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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]

We derive several new convolution identities for the Stirling numbers of the first kind. As a consequence we obtain a new linear recurrence relation which generalizes known relations. [ABSTRACT FROM AUTHOR]

We opt to study the convergence of maximal real roots of certain Fibonacci-type polynomials given by Gn == xkG n-- 1 ++ G n-- 2. The special cases k == 1 and k == 2 were done by Moore, and Zeleke and Molina, respectively. [ABSTRACT FROM AUTHOR]

We consider the values of the consecutive minima of the quantities . W. Schmidt, in 1958, calculated the first and second minimum for j == 1 and d == 2. Schweiger, in 1975, considered the case j == 1 for any d ≥≥ 2. This note is a continuation of these investigations. [ABSTRACT FROM AUTHOR]

This paper presents a new proof that if kαα is irrational then the sequence {⌊⌊ αα ++ log k n⌋⌋} n≥≥1 is not k-regular. Unlike previous proofs, the methods used do not rely on automata or language theoretic concepts. The paper also proves the stronger statement that if kαα is irrational then the generating function in k non-commuting variables associated with {⌊⌊ αα ++ log k n⌋⌋} n≥≥1 is not algebraic. [ABSTRACT FROM AUTHOR]