HOW THIS PROGRAM IS SPECIAL (C)
by Jerry Martin
Sudoku as a Teaching Tool, an innovative new program, is special because:
It is the product of three different sources: education, psychology and a common puzzle (Sudoku). Understanding how these three specialties conjoin into a useful tool for the benefit of many humans is creative. This program answers the WHY? and HOW? questions of doing Sudoku.
Sudoku, by its very nature, requires (and can be used to teach) the logical, critical thinking and rational problem solving that are defining characteristics of intelligence. Dr. Howard Gardner’s theory of human intelligence, presented in his book Frames of Mind: The Theory of Multiple Intelligences, recognizes seven intelligences possessed by humans. Two of these seven, logical and verbal, comprise IQ tests. Sudoku requires and teaches pure logic, half of what scores high on IQ tests.
This program makes the connection, previously unmade, between a simple, cheap, common puzzle and a valuable, easy, practical, lowtech teaching tool. It uses Sudoku to benefit many people. At present I’m using it with public elementary students, but it could also be used with senior citizens and on cruise ships and even with prison inmates.
A few specifics:
About 50 times in each puzzle it is necessary to distinguish between the relevant and irrelevant. Usually there are 36 numbers that are relevant; usually there are many more numbers that are irrelevant and should be ignored. Learning to do this helps develop critical thinking.
About 50 times in each puzzle it is necessary to determine the several causes of each cell’s solution (the effect), which is always one particular number in one particular place (cell). Learning to determine cause and effect is also helpful in all problem solving logically.
Accuracy is totally necessary, and there are about 50 opportunities in each puzzle for inaccuracies. There is only one correct answer for each cell, so about 50 correct answers for the whole puzzle are required for a successful completion. If you make a mistake at any time, it usually won’t be discovered until much later in your work. Accuracy is a cornerstone of logical thinking.
Pattern recognition, taught in many ways by Sudoku, is an important element of everyone's education. Patterns dictate the behavior of nature and mankind's many creations. Science defines patterns, as does psychology.
As I teach Sudoku now to children in an after school program by projecting a large puzzle on to the white board (common in all American elementary school classrooms), children have many opportunities to solve a cell in front of the class. This experience, performed several times by each child each week, is a low stress opportunity to practice public speaking. If done often when young, most humans overcome their fear of public speaking. Successfully solving a puzzle also raises selfesteem and provides intrinsic motivation.
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by Jerry Martin
Sudoku as a Teaching Tool, an innovative new program, is special because:
It is the product of three different sources: education, psychology and a common puzzle (Sudoku). Understanding how these three specialties conjoin into a useful tool for the benefit of many humans is creative. This program answers the WHY? and HOW? questions of doing Sudoku.
Sudoku, by its very nature, requires (and can be used to teach) the logical, critical thinking and rational problem solving that are defining characteristics of intelligence. Dr. Howard Gardner’s theory of human intelligence, presented in his book Frames of Mind: The Theory of Multiple Intelligences, recognizes seven intelligences possessed by humans. Two of these seven, logical and verbal, comprise IQ tests. Sudoku requires and teaches pure logic, half of what scores high on IQ tests.
This program makes the connection, previously unmade, between a simple, cheap, common puzzle and a valuable, easy, practical, lowtech teaching tool. It uses Sudoku to benefit many people. At present I’m using it with public elementary students, but it could also be used with senior citizens and on cruise ships and even with prison inmates.
A few specifics:
About 50 times in each puzzle it is necessary to distinguish between the relevant and irrelevant. Usually there are 36 numbers that are relevant; usually there are many more numbers that are irrelevant and should be ignored. Learning to do this helps develop critical thinking.
About 50 times in each puzzle it is necessary to determine the several causes of each cell’s solution (the effect), which is always one particular number in one particular place (cell). Learning to determine cause and effect is also helpful in all problem solving logically.
Accuracy is totally necessary, and there are about 50 opportunities in each puzzle for inaccuracies. There is only one correct answer for each cell, so about 50 correct answers for the whole puzzle are required for a successful completion. If you make a mistake at any time, it usually won’t be discovered until much later in your work. Accuracy is a cornerstone of logical thinking.
Pattern recognition, taught in many ways by Sudoku, is an important element of everyone's education. Patterns dictate the behavior of nature and mankind's many creations. Science defines patterns, as does psychology.
As I teach Sudoku now to children in an after school program by projecting a large puzzle on to the white board (common in all American elementary school classrooms), children have many opportunities to solve a cell in front of the class. This experience, performed several times by each child each week, is a low stress opportunity to practice public speaking. If done often when young, most humans overcome their fear of public speaking. Successfully solving a puzzle also raises selfesteem and provides intrinsic motivation.
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AN UNUSUAL COMPETITION (c)
by Jerry Martin
Humans have created many forms of competition which match humans against humans, either one against one or teams opposing each other. Most are athletic, such as football, tennis, golf and marathons. Others are more cerebral, such as chess, poker or cribbage. But all have humans as opponents, usually resulting in as many losers as winners. And a player's success or failure is partially determined by the opponent, who is playing defense. In most games, every move I make is good or bad, depending on my opponent's reaction.
Many games have an element of luck. If there are dice or cards or a football (funny bounces out of anyone's control), there are things that effect the outcome that can't be managed by humans. Luck partially determines one's success, in these cases.
Sudoku is a competition that lacks all these characteristics. It is a competition between a human and a puzzle. Team Sudoku is a competition between 3 or 4 humans and a puzzle. The puzzle plays no defense and has already made all his moves. My success or failure depends solely on my own actions. This is true for my team, as well. We all sink or swim together, meeting the same fate. If we make one mistake, we have constructed a plane that won't fly, a boat that won't float, and it's never the puzzle's fault. And luck plays no role.
Also, Sudoku can be done by a very wide range of humanity. Gaining popularity, solo Sudoku is done in many places in the world. Soon, I expect it will be done by teams, which started here in Nevada County. All ages, 6 to 96, enjoy the challenge. Both genders like it equally and demonstrate equal ability. It works with any language. Four players on one team who speak four different languages can be successful solving puzzles. There's no need for athleticism, which makes it fun for more people. And injuries are nonexistent. It's very cheap, requiring no more than a pencil and flat place to put the puzzle. No high tech digital instruments, no batteries, and it folds up in your pocket, traveling well, so not restricted to time limits. It's also very convenient, being done alone or with others.
