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

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.

The studies were made using computer programs for which the language JavaScript proved adequate.

The rules for numerical integration are more precise than earlier versions. The conjecture associated with card shuffling appears to be novel.

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.

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.