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February 27th, 2011, 07:47 AM   #1
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Knight covers the chessboard

Is there any way how to cover 6*6 chessboard with numbers 1 to 36 with these rules?

Rules are. On the start, there are only zeroes on chessboard. You can pick two fields between which knight can jump (for example a1 and b3) and write there a number one greater.
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February 27th, 2011, 09:04 AM   #2
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Re: Knight covers the chessboard

Is the order of 1 to 36 fixed? If yes, then most likely it is not possible. If the ordering does not matter, then it is possible: If you know, that you can find a knights tour ( just pick one and number the squares by the step of this tour the knight will step on them. (Careful: This will not be the number on this square in the end.) We note, that by your rule we can add the same number to successive steps.

So first we add to the 1st and 2nd square 1 to the 3rd and forth square 3 and so forth, until you add to the 35th and 36th square 35. After that is done, we have to add to the 2nd and 3rd square 1 then to the 6th and 7th square one and so forth always adding to two squares and then leaving two as they are until you add one to the 34th square and the 35th one.

So after the first step you have: (in the described ordering)
1,1,3,3,5,5,7,7,9,9,11,11,13,13,15,15,17,17,19,19, 21,21,23,23,25,25,27,27,29,29,31,31,33,33,35,35 and after the second step you arrive at:
1,2,4,3,5,6,8,7,9,10,12,11,13,14,16,15,17,18,20,19 ,21,22,24,23,25,26,28,27,29,30,32,31,33,34,36,35 as required.
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