Anyone who can count to nine can do Sudoku. Unlike crosswords, which require spelling and a familiarity with trivia, Sudoku requires none of these. It's a poor man's education, teaching how to think, rather than what to think. Though it doesn't require much education, it can be devilishly difficult. There's nothing that is political, religious, cultural, racial or ethnic, so Sudoku bypasses these areas that sometimes separate us.
But none of these unusual characteristics explain my enthusiasm. Sudoku teaches and trains several useful thinking habits, called critical thinking, or logic. Solving each cell, of which there are about 50 per puzzle, repeats an educating process of discerning relevance, recognizing useful patterns, operating under very strict demands for total accuracy (truth) and the necessity of withholding a final decision until there's enough necessary information. Guessing never works, and subjective feelings are counterproductive. It also develops focus and concentration in children and produces perseverance necessary for success. This is probably why most children under about 6 years are too young to succeed.
Team Sudoku, historically created here in Nevada County and held, for the first time in a tournament here last April, teaches children important social skills. Unlike most team sports, there is no division of labor. All teammates do the same things; find solutions. Experienced teams must employ good communication and cooperation, holding personal feelings back. There is only one correct answer and eight incorrect answer with every cell. So whenever a player has a solution, the teammates should (if they want to win and avoid mistakes) demand, "Prove it". Proof can be easily done if the proposed solution is correct. If not, it must be rejected until more information is determined. This eliminates long disagreements, all of which can be settled decisively, with universal agreement. The answer is either right or wrong; there's no gray in between answer. All arguments are over quickly.
Team Sudoku simulates challenging natural disasters, such as fires, floods, famine, earthquakes and pandemics that bring humans together to combat a mutual threat. A team solving a Sudoku puzzle must communicate and collaborate in order to succeed, in much the same way we did recently when fire destroyed parts of our community. This brought out the better angels of our nature, as it always does. But Sudoku does it without the danger and stress and life changing losses. It's good practice in togetherness.
I am reminded of a quote from Virgil, the Roman poet: "As the twig is bent the tree's inclined." If we can give our children good thinking skills, forming good mental habits, they will develop into rational thinkers. Sudoku and Team Sudoku are practical means to achieving this end. And kids love it after they learn what it is.
by Jerry Martin
Humans have created many forms of competition which match humans against humans, either one against one or teams opposing each other. Most are athletic, such as football, tennis, golf and marathons. Others are more cerebral, such as chess, poker or cribbage. But all have humans as opponents, usually resulting in as many losers as winners. And a player's success or failure is partially determined by the opponent, who is playing defense. In most games, every move I make is good or bad, depending on my opponent's reaction.
Many games have an element of luck. If there are dice or cards or a football (funny bounces out of anyone's control), there are things that effect the outcome that can't be managed by humans. Luck partially determines one's success, in these cases.
Sudoku is a competition that lacks all these characteristics. It is a competition between a human and a puzzle. Team Sudoku is a competition between 3 or 4 humans and a puzzle. The puzzle plays no defense and has already made all his moves. My success or failure depends solely on my own actions. This is true for my team, as well. We all sink or swim together, meeting the same fate. If we make one mistake, we have constructed a plane that won't fly, a boat that won't float, and it's never the puzzle's fault. And luck plays no role.
Also, Sudoku can be done by a very wide range of humanity. Gaining popularity, solo Sudoku is done in many places in the world. Soon, I expect it will be done by teams, which started here in Nevada County. All ages, 6 to 96, enjoy the challenge. Both genders like it equally and demonstrate equal ability. It works with any language. Four players on one team who speak four different languages can be successful solving puzzles. There's no need for athleticism, which makes it fun for more people. And injuries are nonexistent. It's very cheap, requiring no more than a pencil and flat place to put the puzzle. No high tech digital instruments, no batteries, and it folds up in your pocket, traveling well, so not restricted to time limits. It's also very convenient, being done alone or with others.
Anyone who can count to nine can do Sudoku. Unlike crosswords, which require spelling and a familiarity with trivia, Sudoku requires none of these. It's a poor man's education, teaching how to think, rather than what to think. Though it doesn't require much education, it can be devilishly difficult. There's nothing that is political, religious, cultural, racial or ethnic, so Sudoku bypasses these areas that sometimes separate us.
But none of these unusual characteristics explain my enthusiasm. Sudoku teaches and trains several useful thinking habits, called critical thinking, or logic. Solving each cell, of which there are about 50 per puzzle, repeats an educating process of discerning relevance, recognizing useful patterns, operating under very strict demands for total accuracy (truth) and the necessity of withholding a final decision until there's enough necessary information. Guessing never works, and subjective feelings are counterproductive. It also develops focus and concentration in children and produces perseverance necessary for success. This is probably why most children under about 6 years are too young to succeed.
Team Sudoku, historically created here in Nevada County and held, for the first time in a tournament here last April, teaches children important social skills. Unlike most team sports, there is no division of labor. All teammates do the same things; find solutions. Experienced teams must employ good communication and cooperation, holding personal feelings back. There is only one correct answer and eight incorrect answer with every cell. So whenever a player has a solution, the teammates should (if they want to win and avoid mistakes) demand, "Prove it". Proof can be easily done if the proposed solution is correct. If not, it must be rejected until more information is determined. This eliminates long disagreements, all of which can be settled decisively, with universal agreement. The answer is either right or wrong; there's no gray in between answer. All arguments are over quickly.
Team Sudoku simulates challenging natural disasters, such as fires, floods, famine, earthquakes and pandemics that bring humans together to combat a mutual threat. A team solving a Sudoku puzzle must communicate and collaborate in order to succeed, in much the same way we did recently when fire destroyed parts of our community. This brought out the better angels of our nature, as it always does. But Sudoku does it without the danger and stress and life changing losses. It's good practice in togetherness.
I am reminded of a quote from Virgil, the Roman poet: "As the twig is bent the tree's inclined." If we can give our children good thinking skills, forming good mental habits, they will develop into rational thinkers. Sudoku and Team Sudoku are practical means to achieving this end. And kids love it after they learn what it is.
DISCLOSURES:
These explanations below will be much more easily understood by people who do Sudoku and are reasonably familiar with the solving process.
I am not now a pedagogue, formally, and spend little time reading and discussing their issues using pedagogical vernacular. Therefore, I apologize if my language below doesn’t accurately conform with modern educational language. I hope, however, that I am able to describe Sudoku and its value as a teaching tool accurately despite my inability to use commonly accepted terminology.
HOW SUDOKU CAN IMPACT COMMON CORE STANDARDS FOR MATHEMATICAL PRACTICE (C)
by Jerry Martin
MP.1 Make sense of problems and persevere in solving them.
