Math and Logic Puzzles That Make Kids Think!

Grades 6-8

Jeffrey J. Wanko

• Publisher: Routledge
• Year: 2021
• ISBN:
  9781000494372
• Edition: 1
• Topic: Education
• Subtopic: Education General

Kids love exploring new ways of solving problems, especially in fun and challenging puzzle formats. In Math and Logic Puzzles That Make Kids Think!, the author presents several variations on Sudoku—the most well-known type of logic puzzle—in an easy-to-use, exciting format perfect for any math classroom. These language-independent logic puzzles provide kids with great problems to stretch how they think and reason.Each puzzle variation utilizes some of the basic strategies of Sudoku puzzles, but each one also draws upon other areas of mathematics—ordering of numbers, properties of geometric shapes, basic operations, or enriched number sense. This book provides teachers with puzzles arranged by difficulty level that can be used to support and enhance students' mathematical investigations. It also provides a new and exciting context for the development of students' deductive reasoning skills, which can lay the foundation for further mathematical exploration.Grades 6-8

Chapter 1
Introduction

DOI: 10.4324/9781003236368-1

Sudoku puzzles have been a mainstay in the United States since 2005. They are based on Latin Squares, which were popularized by Leonard Euler in the 18th century. Latin Squares were simply arrangements of numbers or letters (originally, they contained Latin characters which gave rise to their name) in a square so that each character appeared exactly once in each row and column (see Figure 1). These were not necessarily considered puzzles—they were actually a mathematical curiosity that Euler observed and made use of in different areas of mathematics.

Figure 1. Example 9 × 9 Latin Square.

In 1979, Howard Garns published the first Sudoku puzzle—then called Number Place—in which the basic rules of Latin Squares are used with the additional constraint of bordered regions, which also contain each character exactly once in the solution (see Figure 2). In his puzzle, he omitted some of the numbers and challenged the solver to deduce the missing numbers so that the rules of the puzzle were met (each number had to appear exactly once in each row, each column, and each bordered region).

Figure 2. Example 9 × 9 completed Sudoku.

By 1986, the Japanese puzzle magazine Nikoli had adapted the Number Place puzzles—giving them some additional constraints (like limiting the amount of starting numbers and arranging the starting numbers so that there is rotational symmetry—see Figure 3) and renaming them Sudoku (translated as “single number”). When computer programs were developed in the 1990s to generate Sudoku puzzles, they became more popular and were soon featured in newspapers and puzzle books around the world.

Figure 3. Example 9 × 9 Sudoku with starting numbers arranged in a rotationally symmetric pattern (see the answer key for the solution to this puzzle).

Sudoku puzzles have an undeniable appeal. Their rules are easy to understand, yet puzzles can range from very easy to extremely difficult. They also are a type of language-independent and culture-independent logic puzzle—that is, once the basic rules are understood, neither language nor culture is a barrier to the solver. This is very different from the most popular type of puzzle that exists—the crossword puzzle. With crosswords, the solver must know not only the language of the puzzle to read the clues and provide answers, but also aspects of the appropriate culture. Sudoku puzzles transcend language and culture, enabling the solver to pick up a Sudoku puzzle from Japan, Germany, India, or any place in the world and know the goals and rules of the puzzle.

Language-independent logic puzzles have gained in popularity throughout the world over the past few decades. Today, there is even a World Puzzle Federation (WPF), an organization that advocates language-independent puzzles across the globe. The WPF sponsors a World Puzzle Competition every year in which teams of solvers from a number of countries meet to solve puzzles in friendly competition. One popular type of puzzle that appears in the competition every year is the Sudoku puzzle and its many variations.

Some people mistakenly believe that Sudoku puzzles are mathematical because they use numbers. But although numbers are the most typical characters used in Sudoku puzzles, they are not at all necessary. There are Sudoku puzzles that use letters, symbols, or even pictures as the elements that are placed in the grid (see Figure 4 for a Word Sudoku puzzle in which a nine-letter word appears in the solution). Nevertheless, Sudoku puzzles are extremely mathematical—in the number of possible placements of characters, the symmetry of the starting grids, and in the deductive reasoning that is used in finding the solution to a puzzle.

Figure 4. Word Sudoku puzzle (see the answer key for the solution to this puzzle).

