Two mathematical notes - new homogenised Simpson's rules and a riffle shuffle conjecture

Purpose - Seeks to derive a class of "homogeneous" rules for numerical integration from earlier results and empirical findings to treat the apparently magical reordering of a pack of cards after successive shuffles, previously discussed by Zeeberg, from a new angle and to form a mathematical conjecture.Design methodology approach - The studies were made using computer programs for which the language JavaScript proved adequate.Findings - The rules for numerical integration are more precise than earlier versions. The conjecture associated with card shuffling appears to be novel.Practical implications - Improved methods of numerical integration have practical value in many areas. The conjecture is in the field of number theory, with no obvious immediate applications.Originality value - The findings and methods are original. The demonstration of a plausible mathematical conjecture may provoke further studies aimed at its proof as a theorem, or its refutation.