CVector

CVector is an unusual variant, though arguably no more unusual than Ultima, which is popular and successful as chess variants go. While the concept is simple, keeping track of what's going on on the board at any given time can be quite tricky. The idea comes from William James Tychonievich; CVector is based on his much simpler game Troika.

The essence of CVector is that the pieces don't have pre-defined moves. Instead, the type of movement available to each piece on the board depends on the positions of other pieces, which of course change as the game goes on. The mechanism for this simple and straightforward, but the results are not. To ease the learning curve slightly, CVector is designed so as to be played with an ordinary chessboard and pieces. (A stack of checkers or other flat markers will come in handy.) To ease it further, a number of tutorial problems are included below.

The object of the game in CVector is to capture the enemy king or leave the enemy with no legal moves. A stalemate is a win, not a draw. The distinction between checkmate and the actual capture of the king is not entirely academic: Unless you're careful, it's entirely possible in CVector to blunder into a position where your king is threatened, and not realize it. Fortunately, your opponent may not realize it either. This situation is not illegal, and does not require that moves be retracted and replayed.

Each player starts the game with the standard lineup of non-pawn pieces, but no pawns. (The advanced version of the game, described in the last section of this page, also includes two pieces called princes, which can be represented by pawns. It's probably a bad idea to try playing the advanced version before you've mastered the basic version. You're almost certain to get terminally confused and give up. I nearly did.)

The opening setup is shown in Figure 1. It's important to understand at the outset that while we're using pieces with familiar names and abbreviations, the only resemblance between them and traditional chess pieces is that they occupy squares. Oh, and capturing the king ends the game. Aside from that, we could just as easily call them quotas, nodules, rustics, and barbies.

Figure 1. The opening setup in CVector.

 _ _ _ _ _ _ _ _
|_|_|n|q|k|b|_|_|
|_|_|b|r|r|n|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|B|R|R|N|_|_|
|_|_|N|Q|K|B|_|_|

Vectors

In order to explain how the pieces move in CVector, we need to define the term "vector." In moving, pieces must always follow certain vectors. A vector is the geometrical distance between any two squares, and any two squares define a vector. Here, for instance, are two squares that define a particular vector:

 _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|x|_|_|_|
|_|_|_|_|_|_|_|_|
|_|x|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|

We might describe this vector as "move three squares in an orthogonal direction, then turn at right angles and move two more squares." (Note: "Orthogonal" is the opposite of "diagonal." It means "either vertical or horizontal.") We might also describe it as "two squares that are at opposite corners of a 3x2 rectangle," or simply as "3/2." I've found it easier to visualize vectors by looking, when possible, at the inner rectangle that lies between the two squares. I find that the 1x2 rectangle lying between the two squares in the vector shown above is easier to see when examining a crowded board position than the bigger 3x2 rectangle.

However you choose to visualize it, a vector can be rotated by 90, 180, or 270 degress, and also mirrored. On the board below, the queen can reach any of the eight squares marked 'x' using the vector above.

 _ _ _ _ _ _ _ _
|_|x|_|_|_|x|_|_|
|x|_|_|_|_|_|x|_|
|_|_|_|_|_|_|_|_|
|_|_|_|Q|_|_|_|_|
|_|_|_|_|_|_|_|_|
|x|_|_|_|_|_|x|_|
|_|x|_|_|_|x|_|_|
|_|_|_|_|_|_|_|_|

Some vectors, like this one, define eight possible moves. Others, which are defined by pairs of squares that lie on the same row, column, or diagonal, define only four. In the case of long vectors, many of the possible destination squares for a move may lie off the edge of the board, leaving fewer actual moves for a piece that is currently able to use the vector.

Piece Movement

None of the pieces in CVector has a move of the "one or more squares in such-and-such a direction" type; all moves consist of moving directly from the starting square to an end square and stopping. All pieces in CVector jump from their starting square directly to their ending square: Moves are never blocked by intervening pieces.

