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
Famous quotes containing the word solution:
“All the followers of science are fully persuaded that the processes of investigation, if only pushed far enough, will give one certain solution to each question to which they can be applied.... This great law is embodied in the conception of truth and reality. The opinion which is fated to be ultimately agreed to by all who investigate is what we mean by the truth, and the object represented in this opinion is the real.”
—Charles Sanders Peirce (1839–1914)
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)
“To the questions of the officiously meddling police Falter replied absently and tersely; but, when he finally grew tired of this pestering, he pointed out that, having accidentally solved “the riddle of the universe,” he had yielded to artful exhortation and shared that solution with his inquisitive interlocutor, whereupon the latter had died of astonishment.”
—Vladimir Nabokov (1899–1977)