1. 15 Schoolgirls Walk in Space Bill Cherowitzo University of Colorado Denver MAA Rocky Mountain Section April 17, 2010

2. A General Problem 7 8 9 2 6 10 4 5 11 1 3 13 12 14 15 Given a finite set X of size n and the set, F, of all the subsets of X of size t ( t ≤ n), a subset of F is a partition of X if every element of X is contained in exactly one element of F. t must divide n for a partition to exist. A parallelism is a partition of F by disjoint partitions of X. The same divisibility condition (t|n) is a necessary and sufficient condition for the existence of a parallelism. Parallelisms exist in many areas of combinatorics.

3. In Graphs 1 8 10 3 7 11 5 6 12 2 4 14 9 13 15 A partition of the vertices of K n by edges is called a 1-factor (perfect matching) and requires that n be even. A parallelism of K n is a partition of the edges of the graph by 1-factors and is called a 1-factorization. 1- Factorization of K 6

4. In Designs 2 8 11 1 4 12 6 7 13 3 5 15 9 10 14 A partition of the elements of a block design by blocks is called a parallel class of blocks. Clearly the size of a block, k, must divide the number of elements, v. A parallelism in a block design is a partition of the blocks by parallel classes which is called a resolution. A block design with a resolution is called a resolvable design.

5. The Rev. Kirkman 4 6 9 3 8 12 2 5 13 1 7 14 10 11 15 Fifteen young ladies of a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk abreast more than once. The Reverend Thomas Penyngton Kirkman (1806-1895), vicar of the Parish of Southworth, Lancashire and mathematician. Most famous for a minor contribution of his which appeared as Query 6 on page 48 of the Lady’s and Gentlemen’s Diary of 1850.

6. Kirkman Systems 1 5 9 3 4 10 2 7 12 6 8 15 11 13 14 The solution to Kirkman’s 15 schoolgirl problem is a resolution of a block design on 15 points with blocks of size 3 such that every pair of points is contained in a unique block. These block designs are called Steiner Triple Systems (on 15 points). A resolvable Steiner Triple System is called a Kirkman Triple System and there are 7 non- isomorphic Kirkman Triple Systems on 15 points.

7. In Geometry 5 7 10 4 8 13 3 6 14 1 2 15 9 11 12 In an affine plane, a partition of the points by lines is a parallel class. A partition of the lines by parallel classes is a parallelism (the origin of the term). In a projective space (dim > 2), a partition of the points by lines is a (line) spread. A parallelism is a partition of the lines of the space by spreads, called a (spread) packing.

8. PG(3,2) 2 3 9 1 6 11 5 8 14 4 7 15 10 12 13 The projective space PG(3,2) can be described in terms of subspaces of the 4-dimensional vector space V over the binary field. Points of PG(3,2) are the 1-dim subspaces of V. Lines of PG(3,2) are the 2-dim subspaces of V. Planes of PG(3,2) are the 3-dim subspaces of V. Incidence is given by containment. There are 15 points, 35 lines and 15 planes in PG(3,2). Two lines meet iff they lie in a plane. Lines which do not meet are called skew. Five mutually skew lines form a spread.

9. Going for a Walk Since there are 3 points on a line, and two points determine a line, a spread corresponds to 15 points lined up by 3’s on 5 skew lines. A packing of this space is a set of 7 disjoint spreads, and every line is in a unique spread of the packing. Clearly, this is a solution to Kirkman’s Schoolgirl problem. There are two packings of PG(3,2). 7 8 9 2 6 10 4 5 11 1 3 13 12 14 15 1 8 10 3 7 11 5 6 12 2 4 14 9 13 15 2 8 11 1 4 12 6 7 13 3 5 15 9 10 14 4 6 9 3 8 12 2 5 13 1 7 14 10 11 15 1 5 9 3 4 10 2 7 12 6 8 15 11 13 14 5 7 10 4 8 13 3 6 14 1 2 15 9 11 12 2 3 9 1 6 11 5 8 14 4 7 15 10 12 13

10. And More … Take a quadric in a projective 3-space (elliptic quadrics, hyperbolic quadrics and quadratic cones) and consider the planes which intersect the quadric in a conic section. A partition of the points* of the quadric by these conics is called a flock of the quadric. Theorem (N.Johnson, WEC 2010(?)): Every flock of a hyperbolic quadric or a quadratic cone is contained in a parallelism of flocks in PG(3,K) for any field K.