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MASTER SUDOKU

By Jack Cecil, Jr.

AuthorHouse

Copyright © 2010 Jack Cecil
All right reserved.
ISBN: 978-1-4490-0276-3

Chapter One

Master Sudoku

Introduction

As a field engineer, business consultant, and construction project manager, it seems as though my entire life has revolved around problem solving. Even in my free time, I enjoy mathematics and recreational puzzle solving.

Over the past few years, I've become enamored with Sudoku puzzles. After working several puzzles and becoming frustrated with the slow process of solving them through trial and error, I wondered if there was a systematic way to work these little beasts that would effectively speed up an otherwise tedious activity. Or, at least, save me a lot of frustration.

After many hours of analysis I developed a systematic method, which accelerates the ability to solve a Sudoku puzzle compared to the old trial and error method. Granted, trial and error will be necessary to a certain extent, but my methods reduce the time required to solve these little monsters and, consequently, reduce the frustration of Sudoku puzzle solving.

In this book you will learn how to analyze these puzzles, what to look for when "attacking" a puzzle, and how to dramatically reduce your solving time by selecting the right starting point.

If you are a seasoned Sudoku player, brush up on the fundamentals in the first part of the book and then concentrate mainly on the Speed Sudoku and Creating Your Own Sudoku sections

As you can see below, a Sudoku puzzle is comprised of 81 small squares or cells, which is then divided into 9 larger squares. All 81 squares must be filled in so that each larger square or box, each row, and each column contain the digits 1 through 9. The digits are never duplicated. Therefore, there never is more than one 9 or one 6, for example, in any row, column or box.

At the outset, you will notice in a regular Sudoku puzzle that 20 to 40 of the 81 cells have already been filled in with a digit ranging from 1 to 9. We will call these filled in cells given clues, also called hard clues. Each of the given clues becomes an integral part of the puzzle solving process.

Why Solve Puzzles?

For most of us, a great part of life consists of solving problems. Whether we are students or workers, retired, young or old, our problem-solving acumen is directly proportional to mental skills, such as logic, perspicacity, patience, powers of observation, data gathering, skills, memory, and concentration. Solving puzzles of all types facilitates and actually improves these mental skills so that we can be more effective problem solvers in real life situations.

Research has proven that puzzle solving and other mentally challenging activities can enhance and even improve mental function.

There is no expedient to which man will not resort to avoid the real labor of thinking.

- Sir Joshua Reynolds (1723-1792)

This quotation hung on the wall of our business office for almost fifty years and is indelibly etched into my mind. You can witness the accuracy of this quotation everywhere you look, from the popularity of mindless movies and music to the continued decline in the number of hours per capita people spend reading or pursuing other mentally enriching interests. Puzzle solving is a wonderful way to enhance and increase mental capacity and concentration-activities which are invaluable in successful living.

Getting Started

In beginning to solve a puzzle, it is helpful to determine the weaknesses in the puzzle and to attack its weakest point. This is especially true when working on the easier Sudoku puzzles such as the one on the next page.

Remember, the more given clues in a puzzle, generally speaking, the easier it is to solve. If a puzzle has thirty-six given clues, that means it is a relatively easy puzzle. In order to solve this critter, employ elementary logic or a system I call Solve the Simple Square, or S.S.S.

Listed below are some basic terms used in this guide to refer to different parts of a Sudoku puzzle.

Cell - One of the 81 squares in a Sudoku puzzle.

Box - One of the nine larger sections of a Sudoku puzzle. Each box contains nine cells.

Column - There are nine columns. Each column contains nine cells.

Row - There are nine rows. Each row contains nine cells.

Solve the Simple Square

Begin with the large box that contains the most given clues. Since all of the large boxes in Figure 2.1 contain the same number of clues, begin in the upper left hand corner and the first empty cell in the top row.

The numbers 3, 2, 8 and 7 are given in the upper lefthand box. The missing numbers are 1, 4, 5, 6 and 9. Notice that 1's have been placed as given clues in the large boxes below the target box and the large box in the upper right hand corner. This leaves only one option for placing the 1 in the top left-hand box. See Figure 2.2 below.

Since the cell in the lower right hand corner of our target box is the only possible place the 1 will fit within the Sudoku rules, this cell is a solid "1." Use this number as a solid clue to help us solve the other cells.

Next, find the spot for the next missing number in the box in the upper left hand corner, the 4. Notice Figure 2.3 has two possible locations for the number 4 based on the clues given, so place a small number 4 in both of them.

