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9
Consider the straightforward ordering:
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 62 63 64
Adjacent squares in the same row have difference 1, adjacent squares in different rows have difference 7, 8 or 9.
So, for this ordering, D = 9.
If we take any two squares on a chessboard there is a path from one to the other of length at most 7, where each step of the path is to an adjacent square [if the squares are at opposite corners of an m x n rectangle with m >= n, then take m - 1 steps, n - 1 of them diagonal and the rest along the longest side.] So there is a path of at most 7 steps from 1 to 64. At least one step of that path must have a difference of at least 9 (since 7 x 9 = 64 - 1).
Thus, we always have D ≤ 9, and the minimal D is 9. |
