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A226698
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Central symmetric closed knight's tour on an 8x8 board, attributed to Euler. Position after n-th move.
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1
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33, 50, 60, 54, 64, 47, 37, 43, 58, 52, 62, 56, 39, 45, 35, 41, 51, 57, 42, 36, 46, 40, 55, 61, 44, 34, 49, 59, 53, 63, 48, 38, 32, 15, 5, 11, 1, 18, 28, 22, 7, 13, 3, 9, 26, 20, 30, 24, 14, 8, 23, 29, 19, 25, 10, 4, 21, 31, 16, 6, 12, 2, 17, 27
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OFFSET
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0,1
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COMMENTS
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a(n) is the position of the knight on the 8x8 board after the n-th move (a(0) gives the starting position) if one numerates the squares from left to right, top to bottom, from 1 to 64.
If the board is considered as an 8x8 matrix the square numbered N appears as element (n,m) = (floor((N-1)/8)+1, N - 8*floor((N-1)/8)), N = 1, ..., 64. Therefore, a(0) = 33, the knight's starting position is with N = 33: (5,1). The last position is with N = 27: (4,3).
a(n-1) is the inverse of A226697 read as a sequence: A226697(a(n-1)) = n, n=1, 2, ..., 64.
For the board see the example for A226697. Observe there the central symmetry with absolute difference constant 32.
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REFERENCES
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Martin Gardner, Mathematical Magic Show, The Math, Assoc. of Am., Washington DC, 1989, Ch. 14, Knights of the Square Table,Fig. 86, p. 191. German Translation: Mathematische Hexereien, Ullstein, 1977, Abb. 86, S. 186.
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LINKS
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EXAMPLE
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a(1) = 50 because after the first move the knight is on the square N = 50, or considered as matrix position at square (7, 2). The path starts at square a(0) = 33, or (5, 1). It ends after 63 moves on square a(63) = 27, or (4, 3). The next move can close the Hamiltonian path.
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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