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V. Reinhart wrote on Sun, Jul 23, 2017 04:41 AM UTC:

As mentioned before, I completed a calculation of the "Material Power Density" for "Chess and a Half".

First, to explain what this ratio is - it is simply the sum of the value of all chess pieces on the board, divided by the playing area (number of squares).

This paramater may give some prediction of the style of play that can be expected from a game. Higher density usually means pieces have more attacking possibilities, and games can enter into dynamic attacks quickly - requiring fast responses from the other side. A lower density means that the opening development may last longer, with a stronger focus (for example) on the placement of pawns.

From lowest to highest is the "Material Power Density" of several games I've analyzed, with "Chess and a Half" now added in the list.

                     Board         Number   Piece    Power    Ratio to
                     Size (sqs)    of Pcs.  Density  Density  Classical Chess
———————————————————— ————————————  ———————  ———————  ———————  ———————————————
Classical Chess      8 x 8 =  64   32       50%      1.34     1.00
Infinite Plane(1)            360   76       21%      0.56     0.42
A Chess Endgame(2)   8 x 8 =  64    7      (11%)     0.64     0.48
Chess and a Half     12 x 12= 144  48       33%      1.04     0.78
Bulldog Legacy Chess 10 x 8 = 80   36       45%      1.10     0.82
Bulldog Chess        10 x 8 = 80   40       50%      1.31     0.98
Bulldog Chess(Witch) 10 x 8 = 80   40       50%      1.35     1.01
Janus Chess          10 x 8 = 80   40       50%      1.53     1.14
Capablanca Chess     10 x 8 = 80   40       50%      1.55     1.16
Seirawan Chess(3)    8 x 8 =  64   36      [56%]     1.88     1.40
Musketeer Chess(4)   8 x 8 =  64   36      [56%]     1.88     1.40
Chu Shogi            12 x 12= 144  92      [64%]     1.91     1.43
Amsterdam Chess      11 x 8 = 88   44       50%      1.95     1.46
Waterloo Chess       10 x 10= 100  60      [60%]     2.72     2.03


(1) - Chess on an Infinite Plane: Play is assumed to be in an 18 x 20 range. This is the horizontal span and 2 ranks less than the vertical span of the outermost pieces (starting position). Little play is presumed to take place in the outermost ranks except for pieces moving inward.
(2) - A chess endgame: A sample 7-piece ending with KQRR vs. KQR.
(3) - Seirawan and Musketeer Chess: Data assumes all pieces are on board. Some pieces are introduced early or later in the game.
(4) - Musketeer Chess: Assumes game with archbishop and chancellor played as the new pieces.

Chess and Half (4th in list) is interesting in that is has a very low piece density (48 pieces on 144 squares), and the material density is 78% compared to that of chess (100%). This is in the range of games that have become my favorite to play. They usually feature an opening with time for tactical development, rather than the players always reacting only to exact threats from the opponent.

I've seen plenty of variants where the dynamics suffer from too much power concentrated into a normal size chess board.  I would really like to try this game sometime, and as mentioned, if it's ever withing the scope of a chess engine, see if a human can win, or how two engines do against each other.


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