Making sense of Sudoku is easy. One must fill in the empty cells (small boxes) with numbers, 1 through 9, according to a simple set of rules. There are many starting points, all equally effective.
By following the five steps that appear on the back of each puzzle paper that’s distributed in my program, one can solve the cells, one by one. This clearly defined process establishes a method for eventually completing the puzzle.
Each time a cell is successfully filled a small feeling of satisfaction is experienced. This feeling is repeated many times, encouraging the child to continue until the total puzzle is completed. There is an intrinsic incentive to completing the puzzle, rewarded with each small step towards completion.
MP.2 Reason abstractly and quantitatively
When solving any Sudoku puzzle, the questions of who, when and why should be ignored, as they are all irrelevant. The question of how can be reduced to solving the what and where of each puzzle. Only these two questions, what and where, are relevant and need to be considered. Considering them is how we solve every puzzle. This is the abstract reasoning required by Sudoku.
The what is usually any one of nine numbers, 1 through 9, though it could be nine letters of the alphabet or any nine written symbols. Each of these nine numbers must be placed correctly in each family (of nine numbers). For example, if we decide to work on the number 3 (what), we try to determine the only cell in a particular family (where) the 3 can go. If we find such a place, we can enter it as a solution to a cell. And there is only one correct solution in each cell.
Or we can try a different approach. The where can be any empty cell in any incomplete family. We can choose a particular cell (where) and decide if there is only one number (what) that can fit in that cell. If we find that number (what), we should write it into that cell, thus partially solving the puzzle.
The symbols used, usually numbers 1 through 9, are all equal in value, and since any nine symbols could be used, including letters of the alphabet, there are no quantities involved in Sudoku. The nine numbers are never manipulated with mathematical processes; no addition, subtraction, division or multiplication. Consequently, quantitative reasoning is not addressed by Sudoku. Sudoku is not math; it is logic. This is a very important distinction that is often misunderstood by people unfamiliar with Sudoku.
MP.3 Construct viable arguments and critique the reasoning of others.
Sudoku is excellent at giving children many opportunities for arguing solutions and refuting faulty reasoning of others. As there is only one correct solution to each cell (of many in each puzzle), and several incorrect solutions, and each correct solution is subject to visible proofs, Sudoku is clearly set up for possible debates about solutions. It’s always easy to prove or disprove every solution, right or wrong.
MP.4 Model with mathematics
As Sudoku is not mathematics (it’s logic), there is almost nothing about Sudoku or my program that addresses this objective. However, Sudoku does offer many opportunities to “Simplify a complex problem….”. It’s necessary to identify which numbers are relevant in order to solve each empty cell, while ignoring those numbers that are irrelevant. This simplifies a complex problem. This process is required many times to solve a whole Sudoku puzzle.
However, though unnecessary for solving puzzles, an enterprising teacher could find many examples in every puzzle that could be used for teaching percentages. For example, if one family has only two empty cells, needing a 2 and an 8, and we don’t have enough information to determine which is which, we could say there’s a 50% chance each number could belong in either empty cell.
MP.5 Use appropriate tools strategically
When solving Sudoku puzzles, there are many different tactics and strategies required, particularly for harder puzzles. Basically, we can try to solve a cell using different tactics that might work to learn which number goes there (what goes where). Or we can try to enter a number by determining which cell it should go into (where goes what). There are many different tactics to determine either solution. A large part of Sudoku is learning a variety of tactics to be used to fill a cell with the appropriate number. Of course, each tactic can only be used in appropriate circumstances.
MP.6 Attend to precision
One of the most valuable characteristics of Sudoku is its constant requirement for accuracy. As there is only one correct answer to each cell’s solution, and only one correct solution to the overall puzzle, any mistake will result in unsuccessful results. Every puzzle repeatedly persuades children that it’s necessary to be accurate with every cell solution. This persuasion occurs about 50 times each puzzle. One inaccuracy will always produce failure, and you may not discover an earlier mistake until almost at the end of the puzzle. Sudoku teaches precision. Imprecision always fails.
MP.7 Look for and make use of structure
Sudoku is excellent for increasing the observation of each puzzle’s structure. Anyone doing Sudoku constantly looks for certain patterns and locations of numbers and how they can be used to solve cells. As one learns and practices Sudoku, one becomes better at making these observations. A puzzle can only be solved by determining the positions of the relevant given numbers, which are used to place the correct numbers in the empty cells.
MP.8 Look for and express regularity in repeated reasoning.
Sudoku requires repeated reasoning about 50 times in solving each puzzle. We look for “eliminators”, concluding that a number cannot go there or there or there, so it must go here, in a particular cell. If 1,2,3,4,5,6,7 and 8 cannot go in a cell, only a 9 will work. Or we can solve a cell with the other approach, determining that only a certain cell can correctly contain a particular number. Either of these two strategies are repeated continuously, reenforcing logical processes.
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FIVE STEPS FOR SOLVING EACH CELL (C)
by Jerry Martin
1. Identify the family you will work on
2. Say which numbers are needed in that family
3. Say which number you will solve
4. Identify eliminators
5. Solve the cell with a number
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These explanations below will be much more easily understood by people who do Sudoku and are reasonably familiar with the solving process.
I am not now a pedagogue, formally, and spend little time reading and discussing their issues using pedagogical vernacular. Therefore, I apologize if my language below doesn’t accurately conform with modern educational language. I hope, however, that I am able to describe Sudoku and its value as a teaching tool accurately despite my inability to use commonly accepted terminology.
HOW SUDOKU CAN IMPACT COMMON CORE STANDARDS FOR MATHEMATICAL PRACTICE (C)
by Jerry Martin
MP.1 Make sense of problems and persevere in solving them.
Making sense of Sudoku is easy. One must fill in the empty cells (small boxes) with numbers, 1 through 9, according to a simple set of rules. There are many starting points, all equally effective.
By following the five steps that appear on the back of each puzzle paper that’s distributed in my program, one can solve the cells, one by one. This clearly defined process establishes a method for eventually completing the puzzle.
Each time a cell is successfully filled a small feeling of satisfaction is experienced. This feeling is repeated many times, encouraging the child to continue until the total puzzle is completed. There is an intrinsic incentive to completing the puzzle, rewarded with each small step towards completion.