The basic method for figuring out how a given CVector piece can move at a given moment is to look at the vector formed by the current positions of two other pieces. Let's start with the knights and bishops, as their movement definition is simplest. Each of a player's knights moves using the vector defined by the current positions of the player's king and the player's other knight. Likewise, each bishop moves using the vector defined by the positions of the king and the other bishop.

Let's see how this works in practice. The knight on c2 in Figure 2a, below, can move two squares in any orthogonal direction, as defined by the relation between the king (f3) and the other knight (f5). Its possible moves are marked 'x' and '*'. The possible moves of the f5 knight are marked '+' and '*'. The squares marked '*' can be reached by both knights. It's a general feature of this way of defining movement that the two pieces in a pair will always be able to reach one or more of the same squares (though in some positions those common squares may be off the edge of the board). The second and third diagrams have been marked in a similar way: squares that can be reached by both knights are marked '*'.

Figure 2a. The squares to which either knight can move are marked '*'. In addition, the f5 knight can move to the squares marked '+', while the c2 knight can also move to the square marked 'x'.

 _ _ _ _ _ _ _ _
|_|_|_|_|+|_|+|_| 8
|_|_|_|_|_|_|_|_| 7
|_|_|+|_|_|_|_|_| 6
|_|_|_|_|_|N|_|_| 5
|_|_|*|_|_|_|_|_| 4
|_|_|_|_|_|K|_|_| 3
|x|_|N|_|*|_|+|_| 2
|_|_|_|_|_|_|_|_| 1
 a b c d e f g h

Figure 2b. Two more examples of how the movement vectors of a pair of knights are defined. As in Figure 2a, squares that both knights can reach are marked '*', while those that can be reached by one or the other of the knights are marked '+' or 'x'.

 _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_| 8
|_|+|_|_|_|_|_|_| 7
|_|_|_|_|_|_|_|_| 6
|_|_|_|_|_|N|_|_| 5
|_|_|_|_|_|_|K|_| 4
|_|*|_|x|_|_|_|_| 3
|_|_|N|_|_|_|_|_| 2
|_|x|_|*|_|_|_|+| 1
 a b c d e f g h
 _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_| 8
|_|_|_|_|_|_|_|_| 7
|_|_|_|_|_|+|_|_| 6
|_|_|_|_|+|N|+|_| 5
|_|_|_|_|_|*|K|x| 4
|_|_|_|_|_|_|N|_| 3
|_|_|_|_|_|x|_|x| 2
|_|_|_|_|_|_|_|_| 1
 a b c d e f g h

Note that when either knight moves, the movement vector of the other knight will be redefined. The definition may be the same as before, or not -- for example, if the g3 knight in the diagram just above moves to h4 or f4, the vector used by the f5 knight will not change. But if the g3 knight moves to f2 or h2, the f5 knight's movement vector will change. Likewise, if the king moves, the movement vectors of both knights (and in fact of all the pieces in the king's army) are likely to be redefined. It's the essence of CVector that the nature of piece movement is in a constant state of flux.

In general, the further a piece is from the king, the more likely it becomes that the moves available to its partner will be restricted because some or most of them will be off the edge of the board. In consequence, it's not a good idea to leave the king in a protected position in the rear while one advances one's other forces. The king will need to be more or less in the thick of the battle. (Since the king is a powerful piece, this is not necessarily a bad thing.) The more concentrated your forces are, the more moves each piece is likely to have.

The movement of the bishops is defined exactly like the movement of the knights, so there's no difference in value between knights and bishops.

Visualizing the moves is difficult. Let's see if we can make it just a tiny bit easier. It's a general feature of piece pair moves on the CVector board that when any pair of pieces is on a vertical, horizontal, or diagonal, if you look at the positions of the king and that pair of pieces as occupying three corners of either a symmetrical trapezoid or a parallelogram, either piece can move to the fourth corner. There will be two such corners, one in which the king/vacant corner side is parallel to the piece pair side, and another opposite the king. In all four diagrams below, both pieces of the pair can move to the square or squares marked '.' Bear in mind that in some positions the fourth corner might be off the edge of the board.

