With 4-6 inches of snow and only 6 plows, icy Atlanta transformed to Hoth-lanta last week, resulting in a 5-day extended winter break for University students. Although there was a delay in the start of the semester, there was no delay in discussing interesting topics in my Combinatorics course this week.
A Steiner system with parameters l, m, n such that l<m<n is an n-element set S with an m-element subset of S (called blocks) such that any l-element subset of S is contained in exactly one block. To simplify, let l=2 and m=3 to form a Steiner triple system S(2,3,n). For what n does a Steiner triple system exist?
This question inspired Reverend Thomas Kirkman and he solved a special case of this problem in 1847. His solution is widely known as the Kirkman Schoolgirl Problem. So in the spirit of a fresh, new semester, here is the riddle for you: 15 girls walk to school each day in five groups of three. How many days does it take for each pair of girls to walk together exactly once? Note this is a Steiner triple system S(2,3,n) where n is the number of days. Hint: See picture.
A Steiner system with parameters l, m, n such that l<m<n is an n-element set S with an m-element subset of S (called blocks) such that any l-element subset of S is contained in exactly one block. To simplify, let l=2 and m=3 to form a Steiner triple system S(2,3,n). For what n does a Steiner triple system exist?