MP.2 Reason abstractly and quantitatively
When solving any Sudoku puzzle, the questions of who, when and why should be ignored, as they are all irrelevant. The question of how can be reduced to solving the what and where of each puzzle. Only these two questions, what and where, are relevant and need to be considered. Considering them is how we solve every puzzle. This is the abstract reasoning required by Sudoku.
The what is usually any one of nine numbers, 1 through 9, though it could be nine letters of the alphabet or any nine written symbols. Each of these nine numbers must be placed correctly in each family (of nine numbers). For example, if we decide to work on the number 3 (what), we try to determine the only cell in a particular family (where) the 3 can go. If we find such a place, we can enter it as a solution to a cell. And there is only one correct solution in each cell.
Or we can try a different approach. The where can be any empty cell in any incomplete family. We can choose a particular cell (where) and decide if there is only one number (what) that can fit in that cell. If we find that number (what), we should write it into that cell, thus partially solving the puzzle.
The symbols used, usually numbers 1 through 9, are all equal in value, and since any nine symbols could be used, including letters of the alphabet, there are no quantities involved in Sudoku. The nine numbers are never manipulated with mathematical processes; no addition, subtraction, division or multiplication. Consequently, quantitative reasoning is not addressed by Sudoku. Sudoku is not math; it is logic. This is a very important distinction that is often misunderstood by people unfamiliar with Sudoku.
MP.3 Construct viable arguments and critique the reasoning of others.
Sudoku is excellent at giving children many opportunities for arguing solutions and refuting faulty reasoning of others. As there is only one correct solution to each cell (of many in each puzzle), and several incorrect solutions, and each correct solution is subject to visible proofs, Sudoku is clearly set up for possible debates about solutions. It’s always easy to prove or disprove every solution, right or wrong.
MP.4 Model with mathematics
As Sudoku is not mathematics (it’s logic), there is almost nothing about Sudoku or my program that addresses this objective. However, Sudoku does offer many opportunities to “Simplify a complex problem….”. It’s necessary to identify which numbers are relevant in order to solve each empty cell, while ignoring those numbers that are irrelevant. This simplifies a complex problem. This process is required many times to solve a whole Sudoku puzzle.
However, though unnecessary for solving puzzles, an enterprising teacher could find many examples in every puzzle that could be used for teaching percentages. For example, if one family has only two empty cells, needing a 2 and an 8, and we don’t have enough information to determine which is which, we could say there’s a 50% chance each number could belong in either empty cell.
MP.5 Use appropriate tools strategically
When solving Sudoku puzzles, there are many different tactics and strategies required, particularly for harder puzzles. Basically, we can try to solve a cell using different tactics that might work to learn which number goes there (what goes where). Or we can try to enter a number by determining which cell it should go into (where goes what). There are many different tactics to determine either solution. A large part of Sudoku is learning a variety of tactics to be used to fill a cell with the appropriate number. Of course, each tactic can only be used in appropriate circumstances.
MP.6 Attend to precision
One of the most valuable characteristics of Sudoku is its constant requirement for accuracy. As there is only one correct answer to each cell’s solution, and only one correct solution to the overall puzzle, any mistake will result in unsuccessful results. Every puzzle repeatedly persuades children that it’s necessary to be accurate with every cell solution. This persuasion occurs about 50 times each puzzle. One inaccuracy will always produce failure, and you may not discover an earlier mistake until almost at the end of the puzzle. Sudoku teaches precision. Imprecision always fails.
MP.7 Look for and make use of structure
Sudoku is excellent for increasing the observation of each puzzle’s structure. Anyone doing Sudoku constantly looks for certain patterns and locations of numbers and how they can be used to solve cells. As one learns and practices Sudoku, one becomes better at making these observations. A puzzle can only be solved by determining the positions of the relevant given numbers, which are used to place the correct numbers in the empty cells.
MP.8 Look for and express regularity in repeated reasoning.
Sudoku requires repeated reasoning about 50 times in solving each puzzle. We look for “eliminators”, concluding that a number cannot go there or there or there, so it must go here, in a particular cell. If 1,2,3,4,5,6,7 and 8 cannot go in a cell, only a 9 will work. Or we can solve a cell with the other approach, determining that only a certain cell can correctly contain a particular number. Either of these two strategies are repeated continuously, reenforcing logical processes.
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FIVE STEPS FOR SOLVING EACH CELL (C)
by Jerry Martin
1. Identify the family you will work on
2. Say which numbers are needed in that family
3. Say which number you will solve
4. Identify eliminators
5. Solve the cell with a number
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DISSECTING A SUDOKUADVANCED ANALYSIS (C)
by Jerry Martin
WHAT AND WHERE; ONE CONSTANT AND SEVERAL VARIABLES (C)
When solving a Sudoku puzzle we consider two questions, what and where. A what is usually any number, 1 through 9, (though it could be nine letters of the alphabet or any other nine symbols). A where is any empty cell or any incomplete family.
When solving a cell, there is always a constant and several variables. Sometimes a number, (a what), is the constant and any of several empty cells are the variables (the where). Or conversely, sometimes a particular cell or family (the where) is the constant and several numbers, (the what), are the variables.
For example, we might choose 4 (a what) as our constant. Then we look at all the empty cells (the wheres), the variables in this case, to see if a 4, and no other number, can go into one particular cell. Or conversely, we might choose an empty cell or incomplete family (a where) as our constant. Then we consider all the numbers, 1 through 9, (the whats) as our variables, to see if a particular number can only fit into one cell in that family.
It’s always what goes where or where goes what. Sometimes the what is the constant and sometimes the where is the constant. But always the other makes up the variables.
This describes the two basic strategies for solving all Sudoku puzzles. There are many different tactics within these two strategies that must be used, particularly for harder puzzles.
FAMILIES
Every individual cell is simultaneously a member of three different families, a horizontal family, a vertical family and a big box family. And a number, 1 through 9, can only appear in each family once.
Given any particular cell, sometimes several numbers will satisfy all three families, but only one will eventually prove itself correct. The other numbers will be eliminated from that cell, as we progress.
Sometimes we look for eliminators in these three families. If a number is eliminated from a particular cell by one or more families, then it must be rejected. Then we might try another number for that same cell.
Families are very useful in solving Sudoku puzzles, but we need to learn how to use them, with critical thinking. Nine horizontal, nine vertical and nine big box families total 27 families. But we usually only consider, at most, three families at a time. Sometimes we only need to consider one or two families. The others are irrelevant, for the moment, and should be ignored, lest they distract us from solving a cell.
