at 384. This, however, as it turns out, would be far too low a guess. The reason for this is the disambiguation rule described in Section C-10 of the appendix of the FIDE handbook describing algebraic notation. If two (or more) identical pieces can move onto the same square, then such a move must be disambiguated by the file of the initial piece, or if necessary its rank, or (in exceedingly rare cases involving promoted pieces) both its rank *and* file. The theoretical possibility of moves such as Nd4e2 increases the total move count by a considerable factor.

To arrive at an exact figure, letís consider the six types of pieces separately.

**Rook moves:**

A rook may move unambiguously to any of the 64 squares on the board (e.g. Rf4). This move may be a capture or not, and may be either check, checkmate, or neither. These combinations lead to 64 x 6, or 384 possible moves.

There are 16 squares on the first and last ranks of the board. A rook moving to any of these spaces may require disambiguation by specifying any of the eight files (e.g. Rce8, Rbb1). Any of these moves may be a capture, but can they all be a check? Think about it. If two rooks of the same color can both move into a corner, a king of the opposite color checked from that corner must already be in check before either rook moves, which is impossibleónor can a rook moving into a corner discover check. This means that for rooks ambiguously moving into corner squares, there are 4 corners x 8 files x 2 capture possibilities, for 64 such moves.

There are 12 non-corner squares on the first and last ranks. A rook may move ambiguously onto any of these squares while delivering check or checkmate (although for moves like Rdd8+ we must assume the check is discovered). This makes 12 squares x 8 files x 6 capture/check combinations, for 576 possible moves.

There are 12 non-corner squares on the first and last files (a and h)ócall them edge squares. A rook moving onto these squares may require disambiguation by specifying any of the eight files or any of the seven ranks other than the rank moved to. (If disambiguation is possible by either rank or file, standard chess notation dictates that file is preferredóthus moves like R3a3 are impossible.) Any of these moves may be a capture.

For moves that involve two rooks on the same file able to move to the same square (like R2a5+), check or checkmate is possible. This works out to: 12 squares x 7 ranks x 6 capture/check possibilities, or 504 squares.

Moreover, rooks moving from one of the b-g files to an edge square (e.g. Rda4++) may cause discovered check or checkmate. The number of possibilities is 12 squares x 6 files x 6 capture/check possibilities, or 432.

Rooks moving ambiguously from one edge to the opposite edge of the board, however, as in Rha4, cannot discover check. Since another rook must already occupy the file theyíre moving to, they can only deliver check (or checkmate) if they capture a piece that was previously blocking the rook on that file from checking the opposing king. Thus, Rhxa4+ is possible, but Rha4+ is not. This makes for 12 squares x 1 file x 4 capture/check possibilities, or 48 more moves.

When a rook moves vertically to an edge square that a rook on a different file could move to, it results in a move naming the same file twice, like Rhh3. Such a move cannot discover check, and can only deliver check if it captures a piece, as above. Thus: 12 squares x 1 file x 4 capture/check possibilities, for another 48 moves.

Finally, if a rook moves ambiguously to a space not on the edge of the board, it can always deliver check or checkmate, whether capturing a piece or not (although checks like Ree3+ must be discovered). Any of the eight files may be used to disambiguate such a move, as may any of the seven ranks not being moved to. This yields 36 squares x 15 files/ranks x 6 capture/check possibilities, for 3240 moves.

TOTAL ROOK MOVES: 5296

**Knight moves:**

A knight may move unambiguously to of the 64 squares on the board while capturing a piece or not, and delivering check, checkmate, or neither. This comes out to 64 x 6 capture/check possibilities, or 384 moves.

A knight moving to any square may require disambiguation by file, such as in Nde6. The edge files, a and h, allow for only two original files each, as in Nba2 and Nca2. This makes 16 squares x 2 original files x 6 capture/check possibilities, for 192 moves. (Note that a move like Ngh8+ must be an example of discovered check.)

The next files in, b and g, allow for three original files each, as in Nab5, Ncb5 and Ndb5. This gives us 16 squares x 3 original files x 6 capture/check possibilities, for 288 additional moves.

