Solution
If the girls are numbered from 01 to 15, the following arrangement is one solution:
| Sun. | Mon. | Tues. | Wed. | Thurs. | Fri. | Sat. |
|---|---|---|---|---|---|---|
| 01, 06, 11 | 01, 02, 05 | 02, 03, 06 | 05, 06, 09 | 03, 05, 11 | 05, 07, 13 | 11, 13, 04 |
| 02, 07, 12 | 03, 04, 07 | 04, 05, 08 | 07, 08, 11 | 04, 06, 12 | 06, 08, 14 | 12, 14, 05 |
| 03, 08, 13 | 08, 09, 12 | 09, 10, 13 | 12, 13, 01 | 07, 09, 15 | 09, 11, 02 | 15, 02, 08 |
| 04, 09, 14 | 10, 11, 14 | 11, 12, 15 | 14, 15, 03 | 08, 10, 01 | 10, 12, 03 | 01, 03, 09 |
| 05, 10, 15 | 13, 15, 06 | 14, 01, 07 | 02, 04, 10 | 13, 14, 02 | 15, 01, 04 | 06, 07, 10 |
A solution to this problem is an example of a Kirkman triple system, which is a Steiner triple system having a parallelism, that is, a partition of the blocks of the triple system into parallel classes which are themselves partitions of the points into disjoint blocks.
There are seven non-isomorphic solutions to the schoolgirl problem. Two of these are packings of the finite projective space PG(3,2). A packing of a projective space is a partition of the lines of the space into spreads, and a spread is a partition of the points of the space into lines. These "packing" solutions can be visualized as relations between a tetrahedron and its vertices, edges, and faces.
Read more about this topic: Kirkman's Schoolgirl Problem
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