            
 _ _ _ _ _ _ _ _     _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_|   |_|_|_|_|_|_|_|_|
|_|_|_|b|_|_|_|_|   |_|_|_|_|_|.|_|_|
|.|_|_|_|_|_|.|_|   |_|_|b|_|_|_|k|_|
|_|_|_|_|_|_|_|_|   |_|_|_|_|_|_|_|_|
|k|_|_|_|_|_|_|_|   |_|_|_|_|_|_|_|_|
|_|_|_|b|_|_|_|_|   |_|.|_|_|_|b|_|_|
|_|_|_|_|_|_|_|_|   |_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|   |_|_|_|_|_|_|_|_|

 _ _ _ _ _ _ _ _     _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_|   |_|.|_|_|_|_|_|_|
|_|_|k|_|_|_|.|_|   |_|_|_|n|_|_|_|_|
|_|_|_|_|_|_|_|_|   |_|_|_|_|n|_|_|_|
|_|_|_|_|_|_|_|_|   |_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|   |_|_|_|_|_|k|_|_|
|_|_|_|_|_|_|_|_|   |_|_|_|_|_|_|_|_|
|_|_|_|n|_|n|_|_|   |_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|   |_|_|_|_|_|_|_|_|

When the pair are not on an orthogonal or diagonal, they can both move to the fourth corner of a parallelogram, the corner opposite the king. Again, the fourth corner may be off the edge of the board.

 
 _ _ _ _ _ _ _ _     _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_|   |_|_|_|n|_|_|_|_|
|_|_|_|_|_|_|_|_|   |_|.|_|_|_|_|_|_|
|_|_|_|_|k|_|_|_|   |_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|b|_|   |_|_|_|_|k|_|_|_|
|_|_|_|_|_|_|_|_|   |_|_|n|_|_|_|_|_|
|_|_|b|_|_|_|_|_|   |_|_|_|_|_|_|_|_|
|_|_|_|_|.|_|_|_|   |_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|   |_|_|_|_|_|_|_|_|

 _ _ _ _ _ _ _ _
|_|_|k|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|n|_|_|_|_|_|_|
|_|_|_|n|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|.|_|_|_|_|_|

The rooks are more powerful than the knights and bishops. Each rook can use, throughout the game, two movement vectors. The rooks' first vector is defined in the same manner as the vectors of the knights and the bishops, with reference to the positions of the king and the other rook. But rooks can also use the vector defined by the current positions of the two enemy rooks. (The position of the enemy king is irrelevant.) The result is that when either player moves a rook, the movement vectors of the other player's rooks will most likely change. This change will need to be taken into account in figuring out whether a rook move is desirable. The rook is the only piece whose movement vector is affected by the positions of enemy pieces.

Figure 3a. The black rooks (lower-case 'r' in the ASCII diagram) can both reach the positions marked '*'. The b6 rook can also reach the squares marked '+' while the f7 rook can reach the squares marked 'x'. The position of the white king is irrelevant to the black rooks, as they use their own vectors and the vector defined by the two white rooks. If you're still not clear about why CVector is a difficult game to play, take a look at Figure 3b, where I've edited this diagram to include the possible moves of the white rooks. It's a mess. The four rooks amongst them threaten more than 20 squares in a completely irregular pattern.

 _ _ _ _ _ _ _ _
|_|_|_|_|+|_|_|_| 8
|+|_|*|_|_|_|r|_| 7
|_|r|_|_|_|k|_|_| 6
|+|_|+|x|_|_|_|_| 5
|_|_|_|_|*|R|_|_| 4
|_|_|_|+|K|_|x|_| 3
|_|_|R|_|_|_|_|_| 2
|_|_|_|_|_|_|_|_| 1
 a b c d e f g h

Figure 3b. The position is the same as in Figure 3a. Squares reachable by the black rooks (r) are marked 'x', and squares reachable by the white rooks (R) are marked '+'. Squares reachable by both white and black rooks are marked '*'.