A knight moving to one of the four central files has four possible original files, for a total of 32 squares x 4 original files x 6 capture/check possibilities, for another 768 moves.

Sometimes two knights on the same file can both move to the same square, in which case their rank is used to disambiguate them. This is impossible for moves to the first and last ranks. For moves to the second and seventh ranks, a pair of knights may be positioned two files away, making a total of 16 squares x 2 ranks x 6 capture/check possibilities, or 192 moves.

For moves to the other four ranks, pairs of knights may be positioned either one or two files away, making for 32 squares x 4 ranks x 6 capture/check possibilities, or 588 moves.

But we arenít done. It is possible, due to underpromotion, for a player to have three or more knights simultaneously on the board, and in extremely rare cases, a knight may move to a square that can be also reached by a knight on the same rank and a knight on the same file. In such cases, it is necessary to disambiguate the move with both rank and file, as in Nb3c5.

This cannot happen for spaces on the edges of the board, as the three knights involved must form the corners of a 3x5 rectangle, with the destination square in the middle. Nor can it happen for the squares near the corners but not on the edgesób2, b7, g2 and g7.

For the 16 squares one space from the edge along one axis (but not both), such as f7 or b4, the rectangle of knights can be oriented in only one way, and the knight can come from any of the four corners of this rectangle. For exampleóNd6f7, Nd8f7, Nh6f7, and Nh8f7. This gives us 16 squares x 4 initial squares x 6 capture/check possibilities, for 384 moves.

Finally, the 16 central squares can each be reached from 8 different squares, and any of these moves may be doubly ambiguous. This gives us another 16 squares x 8 original squares x 6 capture/check possibilities, or 768 moves.

TOTAL KNIGHT MOVES: 3564

**Bishop moves:**

A bishop may move to any of the 64 squares on the board, and may make a capture when doing so, and may also deliver check or checkmate (although a bishop moving into a corner can only give discovered check). This makes for 64 squares x 6 capture/check possibilities, or 384 moves.

If a player underpromotes, it is possible, though quite rare, for two bishops to be able to move to the same space. It isnít simple to count the number of different ranks and files that can be used to disambiguate bishop moves, because not all spaces can be reached by bishops on any file or rank. For example, the move Bgb5 is impossible.

How many possible bishop moves are there in which file is used to disambiguate between two bishops? A bishop moving into a corner cannot do so unambiguously. Of the 60 non-corner squares on the board, 40 of them can be reached by a bishop on any file other than the file being moved to. That makes 40 squares x 7 files x 6 capture/check possibilities; 1680 moves. (Note that bishops moving ambiguously to non-corner squares on the edge of the board can deliver only discovered check or checkmate, but this is always possible, as they are not coming from corners.)

There are 12 squares that can be reached by a bishop on any other file but one. This gives us 12 squares x 6 files x 6 capture/check possibilities, or 432 moves. (Again, some checks may have to be discovered.)

There are 8 squares that can be reached by a bishop on any other file but two. This gives us 8 squares x 5 files x 6 capture/check possibilities, or 240 moves.

Finally, there are 4 squares (a4, a5, h4 and h5) that can be reached by a bishop on any other file but three. This gives us 4 squares x 4 files x 6 capture/check possibilities, or 96 moves.

Now we consider bishop moves that must be disambiguated by rank. In these cases, two bishops must be able to reach the same square from the same file.

We consider first pairs of bishops in a file adjacent to the file being moved to. This is possible for moves to any rank except the first or last, making a total of 48 squares x 2 ranks x 6 capture/check possibilities (some of which may have to be discovered), or 588 moves.

For moves to the middle four ranks, a pair of bishops may be positioned two files away. This yields another 32 squares x 2 ranks x 6 capture/check possibilities, or 384 moves.

For moves to the middle two ranks, a pair of bishops may be positioned three files away. This yields another 16 squares x 2 ranks x 6 capture/check possibilities, or 192 moves.