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by Jerry Martin
WHAT AND WHERE; ONE CONSTANT AND SEVERAL VARIABLES (C)
When solving a Sudoku puzzle we consider two questions, what and where. A what is usually any number, 1 through 9, (though it could be nine letters of the alphabet or any other nine symbols). A where is any empty cell or any incomplete family.
When solving a cell, there is always a constant and several variables. Sometimes a number, (a what), is the constant and any of several empty cells are the variables (the where). Or conversely, sometimes a particular cell or family (the where) is the constant and several numbers, (the what), are the variables.
For example, we might choose 4 (a what) as our constant. Then we look at all the empty cells (the wheres), the variables in this case, to see if a 4, and no other number, can go into one particular cell. Or conversely, we might choose an empty cell or incomplete family (a where) as our constant. Then we consider all the numbers, 1 through 9, (the whats) as our variables, to see if a particular number can only fit into one cell in that family.
It’s always what goes where or where goes what. Sometimes the what is the constant and sometimes the where is the constant. But always the other makes up the variables.
This describes the two basic strategies for solving all Sudoku puzzles. There are many different tactics within these two strategies that must be used, particularly for harder puzzles.
FAMILIES
Every individual cell is simultaneously a member of three different families, a horizontal family, a vertical family and a big box family. And a number, 1 through 9, can only appear in each family once.
Given any particular cell, sometimes several numbers will satisfy all three families, but only one will eventually prove itself correct. The other numbers will be eliminated from that cell, as we progress.
Sometimes we look for eliminators in these three families. If a number is eliminated from a particular cell by one or more families, then it must be rejected. Then we might try another number for that same cell.
Families are very useful in solving Sudoku puzzles, but we need to learn how to use them, with critical thinking. Nine horizontal, nine vertical and nine big box families total 27 families. But we usually only consider, at most, three families at a time. Sometimes we only need to consider one or two families. The others are irrelevant, for the moment, and should be ignored, lest they distract us from solving a cell.
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DEDUCTION AND INDUCTION (C)
by Jerry Martin
One reason Sudoku is such an effective tool for teaching the basics of logic is that it trains people to employ deductive and inductive reasoning. They are the two components of critical thinking and logic. While solving every puzzle we use these two methods more than 50 times. We almost always use both strategies every time we solve an individual cell.
A common strategy is to choose a family, either horizontal or vertical or big box, to work on. Usually it’s wise to choose a family with the fewest empty cells, since those are the easiest families to solve. When working to solve a cell in a family, we ask, “What numbers are needed to complete the family?” This is inductive reasoning. Then, when narrowing these numbers down to the only one number that will fit, having eliminated all other numbers, we employ deduction.This allows us to write the correct number in the correct cell.
As there is only one correct answer each time, it’s critically important to get it right every time. If we make a mistake and write the wrong number in a cell, the puzzle cannot be solved. But we may not learn of our mistake until much later, when it’s impossible to go back and determine where the mistake was made. Sudoku is unforgiving in this way.
@#@#@#@#@#@#@#@#@#@@#@#@#@#@#@#@#@#@#@#@#@
A common strategy is to choose a family, either horizontal or vertical or big box, to work on. Usually it’s wise to choose a family with the fewest empty cells, since those are the easiest families to solve. When working to solve a cell in a family, we ask, “What numbers are needed to complete the family?” This is inductive reasoning. Then, when narrowing these numbers down to the only one number that will fit, having eliminated all other numbers, we employ deduction.This allows us to write the correct number in the correct cell.
As there is only one correct answer each time, it’s critically important to get it right every time. If we make a mistake and write the wrong number in a cell, the puzzle cannot be solved. But we may not learn of our mistake until much later, when it’s impossible to go back and determine where the mistake was made. Sudoku is unforgiving in this way.
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THE IMPORTANCE OF RELEVANCE (C)
by Jerry Martin
One major component of critical thinking when solving any problem is determining what information is relevant and what is irrelevant. Being able to distinguish between the two is necessary for all logical, critical thinking.
Sudoku is excellent at teaching and training this important skill. This skill is repeated and developed about fifty times in each puzzle. Every time we solve a cell (small square) we must find the two or three numbers that are relevant, numbers whose placement allows us to eliminate all possibilities except one. Once we can narrow the choice down to one, we can solve that cell and write in a number solution accurately. If we cannot eliminate all possibilities but one, we should not write in an answer. This only happens by finding the relevant numbers while ignoring the irrelevant information, which are many visible numbers that don’t help to solve that particular cell.
Doing this successfully and repeatedly, an activity essential for success in Sudoku, builds this capacity in anyone. If children can learn the importance of separating relevant information from the multitude of irrelevant information, they are well on their way to dealing with and solving all problems. Without this skill they are doomed to fail.
by Jerry Martin
One major component of critical thinking when solving any problem is determining what information is relevant and what is irrelevant. Being able to distinguish between the two is necessary for all logical, critical thinking.
Sudoku is excellent at teaching and training this important skill. This skill is repeated and developed about fifty times in each puzzle. Every time we solve a cell (small square) we must find the two or three numbers that are relevant, numbers whose placement allows us to eliminate all possibilities except one. Once we can narrow the choice down to one, we can solve that cell and write in a number solution accurately. If we cannot eliminate all possibilities but one, we should not write in an answer. This only happens by finding the relevant numbers while ignoring the irrelevant information, which are many visible numbers that don’t help to solve that particular cell.
Doing this successfully and repeatedly, an activity essential for success in Sudoku, builds this capacity in anyone. If children can learn the importance of separating relevant information from the multitude of irrelevant information, they are well on their way to dealing with and solving all problems. Without this skill they are doomed to fail.
JOURNEY AND DESTINATION (C)
by Jerry Martin
Most full lives have many destinations, and the journeys are how we get there, if we’re successful. Destinations come in many forms. So do journeys.
Going to Rome might be a destination. Getting there, by many different routes and means of travel, meeting new people and seeing new sights, is the journey.
Eating a home made cake might be a destination. Reading the recipe, mixing the ingredients and baking them in the oven until done, is the journey.
Becoming a doctor is a destination. Graduating from college with good grades in the right courses, applying to and being accepted to a med school, studying anatomy and physiology and performing the residency and internship is the journey.
When a carpenter builds a table he must buy the wood, measure it, cut it and nail it together. That’s the journey. The finished product, a table someone will use for years, is the destination.
Riding an elevator to the top of the Empire State Building is a journey. The view is the destination. No sane person would ride the elevator without getting out and enjoying the spectacular panorama.