 _ _ _ _ _ _ _ _
|_|_|_|_|x|_|_|_| 8
|x|+|x|+|_|_|r|_| 7
|_|r|_|_|+|k|+|_| 6
|*|_|x|*|_|_|_|+| 5
|_|_|_|_|x|R|_|_| 4
|+|+|_|*|K|_|x|+| 3
|_|_|R|_|+|_|+|_| 2
|_|+|_|+|_|_|_|+| 1
 a b c d e f g h

The king and queen don't constitute a pair of pieces in the sense described above. Both are both powerful pieces, and they're similar to one another, but with some important differences. Both the king and the queen can use three different vectors at any given time, and they both use the same three vectors. The three vectors are those defined by the positions of the other piece pairs -- the friendly rooks, bishops, and knights. Figure 4 illustrates how the queen moves in one particular board position.

Figure 4a. The queen in this position can reach the two squares marked '*' using the vector defined by the bishops, the four squares marked '.' using the vector defined by the rooks, and the six squares marked 'x' using the vector defined by the knights. Moving any of the other pieces except the king is likely, though not certain, to affect the queen's available vectors. In this position, for instance, if the h3 rook moves to f3 (which it can do using the vector defined by the king and the g3 rook), the two rooks will define the same vector as before, so the king's and queen's moves won't change.

 _ _ _ _ _ _ _ _
|_|*|_|_|_|_|_|*| 8
|B|_|_|_|_|_|_|_| 7
|_|_|_|x|_|x|_|_| 6
|_|N|_|_|_|_|K|_| 5
|_|x|_|_|.|_|_|x| 4
|_|_|_|.|Q|.|R|R| 3
|N|x|_|B|.|_|_|x| 2
|_|_|_|_|_|_|_|_| 1
 a b c d e f g h

Figure 4b. The position is the same as in Figure 4a, but now we're looking at the movement possibilities for both the king and queen (the former marked '+', the latter 'x', and squares that both can reach '*'). I'll leave it for you to complete the diagram by marking the squares that can be reached by the other pieces.

 _ _ _ _ _ _ _ _
|_|*|_|_|_|+|_|*| 8
|B|_|_|_|_|_|_|_| 7
|_|_|_|*|_|x|+|_| 6
|_|N|_|_|_|+|K|+| 5
|_|x|_|+|x|_|+|x| 4
|_|_|_|x|Q|x|R|R| 3
|N|*|_|B|x|+|_|*| 2
|_|_|_|_|_|_|_|_| 1
 a b c d e f g h

There are three differences between the king and the queen. The first is that capturing the enemy queen doesn't end the game. The second is that moving the queen has no effect on the movement vectors of other pieces. The third will have to be deferred until we've discussed capturing.

Capturing

If you've been paying attention, you're probably wondering what happens when one of the pieces in a pair (a bishop, say) is captured. Doesn't that leave the other piece in the pair with no defined vector to use -- in other words, immobile?

No. In CVector, capturing is performed normally, with the capturing piece moving onto the square occupied by an enemy piece. However, captured pieces are not removed from the board. Instead, they're "frozen" on their current square. If you're playing with actual pieces, the fact that a piece is frozen can conveniently be shown by putting a checker or other flat marker under it.

Until the capturing piece moves off of the square, of course, there will be two pieces on the square, one frozen and one still movable. If your chessboard isn't capacious enough to hold two pieces on a single square, doubtless you'll be able to come up with some workaround. For instance, you might position the captured piece and its checker just off the edge of the board, and remember where it's supposed to go when the other piece moves.

The unfrozen piece in a pair continues to define its movement by the vector between its frozen partner and its king, this vector still being subject to change as the king moves.

