Chapter 2: Explorations with Two-Dimensional Tables

The following sections are included:

• Introduction

• Motivating the Construction of the Addition Table

  • Resolving brainteaser

  • Proof of formula (2.1)

• Motivating the Construction of the Multiplication Table

  • Proof of formula (2.3)

• Using a Spreadsheet in the Construction of the Addition and Multiplication Tables

• Constructing a Single Table of Variable Type and Size

• Divisibility Properties of Numbers in the Addition and Multiplication Tables

  • Counting the multiples of two in the addition and multiplication tables

    • Counting in the addition tables

    • Counting in the multiplication tables

  • Finding the sum of multiples of two in an addition table of even size

  • Proof of formula (2.5)

  • Finding the sum of multiples of two in an addition table of odd size

  • Proof of formula (2.7)

• The Sum of the Multiples of Three in the n × n Addition Table

  • Finding formulas for α(n, 3)

  • Proving formulas (2.8)-(2.10)

    • Proof of formula (2.8)

    • Proof of formula (2.9)

    • Proof of formula (2.10)

• The Sum of the Multiples of Two in the n × n Multiplication Table

  • The case n = 2k

  • The case n = 2k − 1

  • Proof of formula (2.11)

  • Proof of formula (2.12)

• The Sum of the Multiples of Three in the n × n Multiplication Table

  • The case n = 3k

  • The case n = 3k + 1

  • The case n = 3k − 1

  • Proof of formula (2.13)