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Part II of the paper which investigates the patterns in mathematician Blaise Pascal's simplex, as well as the mathematical models, is presented. One of the authors described the Star of David theorems, which involve two-dimensional arrays of binomial coefficients, and their three-dimensional analogues involving trinomial coefficients. It says that the authors use the notations for binomial and trinomial coefficients and extend them to four dimensions.

A conference paper about mathematical solutions from the 68th Annual William Lowell Putnam Mathematical Competition is presented. It discusses various mathematical problems and its solutions, which includes positive integer, area of a convex set in the plane, and polynomials. Details concerning mathematical solutions presented at the competition is presented.

Iterations of the form x n + 1 = f (x n) are used throughout mathematics, and convergence (or not) of an iteration to a fixed point x * = f (x *) is always a central question. We present an informal graphical technique for resolving whether a fixed point in one dimension is attracting or repelling. Specifically, given a real-valued function f, we are determining whether | f ′ (x *) | < 1 or | f ′ (x *) | > 1 at a point x * ∈ R for which x * = f (x *) . The former condition on f ′ (x *) classifies the fixed point as attracting, and the iteration as locally convergent; the latter corresponds to a repulsive fixed point, in which the iterations get further away from the fixed point no matter how close you start them off. The technique requires only that we be able to compute points (x , f (x)) on the graph of f (x) and not necessarily on an explicit functional form or its derivative. [ABSTRACT FROM AUTHOR]

Compound miter saws are used to cut crown molding and to make boxes with equally-sloped sides. Given the slope of the molding or the side of the box, we derive the equations for the proper miter and bevel angles to form the correct corner. The equations are generalized to form boxes where the sides have different slopes. [ABSTRACT FROM AUTHOR]

The article announces that mathematicians Daniel Heath, Andrew Beveridge and Stan Wagon have received the Carl B. Allendoerfer Awards from "Mathematics Magazine."

The article presents information on a mathematical problem related to the value of n for the multiplication table for Zn[x] that have 1's only on the diagonal. It reveals that the problem is meaningful as the multiplication table for Zn[x] has finite size. It mentions an arbitrary ring R to study the problem, where the ring is commutative ring with multiplicative identity 1.

The article focuses on disjoint covering systems suggested by Problem 9 from the 2002 American Invitational Mathematics Examination (AIME). A covering system is reportedly a system of k arithmetic progressions whose union includes all integers. It discusses two questions about covering systems and introduces basic definitions and other facts. It offers upper bounds on the number of consecutive integers that needs to be examined to determine if it is a disjoint covering system.

The article discusses the Brussels Sprouts, a variation of the Sprouts which is a two-person pen-and-paper mathematical game. It describes the rules of the game. It notes that the Sprouts, which was introduced by Michael S. Paterson and John H. Conway, has interesting combinatorial and topological aspects. It introduces a generalization of Brussels Sprouts, called Cloves.

The article presents the equal sums of three fourth powers. This concerns the mathematics trivia of the number 1729. It is said that 1729 is the smallest number expressible as the sum of two cubes in two different ways. The infinite sequence of numbers that this number could be expressed is presented in this paper.

The article analyzes an algebraic problem given at the 1988 International Mathematical Olympiad concerning nonnegative integers. Some unexpected depth and complexity can arise from problems that are stated suprisingly simple. One way to solve such problems is to simplify it even more and analyzing the special cases. The paper also provide a number of exercises posed for the interested reader.

The article discusses a new mathematical model for ribbons. Ribbon models are expressed in terms of the Frenet frame, a moving coordinate system constructed at each point of the curve of a strip of paper. Curvature and torsion along a curve characterizes the geometry of the curve. Examples of mathematical ribbons are presented.

Details mathematician Arthur Cayley's papers on abstract group theory, focusing on his formulation of the abstract group concept. Definition of the abstract group concept; Examples of nonabelian groups of higher orders; Cayley's motivation for the abstract group concept; Introduction of the concept of normal subgroup of a group.

Summary–The simple and elegant result known as Lagrange's identity can be used to derive some well-known results in mathematics and statistics. We share four such results that have elementary derivations based on this identity: the Cauchy–Schwarz inequality, the Pearson correlation coefficient inequality, the Neyman allocation in probability sampling, and the method of equal propositions for apportionment of the United States House of Representatives. [ABSTRACT FROM AUTHOR]

We consider the age-old question: how do I win my fortune at poker? First a disclaimer: this is a poor career choice. But thinking about it involves some great math! Of course, you need to know how good your hand is, which leads to some super-fun counting and probability, but you are still left with questions: Should I hold 'em? Should I fold 'em? Should I bet all my money? It can be pretty hard to decide! Here we get some insight into these questions by thinking about simplified games. Along the way we introduce some ideas from game theory, including the idea of Nash equilibrium. [ABSTRACT FROM AUTHOR]

Proves that the Herstein exercise is no longer true when `finite' is replaced by `countable' and contrasts it with the results of this paper. Finite-dimensional linear space over an uncountable field cannot be covered by a countable family of proper linear subspaces.

Explains Vito Volterra'a short and simple answer to why one cannot find a function continuous at each rational point and discontinuous at each irrational one. Discovered in 1881; Appeared in Volterra's paper `Alcune osservasioni sulle funzioni punteggiate discontinue'; Concept of `pointwise discontinuous' functions; Hermann Hankel.

The article discusses Ramsey's theorem. It presents an assertion of the theory, "the party problem," which states that on any group of six people in a party, it is guaranteed that either three of them will all know each other or there will be three of them, of whom no two know each other. The theorem is illustrated in this paper.

The article presents another way to look at the identity of Ramanujan. Ramanujan integers are defined here and presented with a problem equation followed by the conclusion it satisfy. There are two proofs to this claim and explanation given by Hirschhorn. This paper gives another complete explanation.

We give several non-obvious proofs of the fundamental theorem of algebra, coming from different parts of mathematics. These include the idea of homotopy, Hurwitz's theorem, Rouché's theorem, the open mapping principle, and the argument principle. [ABSTRACT FROM AUTHOR]

Summary. We present an example of a result in graph theory that is used to obtain a result in another branch of mathematics. More precisely, we show that the isomorphism of certain directed graphs implies that some trinomials over finite fields have the same number of roots. [ABSTRACT FROM AUTHOR]