Now I can explain the third difference between the king and the queen: When both pieces of a given pair are frozen, the king can no longer use the vector between them to define its movement. Thus, when all six of the paired pieces on a side are frozen, their king is immobile. (And the game is likely, in consequence, to be over before very long.) The queen continues to use frozen pair vectors for movement. Likewise, even when both enemy rooks are frozen, the rooks continue to be able to move using their vector.

Won't there be positions where more than two pieces find themselves occupying a single square? What happens, for instance, when my piece captures yours, and you immediately retake? In order to explain how this scenario unfolds, we need to discuss swapping.

Swapping

When a piece is frozen, it loses the ability to move under its own power for the remainder of the game. It can still be moved, however, by swapping places with another friendly piece, one that still has the power to move. In a swapping move, a piece -- any piece -- moves onto the square occupied by a frozen piece from its own side, just as if it were capturing an enemy piece. At this point, the frozen piece (which is still frozen) transfers to the square from which the swapping piece started its move. The two pieces trade places. The movement vectors defined by the frozen piece are likely to be altered by the swap, just as if the piece had moved using its own power.

The movable piece that initiates the swap uses its own movement vector in order to arrive on the square where the swap is initiated. The movement vector of the frozen piece is irrelevant.

Now I can explain what happens when a series of two or more captures take place on the same square. Consider the position below. (I've removed a few pieces to make it easier to see what's going on.) The black queen, on f1, is already sitting on the same square as the white bishop; the latter is frozen, as indicated by the parentheses. The white rook on c4 is about to be captured by the black bishop. Sorry for the size of the ASCII diagram; to make matters worse, we'll have to look at it several times to see the sequence of events clearly.

 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |  k  |     |  8
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|  B  |     |     |     |     |  b  |     |     |  7
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  6
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |  b  |     |     |     |     |     |     |  5
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |  R  |     |     |     |  N  |     |  4
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  3
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |  N  |     |  K  |     |     |  2
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|  R  |     |     |     |     |  q  |     |     |  1
|     |     |     |     |     | (B) |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
   a     b     c     d     e     f     g     h

When the bishop captures the rook, the rook becomes frozen. That move gives us this position:

 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |  k  |     |  8
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|  B  |     |     |     |     |  b  |     |     |  7
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  6
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  5
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |  b  |     |     |     |  N  |     |  4
|     |     | (R) |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  3
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |  N  |     |  K  |     |     |  2
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|  R  |     |     |     |     |  q  |     |     |  1
|     |     |     |     |     | (B) |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
   a     b     c     d     e     f     g     h

Next, white is going to retake with the d2 knight (which moves, at the moment, like a normal knight because of the vector between the white king and the other knight). When this happens, the frozen white rook is swapped to d2, and the black bishop becomes frozen:

 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |  k  |     |  8
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|  B  |     |     |     |     |  b  |     |     |  7
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  6
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  5
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |  N  |     |     |     |  N  |     |  4
|     |     | (b) |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  3
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |  K  |     |     |  2
|     |     |     | (R) |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|  R  |     |     |     |     |  q  |     |     |  1
|     |     |     |     |     | (B) |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
   a     b     c     d     e     f     g     h

But wait -- we're not finished yet. Observe that the black queen, on f1, can reach c4 using the vector defined by the two black bishops. When the queen captures (freezes) the knight, the black bishop is swapped to f1, where the white bishop is already frozen. Yes, in CVector two frozen pieces can come to occupy the same square.

 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |  k  |     |  8
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|  B  |     |     |     |     |  b  |     |     |  7
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  6
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  5
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |  q  |     |     |     |  N  |     |  4
|     |     | (N) |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |     |     |     |  3
|     |     |     |     |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|     |     |     |     |     |  K  |     |     |  2
|     |     |     | (R) |     |     |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
|  R  |     |     |     |     | (b) |     |     |  1
|     |     |     |     |     | (B) |     |     |
 ----- ----- ----- ----- ----- ----- ----- -----
   a     b     c     d     e     f     g     h

Once two opposing pieces are frozen on the same square, neither can be swapped off to another square for the rest of the game, because neither player can move a piece onto that square. You can't move any of your pieces onto squares occupied by frozen enemy pieces.

