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Piece Density[Subject Thread] [Add Response]

Joe Joyce wrote on Wed, Oct 7, 2009 02:20 AM UTC:

To continue expanding David's concept of attack density out further [and
incidentally explain why I too call it attack density rather than defensive
density], I'll bring up attack fraction once again, this time in
conjunction with mean free path ideas. 

The attack fraction is the percentage of squares, at each incremental
range, that the piece may attack. A rook or bishop attacks 4 squares at any
range, but at range 1, next to them, that's 4 of 8 squares. At range 2,
that's 4 of 16 squares, and at range 10, that would be 4 of 88 squares.
And the slider is blockable by 1 piece. So its attack fraction would be:
.5, .25, .17, .13, .1, .09, .07... at range 1, 2, 3...

Let's look at the knight. Its attack fraction at range 1 is 0.0, at range
2 it's 0.5, at range 3 and beyond, it's 0. At range 2, the knight is
twice as good as the rook or bishop. And it's unblockable except by a
friendly piece in the target square. And, since self-capture games are
rare, the knight is as unblockable as they get. A slider is not guaranteed
to go 2 squares. 

Okay, now consider the mean free path of a piece in the midgame, then look
at the [bent] hero and shaman. The hero's attack fraction is .5; .75; .17,
and the shaman's is .5; .25; .5, for ranges 1, 2, and 3. Each attacks [up
to] 20 squares, 8 unblockably, and the remaining 12 have 2 paths to each
destination. The hero attacks 4 squares at range 1, 12 at range 2, and 4 at
range 3. The shaman attacks 4 squares at range 1, 4 at range 2, and 12 at
range 3. This explains why, even with a range limited to 3 squares, the
bent hero and shaman are more valuable than a rook on an 8x8. 

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Somewhere earlier in this thread, I believe it was commented that larger
boards may require a lower piece density. I would like to second that
statement, and suggest that after considerable experimentation with large
variants of different sorts, I'm getting a sneaking suspicion that at
least half the very best games for very large boards will have very low
densities.