Most things we do have a destination, which is the reward for the journey. For most things, a destination is necessary. For most things, we wouldn’t take the journey without a destination. And we’d never accomplish the destination without making the appropriate journey correctly.
But Sudoku is different. It’s all journey, and the destination, a completed puzzle, a small piece of paper with 81 numbers written on it arranged in parallel rows and columns, is useless, meaningless, artless and without merit. You couldn’t sell a finished Sudoku to anyone; nobody wants it. And you wouldn’t display it above your mantel or framed on a wall. And it would make a lousy Christmas present.
So why do people do Sudoku? There’s no tangible, material reward at the finish line. The answer lies in the difference between extrinsic and intrinsic motivation. Extrinsic motivation requires a reward, an achievable physical destination. We do a job for money, we study hard for good grades in school, we learn the piano so we can play songs worth hearing, we drive a car to get some place.
But intrinsic motivation means we do something for its own sake, for its enjoyment, without any need for a reward or recognition. Intrinsic motivation produces good feelings, a sense of self esteem, a growth in a skill that doesn’t necessarily produce an immediate result. The love of Sudoku is driven intrinsically. Educators recognize that intrinsic motivation is far more effective than extrinsic motivation for developing long term participation in an activity.
While not immediately apparent, Sudoku trains us to think logically and accurately, so even though there is never a material reward like a cake or diploma or vacation in Rome, Sudoku makes a solid contribution to our capacity to achieve a tremendous variety of destinations we will need in the future in order to live a full life. That feels good. And the mental exercise helps prevent dementia and Alzheimer’s. So Sudoku’s destination is intangible, a smarter mind with less mental deterioration over time.
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by Jerry Martin
Most full lives have many destinations, and the journeys are how we get there, if we’re successful. Destinations come in many forms. So do journeys.
Going to Rome might be a destination. Getting there, by many different routes and means of travel, meeting new people and seeing new sights, is the journey.
Eating a home made cake might be a destination. Reading the recipe, mixing the ingredients and baking them in the oven until done, is the journey.
Becoming a doctor is a destination. Graduating from college with good grades in the right courses, applying to and being accepted to a med school, studying anatomy and physiology and performing the residency and internship is the journey.
When a carpenter builds a table he must buy the wood, measure it, cut it and nail it together. That’s the journey. The finished product, a table someone will use for years, is the destination.
Riding an elevator to the top of the Empire State Building is a journey. The view is the destination. No sane person would ride the elevator without getting out and enjoying the spectacular panorama.
Most things we do have a destination, which is the reward for the journey. For most things, a destination is necessary. For most things, we wouldn’t take the journey without a destination. And we’d never accomplish the destination without making the appropriate journey correctly.
But Sudoku is different. It’s all journey, and the destination, a completed puzzle, a small piece of paper with 81 numbers written on it arranged in parallel rows and columns, is useless, meaningless, artless and without merit. You couldn’t sell a finished Sudoku to anyone; nobody wants it. And you wouldn’t display it above your mantel or framed on a wall. And it would make a lousy Christmas present.
So why do people do Sudoku? There’s no tangible, material reward at the finish line. The answer lies in the difference between extrinsic and intrinsic motivation. Extrinsic motivation requires a reward, an achievable physical destination. We do a job for money, we study hard for good grades in school, we learn the piano so we can play songs worth hearing, we drive a car to get some place.
But intrinsic motivation means we do something for its own sake, for its enjoyment, without any need for a reward or recognition. Intrinsic motivation produces good feelings, a sense of self esteem, a growth in a skill that doesn’t necessarily produce an immediate result. The love of Sudoku is driven intrinsically. Educators recognize that intrinsic motivation is far more effective than extrinsic motivation for developing long term participation in an activity.
While not immediately apparent, Sudoku trains us to think logically and accurately, so even though there is never a material reward like a cake or diploma or vacation in Rome, Sudoku makes a solid contribution to our capacity to achieve a tremendous variety of destinations we will need in the future in order to live a full life. That feels good. And the mental exercise helps prevent dementia and Alzheimer’s. So Sudoku’s destination is intangible, a smarter mind with less mental deterioration over time.
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SUDOKU AND TIME (C)
by Jerry Martin
One of the attractions of Sudoku is it’s independence from time. There are no deadlines and all pace of solution is determined by the player. You can start solving a puzzle and complete it in one sitting. Or you can start it and stop and continue later at your convenience. No one cares how long it takes, or even whether it’s completed. Since Sudoku travels so well, a partially completed puzzle can be taken anywhere (except swimming) and be continued anywhere there’s adequate light to see and a flat place to write. Time is rarely a factor, unless you are in a timed competition, an event which is very unusual in 2015. Doing Sudoku, we can ignore time and experience a relaxed “vacation”, away from the hectic hustle and complications that are stressful and frustrating.
This is in contrast to most of our busy lives in the industrial world today. In our increasingly complex lives, most activities have time limits and deadlines. Whether it’s a mortgage that’s due, a job demanding punctuality, an appointment that’s necessary, a roast in the oven or a movie that starts in 10 minutes, our lives are often governed by clocks and calendars. How often do we become agitated by busy traffic which prevents us from arriving on time? How often do we forget, failing to take medications on time? How often do we rush to get the bills paid on time, get the kids to school on time or rush to answer a ringing telephone? Our lives are full of timedetermined target dates and cutoff points.
Sudoku, in it’s calm pace and lack of consequences, provides a respite from anxiety and responsible obligations. Sudoku is a rare activity that allows us to escape any demands of time or goals. That makes it fun.
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by Jerry Martin
One of the attractions of Sudoku is it’s independence from time. There are no deadlines and all pace of solution is determined by the player. You can start solving a puzzle and complete it in one sitting. Or you can start it and stop and continue later at your convenience. No one cares how long it takes, or even whether it’s completed. Since Sudoku travels so well, a partially completed puzzle can be taken anywhere (except swimming) and be continued anywhere there’s adequate light to see and a flat place to write. Time is rarely a factor, unless you are in a timed competition, an event which is very unusual in 2015. Doing Sudoku, we can ignore time and experience a relaxed “vacation”, away from the hectic hustle and complications that are stressful and frustrating.
This is in contrast to most of our busy lives in the industrial world today. In our increasingly complex lives, most activities have time limits and deadlines. Whether it’s a mortgage that’s due, a job demanding punctuality, an appointment that’s necessary, a roast in the oven or a movie that starts in 10 minutes, our lives are often governed by clocks and calendars. How often do we become agitated by busy traffic which prevents us from arriving on time? How often do we forget, failing to take medications on time? How often do we rush to get the bills paid on time, get the kids to school on time or rush to answer a ringing telephone? Our lives are full of timedetermined target dates and cutoff points.
