Kirkman's Tetrahedron and the Fifteen Schoolgirl Problem

Abstract

We give a visual construction of two solutions to Kirkman's fifteen schoolgirl problem by combining the fifteen simplicial elements of a tetrahedron. Furthermore, we show that the two solutions are nonisomorphic by introducing a new combinatorial algorithm. It turns out that the two solutions are precisely the two nonisomorphic arrangements of the 35 projective lines of PG(3, 2) into seven classes of five mutually skew lines. Finally, we show that the two solutions are interchanged by the canonical duality of the projective space.

Additional information

Notes on contributors

Giovanni Falcone

GIOVANNI FALCONE received his B.A. from the University of Palermo, Italy, in 1994 and his Ph.D. from the University of Erlangen-Nürnberg, Germany, in 1998. He currently teaches at the University of Palermo. His research interests include algebraic groups, Lie groups, and discrete mathematics.

Marco Pavone

MARCO PAVONE received his B.A. from the University of Pisa, Italy, in 1982 and his Ph.D. from the University of California, Berkeley, in 1989. He currently teaches at the University of Palermo. He has published articles in number theory, functional analysis, operator algebras, and combinatorial group theory. He is the author of several mathematical games and puzzles. Besides mathematics, he enjoys classical music, choral singing, logic puzzles, and origami.