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We study a card game called Mousetrap, together with its generalization He Loves Me, He Loves Me Not. We first present some results for the latter game, based, on one hand, on theoretical considerations and, on the other one, on Monte Carlo trials. Furthermore, we introduce a combinatorial algorithm, which allows us to obtain the best result at least for French card decks (52 cards with 4 suits). We then apply the algorithm to the study of Mousetrap and Modular Mousetrap, improving recent results. Finally, by means of our algorithm, we study the reformed permutations in Mousetrap, Modular Mousetrap and He Loves Me, He Loves Me Not, attaining new results which give some answers to several questions posed by Cayley and by Guy and Nowakowski in their papers. [ABSTRACT FROM AUTHOR]

This paper investigates representations of real numbers with an arbitrary negative base -- ββ < --1, which we call the (-- ββ)-expansions. They arise from the orbits of the (-- ββ)-transformation which is a natural modification of the ββ-transformation. We show some fundamental properties of (-- ββ)-expansions, each of which corresponds to a well-known fact of ordinary ββ-expansions. In particular, we characterize the admissible sequences of (-- ββ)-expansions, give a necessary and sufficient condition for the (-- ββ)- shift to be sofic, and explicitly determine the invariant measure of the (-- ββ)-transformations. [ABSTRACT FROM AUTHOR]