[Magictour] Pandiagonal 7x7 magic torus

Gunter sent the following message:

torus7.exe found this:

32 25 11 04 60 53 46
64 50 43 36 22 15 01
26 12 05 61 54 40 33
51 44 30 23 16 02 65
13 06 62 55 41 34 20
45 31 24 10 03 66 52
00 63 56 42 35 21 14



it's just


                                                                        =
60
                                                               50 -- -- =
 
                                                      40 -- -- -- -- 61 =
  
                                             30 -- -- -- -- 51 -- -- =
     
                                    20 -- -- -- -- 41 -- -- -- -- 62 =
 
                           10 -- -- -- -- 31 -- -- -- -- 52 -- --
                  00 -- -- -- -- 21 -- -- -- -- 42 -- -- -- -- 63
                  -- -- 11 -- -- -- -- 32 -- -- -- -- 53 -- --
               01 -- -- -- -- 22 -- -- -- -- 43 -- -- -- -- 64
               -- -- 12 -- -- -- -- 33 -- -- -- -- 54 -- --
            02 -- -- -- -- 23 -- -- -- -- 44 -- -- -- -- 65
            -- -- 13 -- -- -- -- 34 -- -- -- -- 55 -- --
         03 -- -- -- -- 24 -- -- -- -- 45 -- -- -- -- 66
         -- -- 14 -- -- -- -- 35 -- -- -- -- 56
      04 -- -- -- -- 25 -- -- -- -- 46
      -- -- 15 -- -- -- -- 36
   05 -- -- -- -- 26
   -- -- 16
06


wrapped around the torus.
The same construction should work for any n which is prime
or not divisible by 2,3 - or such ... hmm, reminds me to the
nonattacking queens-problem again.

So I conjecture:
there is a closed knight's tour on the
n^d torus which is d-ortho-latin and all pan-diagonals are ortho-latin ,
whenever n doesn't divide any prime < 2^d .

Someone please prove or disprove it !


Guenter


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Received on Thu Dec 04 14:24:18 2001