Maximal Bergholtian Symmetry

Bergholtian symmetry is centrosymmetry of the type that passes twice through the centre,
and is possible only on boards with one side odd and the other singly even (e.g. 6×7 or 5×10).

When numbered from the ends of the central cross it has the property that diametrally opposite numbers add to a constant sum. This is in contrast to the more common tours with Eulerian symmetry in which opposite numbers have a constant difference.

On other boards it is possible to construct tours which have partial Bergholtian symmetry, not all of the pairs adding to the constant. An example of this was sent to me by Prof D. E. Knuth, which inspired the following examples, which I think show the maximum amount of Bergholtian symmetry on the 8×8 and 10×10 boards.

The dots which, form two rhombs, mark the ends of the symmetric parts. One rhomb is connected in the opposite way to the other and this is the only asymmetry. If one of the rhombs is reversed this results in a tour with Eulerian symmetry.

The geometrical forms:

The arithmetical forms:

Diametrally opposite numbers add to 62
with the exception of the three pairs in bold.

47 52 45 26 05 24 21 28
44 03 48 53 64 27 06 23
51 46 01 04 25 22 29 20
02 43 54 49 30 63 12 07
55 50 31 62 13 08 19 60
42 33 40 37 58 61 16 11
39 56 35 32 09 14 59 18
34 41 38 57 36 17 10 15

crosslinks c6-e7-f5-d4-f3 and b3-d2-c4-e5-g6

03 14 53 16 55 34 37 40
52 17 02 13 42 39 56 35
31 04 15 54 33 36 41 38
18 51 32 01 12 43 62 57
05 30 19 50 61 64 11 44
24 21 26 29 08 47 58 63
27 06 23 20 49 60 45 10
22 25 28 07 46 09 48 59

crosslinks: d5-f4-h3-g5-e4 and d3-b4-a6-c5-e6

Connecting instead 01-48, 00-47 or 51-98, 50-97
results in a tour with Eulerian symmetry

10×10 example. This linkage will not work on the 8×8 board,
since moves through two corners are prevented.

17 22 89 26 45 38 87 34 31 36
90 25 18 01 88 27 44 37 86 33
21 16 23 46 39 48 03 32 35 30
24 91 00 19 02 43 28 57 04 85
15 20 93 40 47 56 49 84 29 06
92 69 14 99 42 51 58 05 78 83
13 94 41 70 55 96 79 50 07 74
68 63 66 95 98 59 52 75 82 77
65 12 61 54 71 10 97 80 73 08
62 67 64 11 60 53 72 09 76 81

Diametrally opposite numbers add to 98 with the exception of the three pairs in bold.
00 underlined indicates 100.

crosslinks: 01-00-99-98-97 and 47-48-49-50-51