In case it isn't obvious, two live pieces from the same side can't swap places. Swapping can only be done with frozen pieces.

Given this set of rules, it's not possible for two frozen pieces from the same side to ever occupy the same square. Nor is it allowed for an active piece to occupy the same square as a frozen piece from the same side: When the active piece arrives, a swap must take place.

That's about it for rules in standard CVector. Before we go on to take a peek at the deeper level of confusion that reigns in CVector Plus, let's pause for a few CVector problems. A tutorial, if you will. The first few problems are easy; they get progressively harder. In most of the problems I've left some pieces off the board in order to make it easier to see the moves. The answers are at the very bottom of the page.

Problems

Problem 1. White has a move that will threaten both black knights. Can you find it?

 _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_| 8
|_|_|_|_|_|n|_|_| 7
|_|_|k|_|_|_|_|_| 6
|_|_|_|_|_|_|_|_| 5
|_|_|_|K|_|_|_|_| 4
|_|_|_|_|_|_|B|n| 3
|_|_|B|_|_|_|_|_| 2
|_|_|_|_|_|_|_|_| 1
 a b c d e f g h

Problem 2. White to play. White's king is threatened by both black bishops. How can white turn the tables?

 _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_| 8
|_|_|_|_|_|_|_|_| 7
|_|_|_|b|_|_|_|_| 6
|_|_|_|_|_|k|_|_| 5
|_|b|_|_|_|_|_|_| 4
|_|_|N|_|_|N|_|_| 3
|_|_|K|_|_|_|_|_| 2
|_|_|_|_|_|_|_|_| 1
 a b c d e f g h

Problem 3. Find two different moves for white, each of which creates attacks on two black pieces.

 _ _ _ _ _ _ _ _
|_|_|_|_|_|k|_|_| 8
|_|_|_|_|_|_|_|_| 7
|_|_|_|n|_|_|_|_| 6
|_|_|_|_|n|r|_|_| 5
|_|_|r|_|B|_|_|_| 4
|_|_|B|_|_|K|_|_| 3
|_|_|_|_|_|_|_|_| 2
|_|_|_|_|_|_|_|_| 1
 a b c d e f g h

Problem 4. Black's king and one other black piece are safe. Which piece? Is black's position better or worse than white's?

 _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_|
|_|N|_|_|n|_|_|_|
|_|n|b|_|_|_|_|_|
|_|k|K|N|_|_|_|_|
|_|_|B|_|_|_|_|_|
|_|B|b|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|

Problem 5. White to play. Analyze the white position and decide whether it is strong or weak.

 _ _ _ _ _ _ _ _
|_|_|N|_|R|_|_|_|
|Q|_|_|_|_|_|_|_|
|_|n|_|_|b|_|_|_|
|N|_|_|_|_|_|B|_|
|_|_|_|b|_|k|r|r|
|_|_|_|_|n|_|_|R|
|_|_|K|_|q|_|_|_|
|_|_|_|_|_|_|B|_|

Problem 6. White would like to capture the black bishop on d3 using the f1 rook. The black bishop is undefended, and is threatening the white bishop on e4, so the capture seems to be a good idea. Also, white would like to bring his rooks closer together. Is it a safe move? If not, why not?

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|_|_|_|_|_|_|_|_| 8
|_|_|_|_|_|_|b|_| 7
|_|_|B|_|_|k|r|_| 6
|_|_|K|_|_|r|_|_| 5
|_|_|_|_|B|_|_|_| 4
|R|_|_|b|_|_|_|_| 3
|_|_|_|_|_|_|_|_| 2
|_|_|_|_|_|R|_|_| 1
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Problem 7. White would like to capture the black queen, which is undefended, with the d5 rook, but this capture is not safe. Can you see why? How can white check the black king and simultaneously catch the black queen?

