# An Enumeration of Squares and Diamonds Symmetric Tours

Related page: enumeration of the halfboard cases.

## The method of enumeration employed

My first method of enumerating these tours was to note that, by reflection if necessary, we can always arrange for the diamond passing through the top left corner to be in the same orientation. There are then four ways of placing the second diamond:

For a symmetric tour the arrangement of the diamonds in the diametrally opposite quarter is then just the same rotated 180 degrees. The pattern in the other two quarters can be any of the four, giving ten cases: 11, 12, 13, 14, 22, 23, 24, 33, 34, 44. But the pattern can either be reflected left to right (denoted =) or rotated 90 degrees (denoted ~) making 20 separate cases.

In the resulting diagrams there are many forced moves that can be put in due to other choices being blocked or forming short circuits. However the rest of the enumeration, done by hand, using the drawing facility in Lotus WordPro, not by computer programming, proved to be incomplete and six missing cases were later found, mainly by comparison of groups of similar tours. The numbers found in each case are now as follows (H = double halfboard tours, X = other cases, T = totals):

 - 11= 12= 13= 14= 22= 23= 24= 33= 34= 44= T H 8 3 7 12 1 4 2 3 2 4 46 X 20 11 29 14 0 2 2 5 4 4 91 - 11~ 12~ 13~ 14~ 22~ 23~ 24~ 33~ 34~ 44~ - H 3 5 8 2 1 3 3 4 6 1 36 X 5 4 16 1 7 25 1 38 6 0 103

These numbers form no obvious pattern, so it is not clear if some cases might still be missing. A separate enumeration of the halfboard cases confirmed the total of 46 + 36 = 82. However when the diagrams are rearranged according to the patterns of slants crossing the medians (as used for the halfboard enumeration), the resulting numbers show more regularity, falling mostly into groups of 2, 4, 8 and 16.

In the catalogue that follows the pairs of slants along the medians are conveniently named, as in the halfboard case, narrow (N), medium (M) and wide (W). Within each section the diagrams are arranged according to the pattern of the secondary slants, on lines bisecting the horizontal and vertical halfboards. These are labelled a, b, c indicating that they start in the first second or third file or rank from either end of the halfboard.

Where the above conventions do not fix the order, the tours are arranged according to their central angles, or simply according to the similarity of their patterns.

## Diagrams of the 194 symmetric non-halfboard tours of squares and diamonds.

DOUBLES: two pairs of medial slants = 82 tours.

20 N-N:
These all include a central minimal square.
(4 each of the a-a, a-b, b-a, b-b and c-c types).

30 N-W:
These fall into two groups of 15.
The missing 16th tour in each pattern proves to be one of the double halfboard tours.

N-W first batch of 15: (8 a-c, 7 c-a).

N-W second batch of 15 (8 b-c, 7 c-b type)

16 M-M:
(3 a-a, 3 a-b, 3 b-a, 3 b-b and 4 c-c).
These groups of three tours can each be made up to four, all of similar pattern, by including a double halfboard tour,
except in the a-a case where the fourth in the set proves to be a 90 degree rotation of the first in the group. This seems somewhat anomalous.

16 W-W:
(3 a-a, 3 a-b, 3 b-a, 3 b-b and 4 c-c)
These groups of three tours can each be made up to four, all of similar pattern, by including a double halfboard tour.

TRIPLES: three pairs of medial slants = 98 tours.

8 N-NW:
(2 c-aa, 2 c-ab, 2 c-ba, 2 c-bb).

2 NN-W:
(2 cc-c). These are the only tours with a central lozenge.

16 N-MW:
(a-aa, a-ab, a-ba, a-bb), (b-aa, b-ab, b-ba, b-bb), (2 c-ac, 2 c-bc), (2 a-cc, 2 b-cc).

20 NM-W:
(2 aa-a, 2 aa-b), (2 ab-a, 2 ab-b), (2 ba-a, 2 ba-b), (2 bb-a, 2 bb-b), (2 ca-c, 2 cb-c).

4 NW-M:
(2 ca-c, 2 cb-c).

8 N-WW:
(a-aa, a-ab, a-ba, a-bb, b-aa, b-ab, b-ba, b-bb).

4 NW-W:
(2 cc-a, 2 cc-b).

16 M-MW:
(2 a-ca, 2 a-cb, 2 b-ca, 2 b-cb) (2 c-aa, 2 c-ab, 2 c-ba, 2 c-bb).

2 MM-W:
(cc-c)

2 M-WW:
(c-cc)

16 MW-W:
(2 ac-a, 2 ac-b, 2 bc-a, 2 bc-b) (2 aa-c, 2 ab-c, 2 ba-c, 2 bb-c).

QUADRUPLES: four pairs of medial slants = 14 tours.

2 NM-WW:
(cc-ac, cc-bc).

8 NW-MW:
(aa-ca, aa-cb, ab-ca, ab-cb, ba-ca, ba-cb, bb-ca, bb-cb).

4 MM-WW:
(ac-ac, ac-bc, bc-ac, bc-bc).