Note that bishops moving ambiguously from corners cannot deliver check without making a capture, but for every move considered here, a move not from a corner square is possible. For example, the move B8c6+ can (and must) involve a bishop coming from e8, not a8.

Finally, we must consider those absurdly rare cases in which at least two bishops have been underpromoted onto the same color and in which a bishop move is made that requires disambiguation by both rank and file. In such cases, at least three bishops of the same color must form the corners of a square.

A one-space bishop move of this kind can be made to any of the 36 center squares, for 36 squares x 4 initial squares x 6 capture/check possibilities, or 864 moves. (See exception below.)

A two-space bishop move of this kind can be made to any of the 16 center squares, for 16 squares x 4 initial squares x 6 capture/check possibilities, or 384 moves. (See exception below.)

A three-space bishop move of this kind can be made to any of the 4 center squares, for 4 squares x 4 initial squares x 6 capture/check possibilities, or 96 moves. (See exception below.)

As noted before, a bishop cannot move from a corner and deliver check if another bishop on the same file (or rank) can move to the same spaceóunless a capture is madeóbecause the king being checked must already have been checked by one bishop or the other. This makes a number of doubly ambiguous moves that we have already counted impossible, and they must be subtracted from our total. That number is: 12 destination squares x 1 initial corner each x 2 check possibilities (check or checkmate), for a total of 24 moves to be subtracted.

TOTAL BISHOP MOVES: 5316

**Queen moves:**

A queen may move unambiguously to any of the 64 squares on the board (e.g. Qe6). This move may be a capture or not, and may be either check, checkmate, or neither. These combinations lead to 64 x 6, or 384 possible moves.

If a player has promoted a pawn, it may be necessary for a queen move to be disambiguated. Any of the sixteen ranks or files may be used to do so, and any of these moves can still be a capture, check, or checkmate. 64 squares x 16 ranks/files x 6 capture/check possibilities makes 6144 possible moves.

In extremely rare cases, a player may have three or more queens and make a queen move that requires disambiguation by both rank and file. Counting these moves is the most difficult part of this entire problem and requires considering many groups of squares separately.

Queens moving into corner squares in this fashion cannot give check, as all three avenues leading to such a destination square must already be occupied by a queen, meaning that the opponentís king must already be in check. Moreover, these moves must be diagonal ones in order to require double disambiguation. They may be captures, however, which yields: 4 destination squares x 7 original squares x 2 capture possibilities, or 56 moves.

Consider next the 8 squares adjacent to the corners. For each one, there are 14 squares from which a queen can move there with double ambiguity. This excludes one rank or file for each such squareófor example, the move Qb5b1 is impossible, because Qbb1 would always suffice instead. The number of moves thus obtained is 8 destination squares x 14 original squares x 6 capture/check possibilities, or 672 moves.

However, for each of these eight squares, there is exactly one doubly ambiguous move which does not allow check (or checkmate) without a captureóthe one where a queen moves two spaces along the longer diagonal. (Weird but true - the Cluemeister invites you to set it up yourself!) For example, the move Qe3g1+ is impossible. We thus subtract 8 squares x 2 check possibilities, or 16 moves.

Next, we consider the other sixteen squares on the boardís edges. For each of these, the same 14 original squares as above are possible for a doubly ambiguous queen move onto them. As above, the rank or file perpendicular to the adjacent edge of the board is excluded, because a queen moving along that rank or file could be uniquely determined in notation by the use of that rank or file. This yields: 16 destination squares x 14 original squares x 6 capture/check possibilities, for 1344 moves. (Note that, unlike in the previous case, a move like Qd3f1+ is possible because the opposing king may be on h3, shielded from the queen on f3 by a piece on g3.)

Consider next the four squares closest to corners without being on edgesób2, b7, g2 and g7. There are 23 squares from which a queen may move to one of these squares, and in all cases a doubly ambiguous move is possible. This yields 4 destination squares x 23 original squares x 6 capture/check possibilities, for a total of 552 moves.