 _ _ _ _ _ _ _ _
|_|_|_|_|r|_|r|_| 8
|_|q|_|_|_|_|_|_| 7
|_|_|_|_|_|k|_|_| 6
|B|_|_|R|_|_|R|_| 5
|_|_|_|B|_|_|_|_| 4
|_|_|_|_|K|_|_|_| 3
|_|_|_|_|_|_|_|_| 2
|_|_|_|_|_|_|_|_| 1
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Problem 8. White has a seemingly overwhelming material advantage. The only fly in the ointment is that the white king is in check. What is white's best recourse?

 _ _ _ _ _ _ _ _
|_|_|_|_|_|_|_|_| 8
|_|_|_|B|_|_|_|_| 7
|_|_|_|R|_|_|_|_| 6
|_|B|_|_|_|N|_|_| 5
|_|_|_|_|_|_|_|_| 4
|_|_|_|_|_|_|_|N| 3
|_|Q|_|_|r|r|_|R| 2
|_|_|_|_|_|K|k|_| 1
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Problem 9. White has a move that will win the game. Can you find it?

 _ _ _ _ _ _ _ _
|_|n|_|_|N|_|r|_| 8
|_|_|_|k|_|_|_|_| 7
|_|_|B|n|_|_|b|_| 6
|_|b|_|K|_|_|r|_| 5
|_|_|_|_|_|R|_|_| 4
|_|_|_|R|_|_|_|_| 3
|_|q|_|_|_|N|_|_| 2
|_|_|_|_|_|B|_|Q| 1
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CVector Plus

Now that you've mastered CVector (I am permitted ze leetle joke, yes?), CVector Plus may interest you. The differences are easily explained. First, each player has, in addition to the CVector pieces, two princes. The opening setup is shown below:

 _ _ _ _ _ _ _ _
|_|_|n|q|k|b|_|_|
|_|p|b|r|r|n|p|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|_|_|_|_|_|_|_|
|_|P|B|R|R|N|P|_|
|_|_|N|Q|K|B|_|_|

The princes' movement vector is defined by the positions of their own king and queen. The interesting thing about the princes is not their move but the fact that they can bring frozen pieces back to life. This is done by moving onto the square occupied by a frozen piece, as in a normal swap move. The frozen piece is swapped to the prince's starting square, and is unfrozen.

A prince can unfreeze an enemy piece if desired. There is usually no reason to do this, but on rare occasions it might be used as a "sacrifice" to put the prince in position for an attack. Capturing the enemy princes is likely to be a high priority, however, so using a prince for attack will likely be a last resort. If two frozen pieces are on the same square, neither can be unfrozen by a prince.

The king and queen do not use the vector defined by the princes for their movement; they use only the vectors defined by the bishops, knights, and rooks, as in normal CVector.

The other difference between CVector and CVector Plus is more complex. In CVector Plus, any piece except the king that reaches one of the enemy's eight "home squares" (see Figure 5) may, if the player desires, be promoted -- well, it isn't exactly a promotion, so let's call it a mutation -- to become any other piece except a king. The mutation can occur as part of a swap move, which means that a frozen piece can mutate. In this case it will still be frozen. A mutation must be made immediately upon the piece's reaching a suitable square, and can only be revoked by the piece moving off of that square and back onto the same or another mutation square for another mutation. If a player neglects to make an allowed mutation before the opponent's move, the opportunity to mutate is lost.

Figure 5. The squares on which pieces can be mutated in CVector Plus.

