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George Duke wrote on Fri, Jan 25, 2008 05:57 PM UTC:Excellent ★★★★★
On 8x16 there are 19 bounces possible at most, overlaying the Billiards Mutator for Bishop and Queen. Place Bishop at c2 for convenience and assume no obstruction. c2-b1-Bounce to a2-Bounce to b3-c4-d5-e6-f7-g8-Bounce to h7-i6-j5-k4-l3-m2-n1-Bounce to o2-p3-Bounce to o4-n5-m6-l7-k8-Bounce to j7-i6-h5-g4-f3-e2-d1-Bounce to c2-b3-a4-Bounce to b5-c6-d7-e8-Bounce to f7-g6-h5-i4-j3-k2-l1-Bounce to m2-n3-o4-p5-Bounce to o6-n7-m8-Bounce to l7-k6-j5-i4-h2-g2-f1-Bounce to e2-d3-c4-b5-a6-Bounce to b7-c8-Bounce to d7-e6-f5-g4-h3-i2-j1-Bounce to k2-l3-m4-n5-o6-p7-Bounce to o8-Bounce to n7-m6-l5-k4-j3-i2-h1-Bounce to g2-f3-e4-d5-c6-b7-a8. It means a Bishop on c2 can reach a8 in one move, but not very directly. Billiards 'Bishop c2-Bishop-a8' requires the above 19 bounces. Whereas, Elbow Bishop 'c2-a8' is accomplished c2-d3-e4-(90 degrees)d5-c6-b7-a8 in the one change of direction, a pretty direct route. In sum, 6x8 has 7 bounces, 8x8 4 bounces, 8x10 7 bounces, 8x12 13 bounces, 8x14 15 bounces, 8x16 19 bounces. What is the formulaic pattern? '8x14' requires starting at g2 for best result (Hey, Geometria). There the bounces occur successively after g2 starting square at f1, a6, c8, j1, n5, k8, d1, a4, e8, l1, n3, i8, b1, a2, g8, and not possible anymore at arrival square n1. If the Bishop starts on m8 in 8x14, the number of squares actually traversed in that full route, extended back to m8 for maximization, is 77. Shorthand for this size 8x14 might be 'Bishop m8-n1(77 times one-stepping)'.

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