» Vol. 13, Issue 1, 2012 «

Title: Towards Bidding Connect Four Author: Kenny Goodfellow, Juniata College Author Bio     Abstract: Allis has determined that the player who goes first in Connect Four always has a winning strategy. We consider the discrete bidding variation of the game instead of alternating turns. In discrete bidding, each player holds an integer number of chips, and the players bid for the next turn. Whoever wins the bid takes a turn and gives his chips to the other player; thus, the total number of chips stays constant. Introducing bidding to the game alters a player's strategy, as multiple moves in succession are now possible. Develin and Payne have completed an analysis of Tic-Tac-Toe using discrete bidding and have determined a winning strategy. We analyze bidding Connect Two on all board sizes and bidding Connect Three on a 3-by-3 board, which will give us insight into the strategy for Connect Four. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Convex Hulls and the Casas-Alvero Conjecture for the Complex Plane Author: Thomas Polstra, Georgia State University Author Bio     Abstract: It has been conjectured by Casas-Alvero that polynomials of degree n over fields of characteristic 0, share roots with each of its n-1 derivatives if and only if those polynomials have one root of degree n. In this paper, using the analytic theory of polynomials, an equivalent formulation of the Casas-Alvero Conjecture is established for polynomials over the complex plane t ogether with several special cases of it. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: A Factorial Power Variation of Fermat's Equation Author: Matthew J. Green, Towson University Author Bio     Abstract: We consider a variant of Fermat's well-known equation xn+yn=zn. T his variant replaces the usual powers with the factorial powers defined by xn=x(x-1)...(x-(n-1)). For n=2 we characterize all possible integer solutions of the equation. For n=3 we show that there exist infinitely many non-trivial solutions to the equation. Finally we show there exists no maximum n for which xn+yn = zn has a non-trivial solution. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Invariants of Finite Groups Acting as Flag Automorphisms Authors: Dennis Tseng, Massachusetts Institute of Technology Author Bio     Abstract: Let K be a field and suppose that G is a finite group that acts faithfully on $(x1,...,xm) by automorphisms of the form g(xi)=ai(g)xi+bi(g), where ai(g),bi(g) \in K(x1,...,xi-1) for all g \in G and all i=1,...,m. As shown by Miyata, the fixed field K(x1,...,xi-1)G is purely transcendental over K and admits a transcendence basis {\phi1,...,\phim}, where \phii is in K(x1,...,xi-1) [xi]G and has minimal positive degree di in xi. We determine exactly the degree di of each invariant \phii as a polynomial in xi and show the relation d1 ... dm=|G|. As an application, we compute a generic polynomial for the dihedral group D8 of order 16 in characteristic 2. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Structure and Randomness of the Discrete Lambert Map Authors: JingJing Chen, Pomona College Mark Lotts, Randolph-Macon College Author Bio     Author Bio     Abstract: We investigate the structure and cryptographic applications of the Discrete Lambert Map (DLM), the mapping x --> xgx mod p, for p a prime and some fixed g \in (Z/pZ)* The mapping is closely related to the Discrete Log Problem, but has received far less attention since it is considered to be a more complicated map that is likely even harder to invert. However, this mapping is quite important because it underlies the security of the ElGamal Digital Signature Scheme. Using functional graphs induced by this mapping, we were able to find non-random properties that could potentially be used to exploit the ElGamal DSS. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Tiling with Penalties and Isoperimetry with Density Authors: Yifei Li, Berea College Michael Mara, Williams College Isamar Rosa Plata, University of Puerto Rico Elena Wikner, Williams College Author Bio     Author Bio     Author Bio     Author Bio     Abstract: We prove optimality of tilings of the flat torus by regular hexagons, squares, and equilateral triangles when minimizing weighted combinations of perimeter and number of vertices. We similarly show optimality of certain tilings of the 3-torus by polyhedra from among a selected candidate pool when minimizing w eighted combinations of interface area, edge length, and number of vertices. Finally, we provide n umerical evidence for the Log Convex Density Conjecture. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Recent Developments in Perfect Bricks with Dimension Higher than 2 x 2 Authors: Brooke Fox, Northern Arizona University Author Bio     Abstract: A numerical semigroup S is a set of nonnegative integers such that S contains 0, S is closed under addition, and the complement of S is finite. This paper considers pairs (S,I) of a given numerical semigroup S and corresponding relative ideal I such that \mu(I)\mu(S-I) = \mu(I+(S-I)), where \mu denotes the size of the minimal generating set and S-I is the dual of I in S. We will present recent results in the research of such pairs (perfect bricks) with \mu(I) > 2 and \mu(S-I) > 2. We will also show the existence of an infinite family of perfect bricks. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Cold Positions of the Restricted Wythoff's Game Authors: Ryoma Aoki, Hyogo Prefectural Akashi Kita Senior High School Junpei Sawada, Hyogo Prefectural Akashi Kita Senior High School Yuki Miyake, Hyogo Prefectural Akashi Kita Senior High School Hiroaki Fujiwara, Hyogo Prefectural Akashi Kita Senior High School Author Bio     Author Bio     Author Bio     Author Bio     Abstract: Wythoff's game is a kind of 2-pile Nim game, which admits taking the same number of stones from both piles. It differs only a little from the 2-pile Nim game, but their winning strategies are quite different from each other. Amazingly the winning strategy of Wythoff's game is directly related to a real number, specifically the golden ratio. In this paper we add two restrictions to this game, and investigate the winning strategy of the revised game. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: On the Number of 2-Player, 3-Strategy, Strictly Ordinal, Normal Form Games Authors: Austin Williams, Portland State University Author Bio     Abstract: The 2-player, 2-strategy, strictly ordinal, normal form games were originally studied by Anotol Rapoport and Melvyn Guyer in a paper entitled A Taxonomy of 2x2 Games. Their paper appeared in 1966 and included an exact count, an enumeration (that is, a complete listing), and a taxonomy of such games. Since then it has been known that there are 78 such games. If we allow each player access to one additional strategy, however, the number of games explodes to nearly two billion. In this paper we compute the exact number of 2-player, 3-strategy, strictly ordinal, normal form games. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Homology of Hom Complexes Authors: Mychael Sanchez, New Mexico State University Author Bio     Abstract: The hom complex Hom(G,K) is the order complex of the poset composed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible and derive a lower bound on the rank of its homology groups. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP