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Man and Beast 03: From Ungulates Outward. Systematic naming of the simplest Oblique Pieces.
George Duke wrote on 2009-06-16 UTC
Oblique.(4,5) is Rector; he leaps 4 then 5, or 5 then 4, same difference if with oppositely orthogonal start to the path, since it is a leap, like Knight (1,2) or (2,1), which could mean the very same square. 4 and 5 share no multiplicative factor, so mathematically they are coprime, prime to each other. Why leave out pairs like (2,4), which are not coprime because they have common factor of 2? Because (2,4) is a Nightrider square, and we would rather get there another way than a direct leap. So would (2,6) be Camel-rider second step not coprime. Gilman is omitting non-coprime, because they would be redundant. (8,10) would be Rector-rider and needs no other name. Surrounding 13x13 set of squares to its central departure square is illustrative. We figure the zero(0) squares of Rook either way within one quadrant (the Rook is the border) as coprime for convenience. So, within 13x13 we are only omitting, with the order mattering, (2,2) (3,3) (4,4) (5,5) (6,6) (2,4) (2,6) (3,6) (4,6) (4,2) (6,2) (6,3) out of 48 squares one quadrant, and that's 25% of them. As we go farther out, about the same ratio obtains. All together oblique directions increasingly outnumber the mere two radial ones, but the omissions to naming would stay nearly the same percentagewise, because of the rider-phenomenon, also able to be called non-coprime redundancy. So Gilman is going ahead with naming most of the squares in principle (he is not likely to proceed beyond 15- or 17-block) or approving already-existing names. [See Pritchard's 'ECV' under Pieces for ancient schema for 7x7 block of squares around a central starting square]

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