The table above includes a 'Guard', moving like a nonroyal King. Joe Joyce is quite fond of it, even I have been known to use this piece. The (S) column gives one popular set of standard piece values. The (m+M) column is based on a simple pencil and paper calculation, adding the minimum number of possible moves for the given piece (from a corner square) to the MAXIMUM of possible moves (from a central square). The Knight, for example, has 2 moves minimum and 8 moves MAXIMUM, giving a total of 10 moves. Other people,
with more determination, have precisely calculated a grand total of 336 possible moves from all 64 squares on the board , giving an average value of 5.250 possible moves. Dividing 336 by 32 puts 10.500 in the 'Double Average' column, which is surprisingly close to the previous column. From time to time, I play around with piece values on a cubic playing field with 216 cells, content to use an (m+M) column as my source of raw numbers.
What, if any, sense can we make of these numbers? The last two columns measure piece mobility on an empty board, so they indicate the general strength of each piece in the endgame - which I have found the (S) column well suited to. Note that N + B = R in the Double Average column. No great mystery here, the Knight has 60% of the mobility of the Bishop, while the Rook has 160%. Holding the Bishop at 3 points, this column suggests 4.8 points for the Rook, not an unreasonable choice - some writers assign as little as 4.5 points to the Rook. But nobody values the Knight at 1.8 points! To arrive at the 'standard' values, one must make arbitrary changes in the raw numbers, forcing them towards a desired conclusion. 'Knight-moves' need to be counted as more valuable than the moves made by other pieces, perhaps by a 5:3 ratio. The penalty I am inclined to give the Bishop for being colorbound (therefore limited to half the board) needs to be cancelled out by a matching bonus for the fact that every Bishop move either attacks or retreats. The Rook, with its boring sideways moves, usually attacks only a single enemy piece - also it will have only a single line of retreat after capturing that piece. I love Rooks, but am forced to admit that they are superior to Bishops only because they have many move possible moves, on average. The 3D Rook moves up and down along one axis and sideways along two different axes, making it even more 'boring' than the 2D Rook. I am presently re-thinking the entire subject of piece values for 3D chess.
Here is an idea I had one day: recently Joe Joyce and I have been using the Elephant piece, which can move like a Ferz or an Alfil. Let the Grand Rook move like a Rook or an Elephant and let the Chancellor move like a Rook or a Knight. These two pieces, each adding eight shortrange moves to the Rook, should be nearly identical in value on most boards. But I consider a Grand Rook to be worth around half a Pawn less than a Queen on the 8x8 board - contradicting several statements by Ralph Betza (gnohmon) that the Chancellor and Queen are equal in value. This procedure is an art, not a science, and is even more difficult when working with different boards and new pieces. See my Rose Chess XII for a collection of interesting pieces, inspired by the writings of Ralph Betza, plus some theory of their values on a 12x12 board.
The table above includes a 'Guard', moving like a nonroyal King. Joe Joyce is quite fond of it, even I have been known to use this piece. The (S) column gives one popular set of standard piece values. The (m+M) column is based on a simple pencil and paper calculation, adding the minimum number of possible moves for the given piece (from a corner square) to the MAXIMUM of possible moves (from a central square). The Knight, for example, has 2 moves minimum and 8 moves MAXIMUM, giving a total of 10 moves. Other people, with more determination, have precisely calculated a grand total of 336 possible moves from all 64 squares on the board , giving an average value of 5.250 possible moves. Dividing 336 by 32 puts 10.500 in the 'Double Average' column, which is surprisingly close to the previous column. From time to time, I play around with piece values on a cubic playing field with 216 cells, content to use an (m+M) column as my source of raw numbers.
What, if any, sense can we make of these numbers? The last two columns measure piece mobility on an empty board, so they indicate the general strength of each piece in the endgame - which I have found the (S) column well suited to. Note that N + B = R in the Double Average column. No great mystery here, the Knight has 60% of the mobility of the Bishop, while the Rook has 160%. Holding the Bishop at 3 points, this column suggests 4.8 points for the Rook, not an unreasonable choice - some writers assign as little as 4.5 points to the Rook. But nobody values the Knight at 1.8 points! To arrive at the 'standard' values, one must make arbitrary changes in the raw numbers, forcing them towards a desired conclusion. 'Knight-moves' need to be counted as more valuable than the moves made by other pieces, perhaps by a 5:3 ratio. The penalty I am inclined to give the Bishop for being colorbound (therefore limited to half the board) needs to be cancelled out by a matching bonus for the fact that every Bishop move either attacks or retreats. The Rook, with its boring sideways moves, usually attacks only a single enemy piece - also it will have only a single line of retreat after capturing that piece. I love Rooks, but am forced to admit that they are superior to Bishops only because they have many move possible moves, on average. The 3D Rook moves up and down along one axis and sideways along two different axes, making it even more 'boring' than the 2D Rook. I am presently re-thinking the entire subject of piece values for 3D chess.
Here is an idea I had one day: recently Joe Joyce and I have been using the Elephant piece, which can move like a Ferz or an Alfil. Let the Grand Rook move like a Rook or an Elephant and let the Chancellor move like a Rook or a Knight. These two pieces, each adding eight shortrange moves to the Rook, should be nearly identical in value on most boards. But I consider a Grand Rook to be worth around half a Pawn less than a Queen on the 8x8 board - contradicting several statements by Ralph Betza (gnohmon) that the Chancellor and Queen are equal in value. This procedure is an art, not a science, and is even more difficult when working with different boards and new pieces. See my Rose Chess XII for a collection of interesting pieces, inspired by the writings of Ralph Betza, plus some theory of their values on a 12x12 board.