Kirkman and the Educational Times: Groups and Designs

• Thomas Kirkman made many important discoveries in the theory of groups and combinatorics. From 1865 until his death in 1895 he contributed over 100 questions to the Educational Times (ET), many of them related to his own discoveries.  This talk will describe a few typical examples.   In 1868 he wrote several papers on a proposed method for solving the general quintic equation by radicals, and one of them was in the form of a problem for the ET. He described some complicated calculations with symmetric functions, and claimed that the solution of the quintic was a simple consequence.  Later that year he admitted that he was mistaken. In 1869 he answered some questions posed by W. Lea;  in particular, he constructed what we would now call a design  with parameters 4-(11,5,1).  This design is associated with M11, the smallest of the simple Mathieu groups.  Many years earlier Kirkman himself had shown how to construct designs with parameters 2-(p2+p+1, p+1, 1) for any prime p, which we now call projective planes. He would often express his questions in verse (of sorts), and in 1891 he proposed the problem of ‘The Banned Thirteen’, which is based on the projective plane with p =3.