Many members of the club disliked the lack of variety and togetherness at the club. Although the club still had 12 members, some members were threatening to quit because each schedule was so short and there were so few people around each table.

To satisfy their request, the club decided to seat themselves around a big table and create a longer schedule. The twelve members of the club seated themselves in a schedule such that during each block of 55 days, no person was between the same pair of people. How was the schedule constructed?

(Based on The Round Table)

I thought I had found the solution to TCC3 in my former posting.  The rule that nobody should be seated next to somebody he had already as a neighbour still stands.  But the rule should also be applied to the ends of the both groups.  Then the groups can be considered as circles, where the following rule applies:  create 5 seatings of 11 people, such that nobody has the same neighbour twice.  This is reasonably simple, it can even be created using the clockwise/anticlockwise technique described in my former posting.
When you have the five seatings, put them around a circle.  Go to the first number on the circle and move one position clockwise, two positiions anticlockwise, three positions clockwise,...  This is the first seating of the final solution.  Then you do the same, starting from the second number on the circle, then the third,...  Now you have 11 solutionswith 11 people.  Go to the next of your five seatings and repeat the above, this gives you the other 44 solutions.  Now write twelve after each solution and you are done.
Thanks again to Bruce for insisting on the 5 times 11
Here follows the solution:

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

Posted by Hugo on 2004-09-27 04:49:06