

A226698


Central symmetric closed knight's tour on an 8x8 board, attributed to Euler. Position after nth move.


1



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

0,1


COMMENTS

a(n) is the position of the knight on the 8x8 board after the nth 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((N1)/8)+1, N  8*floor((N1)/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(n1) is the inverse of A226697 read as a sequence: A226697(a(n1)) = n, n=1, 2, ..., 64.
For the board see the example for A226697. Observe there the central symmetry with absolute difference constant 32.


REFERENCES

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.


LINKS



EXAMPLE

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.


CROSSREFS



KEYWORD

nonn,fini,full


AUTHOR



STATUS

approved