These small number 4's indicate that it is possible, at this point in solving this particular puzzle, that the 4 could go in either cell but in none of the remaining seven cells. These little 4's are soft clues. The soft clues will become instrumental in helping solve more complex Sudoku puzzles.

Types of Clues

At this point, let's discuss the various types of clues and how to use them to solve a Sudoku puzzle.

First, the given clues are the number filled cells already given.

Secondly, the hard or solid clues are the numbers previously filled, which can be placed nowhere else in the grid.

Thirdly, soft clues are also derived during the puzzle solving process. These are the numbers which could be placed possibly in several places and are instrumental in narrowing the possible locations for the solid clues. In the puzzles they are shown as the small numbers.

In order to reach a solution, convert the soft clues into solid clues. Manage this by filling in more and more cells around the cells containing the soft clues. As the degree of difficulty of the Sudoku puzzles increase, so will the use of soft clues. Many times the cells are filled in with more than one soft clue. Through using logic and the techniques covered later in the book, solving the solid numbers will become a simple process of elimination.

In Figure 2.4 below a solid 1 and two soft 4's are place in the cells. Now, work on the 5. This is found very easily by using the same logic applied when the cell containing the 1 was found.

Apply elementary logic to discover the only possible place the 5 can go in the upper left hand box is in the lower left hand cell.

Now attempt to place the 6 in the upper left hand box. Because the 'given' 6's have been placed in adjacent cells, the 6 could possibly fit in either of the two cells where 'soft' 6's are placed. See Figure 2.5 below.

Since 7 and 8 are already given in the upper left hand box, place two soft 9's in the two cells where the 9 could possibly end up being placed in the final solution. See Figure 2.6 below.

Now move to the upper middle box. Notice that the 1 can only be placed in the upper left hand corner. Therefore, put a hard 1 in this box. See figure 2.7 on the next page.

Next, using the same logic, place a hard 2 in the center cell. See Figure 2.8 below.

It is imperative to be careful when placing the hard numbers-especially when the hard numbers support or replace the soft numbers. Mistakes made in the early part of the puzzle make it extremely difficult to complete without encountering the infamous "Bird's Nest."

The Bird's Nest

A Bird's Nest occurs at the conclusion of the puzzle. It is a situation where only a handful of numbers need to be filled in to complete the puzzle, but you realize you have serious errors preventing you from a successful solution.

Those who work Sudoku puzzles long enough will probably encounter many Bird's Nests, so be careful and accurate when placing the hard numbers.

Reading the Soft Clue

Placed in the large cell above are a series of soft clues in conjunction with the hard clues 9, 3, 6, 8 and 2. In the upper left hand cell the soft clues of 4, 5, 7 and 1 have been placed. In the left center cell 4 and 7 have been placed.

Because the soft 4's are located in the same column and no other soft clues of the number 4 occur in this box, the soft 4's will be helpful in determining the numbers in the same column in the adjoining boxes in the complete Sudoku puzzle.

The same technique with the soft 1's which occur in the same horizontal row can be used. These will be useful in determining placement in the same row in the complete Sudoku puzzle. The soft 4's and soft 1's mentioned above are called pointing pairs.

The soft numbers 5 and 7 are not as useful because they are not isolated to one row or column. Look for soft clues isolated to one row or column. This technique is a vital tool in solving Sudoku puzzles and is known as Pointing Pairs.

The Technique of "Listing"

One of the greatest tools for beginners when solving Sudoku puzzles is the Listing Technique. List the clues, which directly affect the box presently being solved. Start by listing the hard clues and then listing the clues in horizontal rows and vertical columns, which immediately affect the cell in question.

This technique is useful not only for beginners, but also can be extremely useful in overcoming difficult snags in Bird's Nest situations.

Try the following example of how to use this technique by going back to the earlier puzzle.

The hard clues 3, 2, 8, 5, 7 and 1 in the upper left box are given as well as a couple of soft clues, 4 and 6 in the top middle cell, 6 and 9 in the middle left cell and 4 and 9 in the middle right cell.

In the upper middle box we have solid clues 1, 9, 3, 2, 6 and 4. The listing technique will be used now to solve the remaining small cells in the upper middle box, starting with the marked cell in Figure 3.0 above.

Begin by listing the hard clues within the upper middle box, which are either given clues or hard clues already solved. On a scrap of paper write 1, 9, 3, 2, 6 and 4:

Now list the solid clues in the horizontal row you are trying to solve (marked with arrows and shading in Figure 3)-8, 3, 2, 5 and 1:

In the vertical column affecting the targeted cell being solved are the numbers 6 and 2. See the shaded column in Figure 3.1 below:

Add these numbers to the list, as shown below.

