Problem Solving Series: Booklet No. 1. How to, 2. Combinatorics 1, 3. Graph Theory, 4. Number Theory 1, 5. Geometry 1

1991 ◽  
Vol 75 (472) ◽  
pp. 223
Author(s):  
Maureen D. Mclaughlin ◽  
Derek Holton
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Problem Solving

scholarly journals Nonexistence of Almost Moore Digraphs of Diameter Three

10.37236/811 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
J. Conde ◽  
J. Gimbert ◽  
J. Gonzàlez ◽  
J. M. Miret ◽  
R. Moreno
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Algebraic Number ◽  
Directed Graphs ◽  
Algebraic Number Theory ◽  
Cycle Structure ◽  
Spectral Techniques ◽  
Binary Matrices ◽  
Algebraic Multiplicities ◽  
Big Pieces

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound $M(d,k)=1+d+\cdots +d^k$, where $d>1$ and $k>1$ denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when $d=2,3$ or $k=2$. In this paper, we prove that almost Moore digraphs of diameter $k=3$ do not exist for any degree $d$. The enumeration of almost Moore digraphs of degree $d$ and diameter $k=3$ turns out to be equivalent to the search of binary matrices $A$ fulfilling that $AJ=dJ$ and $I+A+A^2+A^3=J+P$, where $J$ denotes the all-one matrix and $P$ is a permutation matrix. We use spectral techniques in order to show that such equation has no $(0,1)$-matrix solutions. More precisely, we obtain the factorization in ${\Bbb Q}[x]$ of the characteristic polynomial of $A$, in terms of the cycle structure of $P$, we compute the trace of $A$ and we derive a contradiction on some algebraic multiplicities of the eigenvalues of $A$. In order to get the factorization of $\det(xI-A)$ we determine when the polynomials $F_n(x)=\Phi_n(1+x+x^2+x^3)$ are irreducible in ${\Bbb Q}[x]$, where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial, since in such case they become 'big pieces' of $\det(xI-A)$. By using concepts and techniques from algebraic number theory, we prove that $F_n(x)$ is always irreducible in ${\Bbb Q}[x]$, unless $n=1,10$. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.

Even Perfect Numbers—An Update

1981 ◽  
Vol 74 (6) ◽  
pp. 460-463
Author(s):  
Stanley J. Bezuszka
Keyword(s):  
Number Theory ◽  
Problem Solving ◽  
History Of Mathematics ◽  
Independent Study ◽  
History Of ◽  
Perfect Numbers ◽  
Excellent Resource

Do you have students who are computer buffs, always looking for a new problem to program efficiently? Do you have students who do independent study projects? If so, motivate them with this topic that is rich in the history of mathematics and number theory—perfect numbers. They provide an excellent resource for theoretical as well as computerized problem solving.

Teaching Teachers to Teach Mathematics

1996 ◽  
Vol 178 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Rika Spungin
Keyword(s):  
Number Theory ◽  
Problem Solving ◽  
Prospective Teachers ◽  
Problem Solving Strategies

Group investigations from the areas of number theory, probability, and geometry. are presented and discussed. By working in groups, sharing ideas, and making and testing conjectures, prospective teachers gain confidence in their own ability to do mathematics and develop a variety of useful problem-solving strategies.

Polygons, Stars, Circles, and Logo

1986 ◽  
Vol 33 (9) ◽  
pp. 6-11
Author(s):  
Bill Craig
Keyword(s):  
Middle School ◽  
Elementary School ◽  
Number Theory ◽  
Problem Solving ◽  
Elementary School Students ◽  
Upper Elementary ◽  
School Students ◽  
Know How ◽  
Elementary And Middle School ◽  
Mathematical Topic

Many teacher are excited about the potential uses of Logo with elementary school students. The language give students access to mathematical topic they have not previouly explored. The following activitie uae Logo in the study of geometry, number theory, and problem solving. The activities assume that tudents are familiar with turtlegraphic commands (FORWARD, BACK, RIGHT, LEFT) and know how to define procedures. The activitie are designed for students in the upper elementary and middle school grades. The star procedure and explorations are adapted from Discovering Apple Logo by David Thornburg. The book contains excellent ideas for the use of Logo as a tool for mathematical explorations. See the Bibliography for additional resources.

Clock Buddies: An Accessible, Engaging Problem-Solving Activity with Rich Mathematical Content

2017 ◽  
Vol 111 (1) ◽  
pp. 16-24
Author(s):  
Debra K. Borkovitz ◽  
Thomas Haferd
Keyword(s):  
Graph Theory ◽  
Problem Solving ◽  
Real World ◽  
Mathematical Content

Explore an accessible, open-ended, real-world scheduling activity that is connected to topics in graph theory.

A Pattern in Number Theory: Example→ Generalization→ Proof

1971 ◽  
Vol 64 (7) ◽  
pp. 661-664
Author(s):  
David R. Duncan ◽  
Bonnie H. Litwiller
Keyword(s):  
Number Theory ◽  
Problem Solving ◽  
Mathematical Induction ◽  
Whole Number

The study of patterns is an integral part of the study of mathematics. As we teach mathematics, we must point out how to search for patterns and how patterns may aid us in problem solving. The following problem is one that combines patterns, ideas from number theory, and mathematical induction: “Prove that it is possible to pay, without requiring change, any whole number of rubles (greater than 7) with banknotes of value 3 rubles and 5 rubles” (Sominskii 1964, p. 19).

Delving Deeper: The Chebyshev Polynomials: Patterns and Derivation

2004 ◽  
Vol 98 (1) ◽  
pp. 20-25 ◽  
Author(s):  
Benjamin Sinwell
Keyword(s):  
High School ◽  
Number Theory ◽  
Graph Theory ◽  
Fourier Series ◽  
High School Teachers ◽  
Chebyshev Polynomials ◽  
Mathematical Community ◽  
Russian Mathematician ◽  
School Teachers ◽  
Mathematical Exploration

Pafnuty Lvovich Chebyshev, a Russian mathematician, is famous for his work in the area of number theory and for his work on a sequence of polynomials that now bears his name. These Chebyshev polynomials have applications in the fields of polynomial approximation, numerical analysis, graph theory, Fourier series, and many other areas. They can be derived directly from the multiple-angle formulas for sine and cosine. They are relevant in high school and in the broader mathematical community. For this reason, the Chebyshev polynomials were chosen as one of the topics for study at the 2003 High School Teachers Program at the Park City Mathematics Institute (PCMI). The following is a derivation of the Chebyshev polynomials and a mathematical exploration of the patterns that they produce.

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