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George Duke wrote on Thu, Jan 24, 2008 06:17 PM UTC:Excellent ★★★★★
Multi-path routing is important in order to reach squares that may be blocked along one or more pathways. Under related Billiards Progressive we pose the puzzle of distribution of triple paths. In standard Billiards(8x8), the following hold. Bishops and Queens have some (potential) three-path squares, reachable three different ways, so long as they are not on an edge square(the answer). Examples: (1) Bishop at b2 can reach g7 by way of b2-a3-b4-c5-d6-e7-f8-g7 bouncing twice, or else b2-c1-d2-e3-f4-g5-h6-g7 bouncing twice, or else b2-c3-d4-e5-f6-g7 not bouncing. Queen likewise. (2) Queen has more such 3-way squares than Bishop, because of Queen's Rook movement. However Queen still cannot do so from an edge square of square boards like 8x8 and 10x10. Queen has three here: Queen b2-c2-d2, and b2-a3-b4-c5-d6-e7-f8-g7-h6-g5-f4-e3-d2 bouncing three times, and b2-c1-d2 bouncing once. Bishop there has only the second and third of those. In 'Multipath Chess Pieces' year 2004 we put forth the proposition that multi-pathing is the norm. It is mere custom that dictates, for example, that standard Rook from c3 to c7 goes by way of c3-c4-c5-c6-c7 only. Our conclusion reads ''Beyond chronology, any Rule of movement writ large, having real-world counterparts allegorically, describes something multiform and multipath, whereof reduction to mere 'Leaper' or 'Rider' is actually the special case.''

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