A multi-computational exploration of some games of pure chance

Abstract

In the spirit of “multi-culturalism”, we use four kinds of computations: simulation, numeric, symbolic, and “conceptual”, to explore some “games of pure chance” inspired by children board games like “Snakes and Ladders” (aka “Chutes and Ladders”) and “gambler's ruin with unlimited credit”. Even more interesting than the many computer-generated specific results described in this paper and its web-site extension, is our broad-minded, ecumenical approach, not favoring, a priori, any one of the above four kinds of computation, but showing that, a posteriori, symbolic computation is the most important one, since (except for simulation) numerics can be made more efficient with the help of symbolics (in the “downward” direction), and, (in the “upward” direction) the mere existence of certain symbolic-computational algorithms imply interesting “qualitative” results, that certain numbers are always rational, or always algebraic, and certain sequences are always polynomial, or C-recursive, or algebraic, or holonomic. This article is accompanied by four Maple packages, and numerous input and output files, that readers can use as templates for their own investigations.