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This item is a game information page
It belongs to categories: Orthodox chess, 
It was last modified on: 1996-01-01
 By Ralph  Betza. Buypoint Chess. Buy your fighting force - each piece costs a number of points.[All Comments] [Add Comment or Rating]
Jörg Knappen wrote on 2013-02-01 UTC
I think a R5 for 4 buy points is a bargain, a R4 would be perfect.

Looking differently on Ralph Betza's old idea expressed here, I take it for granted that a ranging piece may move with some probability one step further.

This gives the following formula for the value of a full rook:

R = R1 * (1 + p + p2+ p3+ p3+ p4+ p5+ p6)

Inserting R=5 and R1=1.5 gives us p=0.73. This averages over everything relevant, no model for crowded board mobility is needed.

The main point is: The magic number p is different for the ranging pieces; for a bishop it is only 0.5 and for the queen it is ≈0.715.

The low number for the bishop comes from the board geometry: The diagonals are on average shorter than the orthogonals. In addition, the bishop has only one way from a1 to g1, and this way goes through the well-guarded centre of the board.

The queens magic number is almost (but not fully) the same as the rook's number. This is very interesting and I interpret it this way: The queen almost lifts all the geometric restrictions of the bishop.

Below are tabulated results for n-step rooks, bishops, and queens. A Q2 is a nice rook-strength piece. All values are in centipawns.

X1 X2 X3 X4 X5 X6 X7 magic number
Rook 150 260 339 398 440 471 494 0.73
Bishop 150 225 262 282 291 296 298 0.5
Queen 300 515 668 777 855 910 950 0.715