The list is now completed. The list created contains all the numbers which cannot logically be placed in the small cell we are solving.

Now comes the fun part. Start by counting from one to nine on the list containing 1, 2, 3, 4, 5, 6, 8 and 9. Note that 7 is missing. Therefore, the only possible number to place in the small cell being solved is 7.

Although you probably could have solved this particular example without using the Listing Technique, it can be a valuable tool in more complex solving situations.

Let's Get Started!

Now we are going to solve some puzzles. We will begin with Puzzle 1.

Cut out the blank grid worksheet on the next page. Using a pencil with a good eraser, follow along step by step filling in the numbers on your worksheet.

Worksheet for Puzzle 1

Cut along the dotted line to remove the worksheet.

Puzzle 1

Now begin working through the sample puzzles.

Cut out the worksheet for Puzzle 1 and follow each step until you get to page 21.

The first four sample puzzles will contain a maximum of seven steps per grid, so that the process will appear as simple as possible. Follow in order steps A through G. Remember, solving a Sudoku puzzle is a progressive process. The solution being attempted depends on the prior cells filled in. In other words, step D many times depends on steps A, B and C.

Note: Always refer to previous grids to verify open cells.

This is a fairly easy puzzle with 36 given clues.

(A) Begin by filling in the soft clues in the puzzle by using the given clues and elementary logic. Notice each column has been marked with the letters A through I and each row with the numbers 1 through 9, allowing easy identification of a particular cell, column or row in the puzzle in order to follow each step.

Because Column A contains a 2 and Row 2 contains a 2, the only possible location for a 2 in the large upper left box is in cell C-1. We say this cell 'must be a 2.'

Using "Can Be, Can't Be"

A critical solving tool in placing soft clues as well as solid clues is the 'can be, can't be' method. You will use this method repeatedly in solving sudoku puzzles no matter what level of player you become. For example, in the circled cell above, your inner dialogue might go like this:

"This cell can be 1 (place a soft 1), can be 2 (place a soft 2), can't be 3, 4, 5, 6, can be 7 (place a soft 7), can't be 8 or 9."

The 'can be, can't be' method is one of the greatest solving tools you can use!

Now the upper left and right boxes are complete so move on to the middle left and right boxes.

Worksheet for Puzzle 1A

Cut along the dotted line to remove the worksheet.

Puzzle 1A

Worksheet for Puzzle 2

Cut along the dotted line to remove the worksheet.

Puzzle 2

This is another fairly simple puzzle. Proceed with steps A - G, referring back to the previous grid to help understand the solving process. Work the process step by step to reach the solution.

With the upper middle box completed, now work on the upper right box.

Worksheet for Puzzle 2A

Cut along the dotted line to remove the worksheet.

Puzzle 2A

Cut along the dotted line to remove the worksheet.

Worksheet for Puzzle 3

Cut along the dotted line to remove the worksheet.

Puzzle 3

This puzzle is more challenging. Follow the solution process step by step. Work in progression from Step A through G, referring back to Grid 1 to verify open cells.

Worksheet for Puzzle 4

Cut along the dotted line to remove the worksheet.

Puzzle 4

Number of given clues - 30. Here is another more complex puzzle. The pace will increase slightly.

Worksheet for Puzzle 5

Cut along the dotted line to remove the worksheet.

Puzzle 5

Worksheet for Puzzle 6

Cut along the dotted line to remove the worksheet .

Puzzle 6

Speed Sudoku

Filling in the hard and soft clues and the listing and counting methods are traditional Sudoku solving techniques that some players use to become proficient in solving Sudoku puzzles. Now, add the dimension of speed to the solving process. The techniques in this section help in finding solutions to puzzles in the quickest way possible.

After introducing the techniques one by one, use each of them in the sample puzzles that follow. The solving tools used are virtually identical to the methods already developed, however, the order these are applied will be slightly different so that the solutions can be reached more quickly - with less time consuming drudgery. Continue to use Pointing Pairs, Pointing Triples, hidden doubles and triples, the "can be can't be" method, and analyzing the columns, rows and boxes.

To begin Speed Sudoku, work through each number, finding as many solid clues, Pointing Pairs and Pointing Triples as you can. Always start with number 1 and work up to number 9. Fill in as many cells as possible using this method. Then you go back, using the new clues to solve the rest of the puzzle in combination with the other methods already learned in this guide.

(Continues...)

Excerpted from MASTER SUDOKUby Jack Cecil, Jr. Copyright © 2010 by Jack Cecil. Excerpted by permission.
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