Kirkman's schoolgirl problem (1850): 15 girls walk in groups of 3 for 7 days. No pair walks together twice. Solution: partition 35 triples of S(2,3,15) into 7 parallel classes (each covering all 15 girls). 7 non-isomorphic solutions exist.
How to Use
View the 7-day schedule:
- Days: 5 groups of 3
- Pairs: All 105 covered
- Verify: No repeats
History
Thomas Kirkman posed this in 1850, before Steiner systems were defined! Arthur Cayley and others worked on it. The complete classification of solutions came much later. It's one of the most famous problems in combinatorics, inspiring centuries of design theory work.
Generalization
Resolvable (v,3,1)-BIBD: exists iff v≡3(mod6). For v=9: resolvable S(2,3,9)=AG(2,3). For v=15: Kirkman. For v=21: multiple solutions. Ray-Chaudhuri-Wilson theorem guarantees existence for all v≡3(mod6).
Step-by-Step Instructions
- 1View schedule.
- 2Check each day.
- 3Verify no pair repeats.
- 4Count all pairs.
- 5Explore solutions.