
The Puzzling Side of ChessJeff Coakley 
Average Mobility: A Puzzling Calculation One way to evaluate the relative strength of the chess pieces is to compare their average mobility on an empty board. The mobility of a piece is measured by its number of possible moves. With a little basic arithmetic, we can determine values for the following terms: R average mobility of a rook The point of the exercise is to answer this question. "Which is greater, R or the sum of B and N?" I think you will be surprised at the answer.
Don't worry, folks. There are some "real puzzles" in the latter part of this column. If you're not in the mood for a math quiz, here are the values for average mobility on an empty board. The calculations are given in the solution section. R = 14 moves So the answer to the question is "neither". R = B + N The average mobility of a rook is exactly equal to the sum of the average mobilities of bishop and knight. This is an amazing coincidence. There is no logical reason why these numbers should combine so harmoniously. Another question: "Which is greater, the average mobility of a rook doubled or the sum of the average mobilities of a queen and a knight?" (2 x R) or (Q + N)? As we all know, a queen has the powers of a rook and a bishop, so her average mobility is the sum of theirs. Average mobility Q = R + B A bit more math will show, perhaps less surprisingly than before, that the average mobility of a rook doubled is equal to the sum of the average mobilities of queen and knight. 2 x R = Q + N We can also derive the following noteworthy equation: Q = (2 x B) + N Of course, calculations like these are not the basis for determining the standard value of the pieces. The soundness of the 95331 counting system has been established by centuries of praxis, not by mathematics. Master games prove that a bishop and a knight together are superior to a rook (3 + 3 > 5). They demonstrate how a queen and a knight can outplay two rooks (9 + 3 > 5 + 5). But still, isn't it strange that there is no actual math underlying these numerical values? In conclusion, one important fact should be stated. The average mobility of a chess piece varies with the size of the board. For an 8 by 8 chessboard, the equation R = B + N is true. However, with different size boards, all the values change. Larger boards favour the rook. On a 10 by 10 board: R = 18 Smaller boards help the minor pieces. On a 6 by 6 board: R = 10 Maybe there is some natural connection between the chess pieces and our good old 8 by 8 board. Before you ponder that for too long, here is a selection of puzzles to occupy your mind. They all feature rooks, bishops, and knights. 1. Triple Loyd #14
Place the black king on the board so that: A. Black is in checkmate. For triple loyds 113 and additional information on this kind of puzzle, see the following columns from last year in the ChessCafe.com Archives: June, July, September, November. 2. Triple Loyd #15
Place the black king on the board so that: A. Black is in checkmate. The rest of the puzzles all begin with this dynamic position.
3. Construction Task #3a
Construct a position with a white king, rook, bishop, and knight against a lone black king so that White has the most mates in one move. Discovered checks are not allowed. For more explanation about construction tasks, see the Eight Officers column from October 2012. Constructed positions must be legal. In other words, they must be reachable from an actual game. To show that a position is legal, find two previous moves (one white, one black) that would lead to the position. The usual difficulty is that Black was in an impossible double check on the previous turn. 4. Construction Task #3b
Construct a position with a white king, rook, bishop, and knight against a lone black king so that White has the most mates in one move. Discovered checks are allowed. (Each different move by a piece that uncovers mate is counted separately.) 5. Construction Task #4a
Construct a position with a white king, two rooks, two bishops, and two knights against a lone black king so that White has the most mates in one move. Discovered checks are not allowed. The two bishops must be placed on oppositecoloured squares. 6. Construction Task #4b
Construct a position with a white king, two rooks, two bishops, and two knights against a lone black king so that White has the most mates in one move. Discovered checks are allowed. The two bishops must be placed on oppositecoloured squares. 7. Independent Piece Placement: 4R + 4B + 4N
Place four rooks, four bishops, and four knights on the board so that none of the pieces attack each other. Two bishops should be on dark squares, and two on light squares. 8. Defensive Loop: 4R + 4B + 4N
Place four rooks, four bishops, and four knights on the board so that each piece is defended exactly once and each piece defends exactly one other piece. Two bishops should be on dark squares, and two on light squares. The defensive chain should form a continuous loop. The first piece guards the second piece; the second guards the third; the third guards the fourth; ...; and the twelfth guards the first. For other defensive loop puzzles, see the Eight Officers columns from October and December 2012. Solutions AVERAGE MOBILITY (on an empty board) R = 14 moves B = 8.75 moves (13 x 4) + (11 x 12) + (9 x 20) + (7 x 28) = 560
N = 5.25 moves (8 x 16) + (6 x 16) + (4 x 20) + (3 x 8) + (2 x 4) = 336
The average mobility of a rook is equal to the sum of the average mobilities of a bishop and a knight!? R = B + N Q = 22.75 moves Average mobility Q = R + B The average mobilty of a rook doubled is equal to the sum of the average mobilities of a queen and a knight. (2 x R) = Q + N K = 6.5625 moves (3 x 4) + (5 x 24) + (8 x 36) = 420 P = ? 1. Triple Loyd #14 J. Coakley 1996
A. Kh8# Rook, bishop, and knight make a great attacking team. 2. Triple Loyd #15 J. Coakley 2010
A. Kd1# It's always pleasant to mate in the centre of the board. 3. Construction Task #3a J. Coakley 2010
Four mates in one: 1.Bc6#, 1.Bf7#, 1.Re7#, 1.Rh8#. There are many ways to achieve the maximum of four
mates. Here is another solution: Ka6 Rb7 Bh2 Nd5 vs. Ka8 4. Construction Task #3b J. Coakley 2010
Fourteen mates in one: any move by the white rook. There are many ways to achieve the maximum of fourteen
mates, but all involve fourteen moves of the rook uncovering a check by
the bishop. Here is another solution: Ka6 Rg2 Bh1 Nd7 vs. Ka8 5. Construction Task #4a J. Coakley 2013
Ten mates in one: 1.Rb6#, 1.Rbe7#, 1.Rhe7#, 1.Rh6#, 1.Nc5#, 1.Nd4#, 1.Nf4#, 1.Ng5#, 1.Bc4#, 1.Bg4#. The maximum of ten mates can be achieved in many ways. 6. Construction Task #4b J. Coakley 2013
Twentynine mates in one: (R14 + B13 + N2) The previous moves could have been 1.Nc5e6+ Kf4e3. There are lots of ways to achieve twentynine mates. I believe that thirty is impossible. How about you? 7. Independent Piece Placement: 4R + 4B + 4N J. Coakley 2010
There are many solutions. The one shown above is an example of independent domination, which means that all empty squares are attacked. It may be more difficult to find a solution in which some empty squares are not attacked, as in the diagram below, where c8 is "safe". Can anyone leave two or more squares unattacked? J. Coakley 2013
For those who enjoy domination, we have a bonus puzzle. 7b. Total Domination: 4R + 4B + 4N Place four rooks, four bishops, and four knights on the board so that all sixtyfour squares are attacked. A piece does not attack the square it stands on, so all occupied squares must be attacked by another piece. One solution to this relatively easy puzzle is Ra1 Rb8 Rc2 Rd7 Ne3 Nf3 Ng3 Nh3 Be5 Bf5 Bg5 Bh5. For more information on board domination problems, see the July 2012 column. 8. Defensive Loop: 4R + 4B + 4N J. Coakley 2013
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© 2013 Jeff Coakley. Illustration by Antoine Duff. All Rights Reserved. A PDF file of this week's column, along with all previous columns, is available in the ChessCafe.com Archives. Comment on this week's column via our official Chess Blog!! 
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