Surprise Maximization

D. Borwein, J.M. Borwein and P. Mar e hal July 25, 1999 The Surprise Examination or Unexpe ted Hanging Paradox has long fas inated mathemati ians and philosophers, as the number of publi ations devoted to it attests. For an exhaustive bibliography on the subje t, the reader is referred to [1℄. 1 Herein, the optimization problems arising from an information theoreti avoidan e of the Paradox are examined and solved. They provide a very satisfa tory appli ation of both the Kuhn-Tu ker theory and of various lassi al inequalities and estimation te hniques. Although the ne essary onvex analyti on epts are re alled in the ourse of the presentation, some elementary knowledge of optimization is assumed. those unfamiliar with this ba kgroundmay simply skip a ouple of proofs and few te hni al details. 2