Sudoku, in it’s calm pace and lack of consequences, provides a respite from anxiety and responsible obligations. Sudoku is a rare activity that allows us to escape any demands of time or goals. That makes it fun.
^&^^&^&^&^&^&^&^&^&^^&^&^^&^^^&^&^&^&^&^&^&^^&^&^&^&^&^&^&^^&
THE TWO FISHES (C)
(Sufficiency and Efficiency)
by Jerry Martin
Sufficiency of Information
Sudoku is excellent at teaching the need to wait until you have a sufficient amount of information before writing a solution into a cell. There is only one correct answer for each of about 50 cells, so accuracy is very important. If you make one mistake you will fail to completely solve the puzzle. Therefore, making a definitive action without being 100% sure, without eliminating all other possibilities, which only happens with sufficient information, will usually produce failure. Never guess; Sudoku teaches to never act on insufficient information.
Efficiency of Effort
Two strategies:
There are two strategies for the most efficient means of solving each cell. The first strategy involves counting the number of empty cells in each incomplete family. The easiest solutions are in families with the fewest empty cells. A family with 1,2,3,4,5,6,7 and 8 already filled, and one empty cell unsolved, is easy to solve. The one empty cell must be a 9. A family with two empty cells would be easier to solve, in most cases, than a family with three or more empty cells. Consequently, experienced solvers work first on the families that are the most complete, with the fewest empty cells. Do the easy families first, as they will provide eliminators for the harder families, making the harder families easier to solve.
The second strategy involves counting the numbers given in an undone puzzle. When completed, all puzzles will have nine of each number, 1 through 9. But at the beginning we will be given different amounts of each number. For example, a new, undone puzzle might give us three 1s and four 2s and five 3s, but no 4s. In this example, the 3s would almost always be easier to solve than the 4s. There would be fewer 3s to solve and the given 3s would be eliminators for the unsolved 3s. Conversely, all nine 4s would have to be solved, and since there are none to act as eliminators, the 4s would be difficult to solve at the beginning.
To implement this strategy, some solvers begin by counting how many 1s are given, how many 2s are given, and so on through 9s. Then, solving the numbers with the most givens should be done first, for maximum efficiency.
Both Strategies Are Similar
Both strategies involve solving the easiest problems first, which will improve efficiency. The families with the least number of empty cells, and the numbers with the most number given, will almost always be easier than the families and numbers with the most unsolved cells and numbers.
^&^&^&^&^&^&^&^&^&^&^^&^&^&^&^&^&^&^&^&^&^&^&^^&^&^&^&^&^&^^
(Sufficiency and Efficiency)
by Jerry Martin
Sufficiency of Information
Sudoku is excellent at teaching the need to wait until you have a sufficient amount of information before writing a solution into a cell. There is only one correct answer for each of about 50 cells, so accuracy is very important. If you make one mistake you will fail to completely solve the puzzle. Therefore, making a definitive action without being 100% sure, without eliminating all other possibilities, which only happens with sufficient information, will usually produce failure. Never guess; Sudoku teaches to never act on insufficient information.
Efficiency of Effort
Two strategies:
There are two strategies for the most efficient means of solving each cell. The first strategy involves counting the number of empty cells in each incomplete family. The easiest solutions are in families with the fewest empty cells. A family with 1,2,3,4,5,6,7 and 8 already filled, and one empty cell unsolved, is easy to solve. The one empty cell must be a 9. A family with two empty cells would be easier to solve, in most cases, than a family with three or more empty cells. Consequently, experienced solvers work first on the families that are the most complete, with the fewest empty cells. Do the easy families first, as they will provide eliminators for the harder families, making the harder families easier to solve.
The second strategy involves counting the numbers given in an undone puzzle. When completed, all puzzles will have nine of each number, 1 through 9. But at the beginning we will be given different amounts of each number. For example, a new, undone puzzle might give us three 1s and four 2s and five 3s, but no 4s. In this example, the 3s would almost always be easier to solve than the 4s. There would be fewer 3s to solve and the given 3s would be eliminators for the unsolved 3s. Conversely, all nine 4s would have to be solved, and since there are none to act as eliminators, the 4s would be difficult to solve at the beginning.
To implement this strategy, some solvers begin by counting how many 1s are given, how many 2s are given, and so on through 9s. Then, solving the numbers with the most givens should be done first, for maximum efficiency.
Both Strategies Are Similar
Both strategies involve solving the easiest problems first, which will improve efficiency. The families with the least number of empty cells, and the numbers with the most number given, will almost always be easier than the families and numbers with the most unsolved cells and numbers.
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PATTERN RECOGNITION (C)
by Jerry Martin
Sudoku is very powerful in teaching and training pattern recognition. The three rules that control all solutions, that require the numbers one through nine be placed in the horizontal, vertical and big box families, are patterns. To solve each individual cell, and there are about fifty cells in each puzzle, one must recognize patterns. For example, the families with the fewest empty cells are usually the easiest to solve, so should be addressed first. For example, all numbers, one through nine, must be used only once in each family. Violating this pattern will produce failure. For example, any number, one through nine, which fills a cell, must be the only one of that number in the three families each cell occupies. We solve cells by finding eliminators, and these eliminators follow certain patterns. Recognizing these patterns helps find these eliminators, which are necessary for success at Sudoku.
There are more Sudoku examples, too complex to describe here.
Patterns are everywhere in our modern civilization and nature. Every busy urban intersection is controlled by traffic lights, red meaning stop (that’s one pattern) and green meaning go (another pattern). Common vehicles usually drive on roads; if they drive off the road, at speed, they will violate this pattern and cause great damage.
In every supermarket, groceries are arranged in patterns, with all cereals on three shelves in one aisle, all dairy products on another aisle and all canned vegetables on four shelves in another aisle. Without groceries organized by these patterns, shopping would be confusing, frustrating and take a lot longer, and stores would sell less and lose money.
If we enter an elevator and want to go up, the recommended pattern is to push a button with a higher number than the floor we entered. If we push a lower number button, we’ll go down, an unwanted result. Another useful pattern.
If a person is consistently late, that's a pattern. If a person speaks with a large vocabulary, that's a pattern. If a person is hypersensitive and gets hurt and angry easily and often, that's a pattern. If a person has memorized Shakespeare and quotes it often, that's a pattern. Recognizing these patterns helps us retain relationships and end others. Recognizing these patterns in ourselves builds interpersonal intelligence and delivers all the advantages that honestly knowing ourselves imparts.