However, for each of these four squares, there is (again) one move which does not allow check (or checkmate) without a captureóthe one where a queen moves two spaces along the long diagonal. For example, the move Qe4g2+ is impossible. We thus subtract 4 squares x 2 check possibilities, or 8 moves.

Consider next the 8 non-edge squares adjacent to these squares, such as c7 and g3. There are 23 squares from which a queen may move to one of these squares, and in all cases but oneóthe square furthest away along a rank or fileóa doubly ambiguous move is possible. (Qh3b3 is not possible, for example.) This yields 8 destination squares x 22 original squares x 6 capture/check possibilities, for a total of 1056 moves.

Then there are the other eight squares one space from the edge, such as d2 and g5. There are 23 squares from which a queen may move to one of these squares, and in all cases but twoóthe two squares furthest away along a rank or fileóa doubly ambiguous move is possible. (Qa5g5 and Qb5g5 are not possible, for example.) This yields 8 destination squares x 21 original squares x 6 capture/check possibilities, for a total of 1008 moves.

Consider next the 4 squares on the long diagonals next to the center squares (c3, c6, f3 and f6). There are 25 squares from which a queen may move to one of these squares, and in all cases a doubly ambiguous move is possible. This yields 4 destination squares x 25 original squares x 6 capture/check possibilities, for a total of 600 moves.

Then consider the 8 other non-central squares, such as d3 or f5. There are 25 squares from which a queen may move to one of these squares, and in all cases but oneóthe square furthest away along a rank or fileóa doubly ambiguous move is possible. (Qd8d3 is not possible, for example.) This yields 8 destination squares x 24 original squares x 6 capture/check possibilities, for a total of 1152 moves.

Finally, consider the 4 central squares. There are 27 squares from which a queen may move to one of these squares, and in all cases a doubly ambiguous move is possible. This yields 4 destination squares x 27 original squares x 6 capture/check possibilities, for a total of 648 moves.

TOTAL QUEEN MOVES: 13,592

**King moves:**

King moves are easy. A king can move to any of 64 squares, and this move can be a capture. It can also deliver discovered check, or even checkmate, for a total of 64 squares x 6 capture/check possibilities, or 384 moves.

Castling can also be considered a king move. A player can castle either kingside (0-0) or queenside (0-0-0), and to do so may cause check or checkmate, yielding 2 x 3 or 6 more moves.

TOTAL KING MOVES: 390

**Pawn moves:**

Pawns can move without capturing to any of the 48 squares not on the first or last rank, and this move may be check or checkmate. This yields 48 x 3 or 144 moves.

Pawns can capture onto any of the 12 edge squares, which may be check or checkmate, for 12 x 3 or 36 moves.

Pawns can capture from either of two initial files onto the 36 central squares of the board, which yields 36 squares x 2 files x 3 check possibilities, for 216 moves.

Pawns can capture other pawns en passant onto edge squares (a3, a6, h3 and h6), which may deliver check or checkmate, for 4 x 3 or 12 moves.

Pawns can capture other pawns en passant onto non-edge files in two ways each: 12 squares x 2 original files x 3 check possibilities, or 72 moves.

Pawns can promote without capturing into any of four types of pieces on any of 16 squares, and this yields 16 squares x 4 types of pieces x 3 check possibilities, or 192 moves.

Pawns may capture onto corner squares while promoting in this many ways: 4 squares x 4 types of piece x 3 check possibilities, or 48 moves.

Pawns may capture onto non-corner promotion squares in this many ways: 12 squares x 2 original files x 4 types of pieces x 3 check possibilities, or 288 moves.

TOTAL PAWN MOVES: 1008

Adding the six possible types of moves together gives us the grand total and the answer to our problem.

#### GRAND TOTAL: 29,166

Congratulations to Laramie S. for making this month's sole guess of 23,000,017. You were off by about three orders of magnitude, but that's good enough for showbiz - there'll be a prize for you at the con.

The Cluemeister now invites you all to join him in sweeping the pieces majestically off the board, flinging the board capriciously through the nearest window, and leaning back in an easy chair to take a well-earned breather.

And when you're ready, check out the