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April 22nd, 2010, 01:28 PM  #1 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  A Chessboard Problem
A delightfully simpleseeming problem. A king is to make a tour of a chess board, visiting each square exactly once and returning to the same square where it begins. The trick is  the king prefers diagonal moves to regular moves. What is the maximal number of diagonal moves?

May 14th, 2010, 06:04 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond  Re: A Chessboard Problem
Graph problem, maybe?

May 16th, 2010, 11:42 AM  #3 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  Re: A Chessboard Problem
Oh, I should have answered this when I was done. It turns out that the king only has to make 8 nondiagonal moves, 2 associated with each corner, which can be verified as soon as you consider a few possibilities.

May 17th, 2010, 10:43 AM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond  Re: A Chessboard Problem
Thanks jason.spade. Difficult problem, I think.

May 17th, 2010, 10:47 AM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A Chessboard Problem
It seemed to be difficult. I guess maybe I needed less depth and more thought than I gave it. I trust this generalizes to n X n boards with n > 1? Or at least even n > 1? 

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