In nature, day follows night follows day follows night. This is a predictable pattern that’s useful to know. Winter is colder than summer, in most places, and spring brings new life and autumn brings a decline from life. These patterns are inviolable, important and relevant to agriculture. Understanding them and using them to farmers‘ advantage makes that farmer successful. Ignorance of them produces fallowness. Water flows down streams to rivers to the ocean, a pattern controlled by gravity, a pattern ignored by humans at their ignorant peril. Rain requires clouds, snow comes with cold moisture, and sunburns come with neither, if exposed skin is outside long enough in the sun. These are all patterns long recognized by humanity, useful for our survival.
Patterns are obvious and useful to seamstresses, engineers, mechanics, astronomers, meteorologists, physicists, chemists, accountants and architects. Without patterns, no two items under construction would match, no parts would be interchangeable. It would be impossible to teach any of these professions without patterns. Science couldn’t exist.
Children need to learn to recognize patterns, both manmade and natural. Doing so will allow them to discover new truths and develop new inventions. Doing Sudoku is wonderful training for recognizing patterns, a skill that will serve them countless times throughout their resourceful lives. Patterns are knowledge.
by Jerry Martin
Sudoku is very powerful in teaching and training pattern recognition. The three rules that control all solutions, that require the numbers one through nine be placed in the horizontal, vertical and big box families, are patterns. To solve each individual cell, and there are about fifty cells in each puzzle, one must recognize patterns. For example, the families with the fewest empty cells are usually the easiest to solve, so should be addressed first. For example, all numbers, one through nine, must be used only once in each family. Violating this pattern will produce failure. For example, any number, one through nine, which fills a cell, must be the only one of that number in the three families each cell occupies. We solve cells by finding eliminators, and these eliminators follow certain patterns. Recognizing these patterns helps find these eliminators, which are necessary for success at Sudoku.
There are more Sudoku examples, too complex to describe here.
Patterns are everywhere in our modern civilization and nature. Every busy urban intersection is controlled by traffic lights, red meaning stop (that’s one pattern) and green meaning go (another pattern). Common vehicles usually drive on roads; if they drive off the road, at speed, they will violate this pattern and cause great damage.
In every supermarket, groceries are arranged in patterns, with all cereals on three shelves in one aisle, all dairy products on another aisle and all canned vegetables on four shelves in another aisle. Without groceries organized by these patterns, shopping would be confusing, frustrating and take a lot longer, and stores would sell less and lose money.
If we enter an elevator and want to go up, the recommended pattern is to push a button with a higher number than the floor we entered. If we push a lower number button, we’ll go down, an unwanted result. Another useful pattern.
If a person is consistently late, that's a pattern. If a person speaks with a large vocabulary, that's a pattern. If a person is hypersensitive and gets hurt and angry easily and often, that's a pattern. If a person has memorized Shakespeare and quotes it often, that's a pattern. Recognizing these patterns helps us retain relationships and end others. Recognizing these patterns in ourselves builds interpersonal intelligence and delivers all the advantages that honestly knowing ourselves imparts.
In nature, day follows night follows day follows night. This is a predictable pattern that’s useful to know. Winter is colder than summer, in most places, and spring brings new life and autumn brings a decline from life. These patterns are inviolable, important and relevant to agriculture. Understanding them and using them to farmers‘ advantage makes that farmer successful. Ignorance of them produces fallowness. Water flows down streams to rivers to the ocean, a pattern controlled by gravity, a pattern ignored by humans at their ignorant peril. Rain requires clouds, snow comes with cold moisture, and sunburns come with neither, if exposed skin is outside long enough in the sun. These are all patterns long recognized by humanity, useful for our survival.
Patterns are obvious and useful to seamstresses, engineers, mechanics, astronomers, meteorologists, physicists, chemists, accountants and architects. Without patterns, no two items under construction would match, no parts would be interchangeable. It would be impossible to teach any of these professions without patterns. Science couldn’t exist.
Children need to learn to recognize patterns, both manmade and natural. Doing so will allow them to discover new truths and develop new inventions. Doing Sudoku is wonderful training for recognizing patterns, a skill that will serve them countless times throughout their resourceful lives. Patterns are knowledge.
THREE IMPORTANT LESSONS (C)
by Jerry Martin
1. RelevanceTo solve any problem it is always necessary to discern the difference between relevant and irrelevant information. When solving any cell in Sudoku it is imperative that we find the two or three relevant numbers and ignore the many irrelevant numbers that are useless, doing nothing except distracting us from our goal. Since we repeat this important process many times, this is very good training for anyone learning the basics of solving all problems.
2. Quality of InformationSudoku is unforgiving. It requires total accuracy in the information used to solve every cell. Since wrong answers in any cell will always yield failure, the information acted upon must be correct. There is no leeway, no fudging, no acceptance of “maybe”. Each answer must be absolutely correct for a successful completion of a puzzle. Never guess.
3. Quantity of InformationOften, when solving any cell, we can correctly conclude that the answer could be one of two (or more) choices. Beginners, in error, sometimes operate on a feeling and will choose one or the other choice. This is a mistake half of the time and will produce failure due to deciding without enough information. It’s wise to wait until all but one of those possible choices can be eliminated before committing to an answer for that cell. Never fill in a cell without enough information that would guarantee a successful conclusion.
by Jerry Martin
1. RelevanceTo solve any problem it is always necessary to discern the difference between relevant and irrelevant information. When solving any cell in Sudoku it is imperative that we find the two or three relevant numbers and ignore the many irrelevant numbers that are useless, doing nothing except distracting us from our goal. Since we repeat this important process many times, this is very good training for anyone learning the basics of solving all problems.
2. Quality of InformationSudoku is unforgiving. It requires total accuracy in the information used to solve every cell. Since wrong answers in any cell will always yield failure, the information acted upon must be correct. There is no leeway, no fudging, no acceptance of “maybe”. Each answer must be absolutely correct for a successful completion of a puzzle. Never guess.
3. Quantity of InformationOften, when solving any cell, we can correctly conclude that the answer could be one of two (or more) choices. Beginners, in error, sometimes operate on a feeling and will choose one or the other choice. This is a mistake half of the time and will produce failure due to deciding without enough information. It’s wise to wait until all but one of those possible choices can be eliminated before committing to an answer for that cell. Never fill in a cell without enough information that would guarantee a successful conclusion.