George Duke wrote on Thu, Apr 13, 2017 07:54 PM UTC:
There could be other sequences used for where a radial piece is allowed to stop. A compound piece of pentagonal, hexagonal, heptagonal and octagonal numbers can move: 5, 6, 7, 8, 12, 15, 18, 21, 22 or 28..., keeping to the lower lengths for 30x30 board. Aligning the primes and Fibonacci and Deficient and others pairwise, both orthogonally and diagonally, may discover hidden relationships just by tooling around, for applicability beyond the chessboards. For example, applying some Knights Tour like V. Reinhart mentions. Another piece can have numbers of distinct integral squares dividing a rectangle: 1, 4, 7, 8, 9, 10, 14, 15, 18. Call that one Squarec, and it is good piece to cross quickly the boundaries of boards either 12x12 or 16x16. With 15 steps specifically allowed it traverses 16x16 diagonally, or orthogonally, all the way without being just plain Queen
There could be other sequences used for where a radial piece is allowed to stop. A compound piece of pentagonal, hexagonal, heptagonal and octagonal numbers can move: 5, 6, 7, 8, 12, 15, 18, 21, 22 or 28..., keeping to the lower lengths for 30x30 board. Aligning the primes and Fibonacci and Deficient and others pairwise, both orthogonally and diagonally, may discover hidden relationships just by tooling around, for applicability beyond the chessboards. For example, applying some Knights Tour like V. Reinhart mentions. Another piece can have numbers of distinct integral squares dividing a rectangle: 1, 4, 7, 8, 9, 10, 14, 15, 18. Call that one Squarec, and it is good piece to cross quickly the boundaries of boards either 12x12 or 16x16. With 15 steps specifically allowed it traverses 16x16 diagonally, or orthogonally, all the way without being just plain Queen