This book contains a few variations of Sudoku puzzles and Latin Squares that have additional mathematical elements in support of school mathematics. Each of the puzzle types included here involves a mathematical concept (or concepts) that contributes to the solution strategies for the puzzles. Some of these solution strategies are discussed in the opening section of each puzzle type, but other strategies are left for you and your students to discover and devise.

The puzzles in each section are assigned general difficulty levels. These levels represent our best attempt at classifications, but individual solvers may approach a puzzle with different strategies that might indicate a different difficulty level. This is a good thing and solvers are encouraged to explore the underlying mathematics and logic of the puzzles to find alternative strategies for solving. With each puzzle, there is exactly one solution—but there may be more than one way to arrive at that solution.

Using the Puzzles in Your Classroom

There are many ways in which these and other language-independent logic puzzles can be used in your classroom. At the most basic level, students can be challenged as solvers to find solutions to the puzzles. This is what we do when we pick up a puzzle magazine and work on puzzles—it’s a solitary endeavor that can lead to some new insight about strategies and methods, but not always.

Another approach is to have students work collaboratively on puzzles or to have conversations about their solving methods. These approaches help students devise problem-solving strategies and develop their deductive and spatial reasoning skills by making arguments (about what they know and how they know it) to their peers. This practice forms the foundation of mathematical proof and argumentation—skills that become more important for students as they interact with more advanced levels of mathematics. With this approach, teachers often allow students to work together or independently on puzzles, while mediating conversations about the students’ solving strategies. Questions that often are used as prompts in these conversations include:

• What is the first number that you filled in and how do you know that it must be placed there?
• Is there a row, column, or bordered region that looks like it might be one of the first ones that you may be able to complete and why?
• What are all of the different starting moves that you can make in a puzzle without knowing any information other than what is originally given?
• Is there a unique strategy that can be used in this puzzle and what about this puzzle allows for this strategy to be possible?

The most common question that is asked in this conversation is simply:

• How do you know that?

This basic prompt indicates to students that they must supply a sound mathematically deductive reason for a particular step. The explanation must be easy to follow, must account for any other possibilities, and must be defensible—conditions that students will be addressing when they are writing formal proofs in geometry and other mathematics classes.

These suggestions address the important logical aspect of the puzzles—the left-brain activities that are a natural part of most mathematical endeavors. But there also is a right-brain aspect of puzzles that can appeal to the creative nature of your students as well. After solving and discussing these puzzles, students also can be encouraged and challenged to create puzzles of their own for their peers to solve. Good puzzle design starts with an understanding of the underlying structure of the particular type of puzzle. From there, puzzle designers are challenged to find new approaches or new variations for solvers to experience. This is how the puzzles in Chapter 3—Shapedoku—came about. My colleague and I were exploring the different aspects of Sudoku variations and proposed a new puzzle type that forced the solver to look at the placements of a set of numbers while considering the geometric shape created by connecting the numbers in that set. We played around with that idea until we had developed Shapedoku—a variation that uses not the values of the numbers themselves (as with the other types in this book), but the figure created by the placement of the numbers.

Students don’t have to create their own variation of Sudoku to use right-brain thinking strategies. They should start by picking one of the Sudoku variations included here and creating new examples. They will learn that the process is not an easy one—that even through their best efforts, it is not uncommon to create a puzzle that has more than one solution or does not have enough information to solve it. Information is included near the end of the introduction in each chapter about how students might approach designing puzzles of that type. As a teacher, you might have students solve one another’s puzzles to check for uniqueness of solutions and devise a rating system for the difficulty of each puzzle. Students’ puzzles could then be collected in a class puzzle book that could be shared with parents and other students.

Chapter 2
Greater Than Sudoku

DOI: 10.4324/9781003236368-2

The first variation is known as Greater Than Sudoku. It uses the basic rules of Sudoku puzzles (each number appears exactly once in each row, each column, and each bordered region) but also capitalizes on the ordering of the numbers that are used. Within each bordered region, greater than (and less than) signs are placed between any adjacent squares, indicating which of t...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Chapter 1: Introduction
• Chapter 2: Greater Than Sudoku
• Chapter 3: Shapedoku
• Chapter 4: Killer Sudoku
• Chapter 5: Calcudoku
• Answer Key
• About the Author