We can label the squares of an 8 x 8 chess board from from 1 to 64 in 64! different ways.

For each arrangement we find D, the largest difference between the labels of two squares which are adjacent (orthogonally or diagonally).

What is the smallest possible D (and how would you prove it)?

Hmmm... this isn't a proof, but here's a solution where D is 9 (along NW-SE lines).

1  9 17 25 33 41 49 57
2 10 18 26 34 42 50 58
3 11 19 27 35 43 51 59
4 12 20 28 36 44 52 60
5 13 21 29 37 45 53 61
6 14 22 30 38 46 54 62
7 15 23 31 39 47 55 63
8 16 24 32 40 48 56 64

Posted by SilverKnight on 2004-02-05 13:29:19