Sequential Reasoning is the third book in the series Developing Mathematical Thinking Through Game-based Learning. Sequential Reasoning is designed to guide the active reader towards accomplishing a goal through a variety of sequential steps. The connections between sequential reasoning and mathematical thinking are clearly displayed.
Sequential reasoning is a broad and perhaps vague term given to the type of reasoning that is at the core of everyday problem solving and mathematical thinking. Sequential reasoning is used in many projects, whether it is in party planning, remodeling a home, or solving complex issues in your work. In mathematics instruction, sequential reasoning is used in problem solving and is fundamental to solving equations and creating proofs.
Sequential Reasoning is intended for a variety of ages and audiences. Sequential Reasoning puzzles can be played as part of your school math curriculum or as home entertainment for game players ages 9 to 90.
Sample Sequential Reasoning Puzzles
Color Network Puzzle
Each circle is to be colored red (R), yellow (Y), or Blue (B). However no two vertices of a triangle are permitted to be the same color. What must be the color of the circle marked with a question mark?
Angle Chase Puzzles
Students rely on their recall of geometry properties, not their protractor, when solving Angle Chase puzzles. The Angle Chase puzzle below uses properties of parallels, vertical angles, isosceles triangle properties, and polygon sum properties. Find the measures of all the angles in the Angle Chase below.
Sudoku Puzzles
A Sudoku puzzle is a logical placement puzzle in the form of a square grid of squares called cells. Some cells contain given numerals. The most common Sudoku puzzle, the puzzle that has been sweeping the planet, is a 9x9 Sudoku. When a 9x9 Sudoku puzzle is completed every row, column, and non-overlapping 3x3 box contains every digit, 1 through 9. The objective is to fill in all the empty cells. Logical sequential reasoning and perseverance is all that is required.
The most common Sudoku is the 9x9. A polyomino Sudoku puzzle is a Sudoku puzzle in which each box is a polyomino region rather than a square box.
Classic 9x9 Sudoku Polyomino Sudoku
NumbrixTM Puzzles
In the fall of 2008, Marilyn vos Savant began publishing a puzzle called NumbrixTM each week in her "Ask Marilyn" column. The puzzle is played on a square array of cells, often a 9x9. Some of the cells contain numbers. The objective in a 9x9 NumbrixTM puzzle is to fill the remaining empty cells with integers from 1 to 81 so that when completed there is a continuous path of numbers from 1 through 81. The path moves horizontally or vertically from cell to cell but not diagonally.
Here are two puzzles I created. Numbrix Puzzle 1 has the given numbers around the outer perimeter. Numbrix Puzzle 2 has the given numbers along the two diagonals. Add the missing numbers to complete the 7x7 puzzle below left and the missing numbers in the 9x9 puzzle below right to create a continuous path in each.
Numbrix Puzzle 1. Numbrix Puzzle 2.
Magic Square
A Magic Square is a square array of distinct integers such that the numbers in any row, column, or main diagonal, have the same sum (called the magic sum). If the n2 numbers in an n by n magic square are the positive integers 1 through n2 then the magic square is a normal magic square. The magic square below is a normal 4x4 magic square.
Magic Square Puzzles. A magic square puzzle is an incomplete magic square. Complete the 5x5 magic square. You will need to calculate the magic sum first.
Lunar Lockout Puzzles
Lunar Lockout is a set of 40 puzzles created by Hiroshi Yamamoto with the assistance of Nob Yoshigahara for Binary Arts (now called ThinkFun.com). Lunar Lockout is played on a 5x5 grid. The objective is to maneuver the red piece onto the center red square.
The Rules of Lunar Lockout. In the Lunar Lockout puzzles to follow, the objective is to maneuver the piece (Spacepod) indicated by the letter Q, with the help of the other helper-bots indicated by the numbers 1-5 until the Q piece lands on the Emergency Entry Port (the center red square in the puzzles below). All pieces (the Q as well as the helper bots 1-5) can only move according to the same two rules.
The Rules for moving pieces:
• All pieces move only horizontally or vertically.
• A piece continues to move until its path is blocked by another piece.
Example: How do you move the Q onto the red square?
The Q (red piece) cannot move straight down because there is nothing to stop it (so it would continue past the red square). However, if the 1-bot moves to the right it is stopped by the 2-bot. The Q can then move down to the red square because it will be stopped by the 1-bot.
We symbolize the movements up, down, right, and left of the pieces by using arrows. One solution is 1 to the right, then Q down.
Symbolically it is written: 
There is another solution. How would you write it?
Lunar Lockout Puzzles. Solve these Lunar Lockout puzzles and write their solutions symbolically.
Lunar Lockout 1. Lunar Lockout 2.
Summary
This article described six different sequential reasoning puzzles (and of course each puzzle has many variations). These sequential reasoning puzzles, played as part of your school mathematics curriculum, will create an exciting change of pace. They are novel ways to introduce the concept of sequential reasoning in problem solving and proof. Because these sequential reasoning puzzles are fun they are also motivational. Yet your students will develop strategies and reasoning skills faster and deeper with these games than with traditional methods. Many parents are looking for opportunities to do mathematics with their children. Send these puzzles home with your students to be played with by the whole family. Sequential reasoning puzzles provide a setting for students and parents to work on problem solving and mathematics together. Try them. I think you’ll see the “reason” for including them in your lesson plans.