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|_|_|x|x|x|x|_|_| 8
|_|_|x|x|x|x|_|_| 7
|_|_|_|_|_|_|_|_| 6
|_|_|_|_|_|_|_|_| 5
|_|_|_|_|_|_|_|_| 4
|_|_|_|_|_|_|_|_| 3
|_|_|x|x|x|x|_|_| 2
|_|_|x|x|x|x|_|_| 1
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The implications of mutation for the movement of other pieces are as follows:

When a player has three (or more) knights, bishops, or rooks, each piece in such a group can use the vectors defined by the relationships between all of the other identical pieces and the king. In the case of rooks, enemy rooks can use any of the vectors defined by pairs of your own rooks, so mutating a piece into a rook will usually be a very bad idea. However, mutating one or, if you can manage it, both of your bishops to knights (or vice-versa) will increase the power of your forces considerably.

When one piece of a pair (rook, knight, or bishop, not prince) is mutated to another piece type, the remaining piece is instantly frozen in place. Mutating another piece to that type does not unfreeze the previously frozen piece. It can be unfrozen by a prince only if there is another friendly piece of the same type (frozen or unfrozen) to define its movement vector. If a piece can't be unfrozen, a prince can swap squares with it exactly as any other piece would.

Ah, but what if the prince was starting on a square where that unpaired piece could be mutated? In this case, the piece can be unfrozen and mutated in a single move by swapping.

The king and queen can use the vectors defined by any two of the pieces within a rook, bishop, or knight group. A group of four knights, for example, defines as many as six vectors. The king can't use vectors defined by frozen pairs of pieces, however. If a third unfrozen piece of the same type is on the board, the king can only use the vectors defined by the unfrozen piece and either of the frozen pieces.

If the king has become immobile because all of its piece pairs are frozen, a queen or prince can mutate to a rook, knight, or bishop in order to allow the king to move.

If a piece is mutated to a queen, the princes can make use of the vectors defined by the king and any queen, but not the vectors defined by a pair of queens.

Answers to Problems

Answer to problem 1: B-d5. This bishop doesn't threaten anything from its new position. However, the other bishop now threatens the h3 knight, while the king threatens the f7 knight (because of the vector defined by the two bishops).

Answer to problem 2: K-c5 puts the black king in triple-check.

Answer to problem 3: B-d4 threatens both knights -- d4xe5 and also e4xd6. K-d4 threatens c3xc4 and e4xf5. (By the way, Kxe5 is suicidal, because the black king protects e5 using the rook vector.)

Answer to problem 4: Black's c6 bishop is not being attacked. All of white's pieces are safe, so the white position is superior.

Answer to problem 5: Overall, white's position appears far weaker, as the white pieces are much too spread out and black has several attacks, including an attack on the white king. However, the black king is threatened by the white queen using the rook vector, and it's white's move, so white has won the game.

Answer to problem 6: The capture is not safe. Once the two white rooks are positioned at a3 and d3, the black rook at f5 will be attacking the white king.

Answer to problem 7: Rxb7 leaves the white king attacked by the f8 rook. The right move is K-e6. This checks the black king with the g5 rook, and also attacks the black queen via the a5 bishop.

Answer to problem 8: White should resign. The white king has only one move -- d3 -- because the white forces are badly arrayed. This square is threatened by the e2 rook, so white has lost.

Answer to problem 9: K-e3. The squares to which the black king can move are marked '.' in the diagram below. After K-e3, g7 is covered by the white queen using the knights' vector, a7 and d4 by the c6 bishop, c2 and e2 by the d3 rook, and f5 by the f4 rook. The e8 knight, meanwhile, is poised to capture the black king. The king has no safe square to which to retreat, and the e8 knight is not under attack. All of these threats are put in place by the white king's move.

 _ _ _ _ _ _ _ _
|_|n|_|_|N|_|r|_| 8
|.|_|_|k|_|_|.|_| 7
|_|_|B|n|_|_|b|_| 6
|_|b|_|_|_|.|r|_| 5
|_|_|_|.|_|R|_|_| 4
|_|_|_|R|K|_|_|_| 3
|_|q|.|_|.|N|_|_| 2
|_|_|_|_|_|B|_|Q| 1
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