Thursday, August 5, 2010
Saturday, April 4, 2009
Why Sudoku makes your brain ache
No wonder Sudoku puzzles give your brain a good work-out. Scientists say solving them depends on neural pathways that even the most powerful computers can't replicate.
They say that by studying how people solve the puzzles, we might be able to develop more intelligent and brain-like computers.
In a paper published on the arXiv physics website, Professor John Hopfield of Princeton University explores the unique brain processes we use when playing Sudoku.
This mathematical puzzle involves filling in a grid of 81 squares with varying combinations of the numbers one to nine, something that sounds simple but can be diabolically hard.
To crack Sudoku our brains use a unique set of neural pathways known as associative memory, Hopfield says, which enables us to discover a pattern from a partial clue.
Although computers can store large amounts of information and process it at great speed, they aren't yet capable of sophisticated associative memory.
Hopfield provides an algorithm of associative memory in his paper, which he says could be implemented in silicon chips.
Patterns
We all recognise the basic pattern of counting from one to nine, yet the task of completing a Sudoku puzzle is confounded because of the large number of possible permutations of this pattern.
But every time we put the right number in the right place it provides us with a clue, which reduces the number of permutations.
In this way Sudoku is based on a combination of logic and intelligent guesswork based on our abilities of associative memory, Hopfield says.
"In neural terms, the signals developed ... can produce a strong and reasonably accurate feeling of correctness of the item retrieved," Hopfield says.
"This fact may account for our strong psychological feeling of 'right' or 'wrong' when we retrieve a memory from a minimal clue."
Brains versus computers
Associate Professor Andrew Paplinski is an Australian computer scientist who specialises in neural networks at Monash University in Melbourne.
He says the process described in Hopfield's paper helps us to remember a name from a fragment or recognise a partially obscured face.
He says applying Hopfield's model could lead to more accurate facial recognition computer technology.
Being able to mimic associative memory would give computers "extreme robustness of pattern recognition", Paplinski says.
For example, for a computer to recognise a partially visible face it would first have to recognise that the face is obscured, then that it is a face, and then it would have to find a match.
"To answer all these questions takes an enormous amount of computation," he says.
He says we can do this in a fraction of a second in a slow computer like our brain. So there were would be significant implications if we can figure out how this is done and design computers that can replicate it.
They say that by studying how people solve the puzzles, we might be able to develop more intelligent and brain-like computers.
In a paper published on the arXiv physics website, Professor John Hopfield of Princeton University explores the unique brain processes we use when playing Sudoku.
This mathematical puzzle involves filling in a grid of 81 squares with varying combinations of the numbers one to nine, something that sounds simple but can be diabolically hard.
To crack Sudoku our brains use a unique set of neural pathways known as associative memory, Hopfield says, which enables us to discover a pattern from a partial clue.
Although computers can store large amounts of information and process it at great speed, they aren't yet capable of sophisticated associative memory.
Hopfield provides an algorithm of associative memory in his paper, which he says could be implemented in silicon chips.
Patterns
We all recognise the basic pattern of counting from one to nine, yet the task of completing a Sudoku puzzle is confounded because of the large number of possible permutations of this pattern.
But every time we put the right number in the right place it provides us with a clue, which reduces the number of permutations.
In this way Sudoku is based on a combination of logic and intelligent guesswork based on our abilities of associative memory, Hopfield says.
"In neural terms, the signals developed ... can produce a strong and reasonably accurate feeling of correctness of the item retrieved," Hopfield says.
"This fact may account for our strong psychological feeling of 'right' or 'wrong' when we retrieve a memory from a minimal clue."
Brains versus computers
Associate Professor Andrew Paplinski is an Australian computer scientist who specialises in neural networks at Monash University in Melbourne.
He says the process described in Hopfield's paper helps us to remember a name from a fragment or recognise a partially obscured face.
He says applying Hopfield's model could lead to more accurate facial recognition computer technology.
Being able to mimic associative memory would give computers "extreme robustness of pattern recognition", Paplinski says.
For example, for a computer to recognise a partially visible face it would first have to recognise that the face is obscured, then that it is a face, and then it would have to find a match.
"To answer all these questions takes an enormous amount of computation," he says.
He says we can do this in a fraction of a second in a slow computer like our brain. So there were would be significant implications if we can figure out how this is done and design computers that can replicate it.
Information We Must Know on How to Solve Sudoku
Learning how to solve Sudoku possibly is not an easy thing to do if we don’t appreciate the objectives of the game. In point of fact Sudoku is a puzzle that consists of a 9X9 grid and the object of the puzzle is to fill up the grid with numbers 1-9. What makes how to solve Sudoku is difficult is that each number just allowed to emerge in each row, column and grid one time. Here are some strategies for how to solve Sudoku puzzle successfully.
Where to Start
In Sudoku game, there is no literal area to start the game. Nevertheless, the best area to start is the center three squares. Within these three rows, find a number that come out twice and start with that number. Look through the columns and then try to fit the third number into that section of the grid. This will give you a starting spot on how to solve Sudoku puzzle.
Use Counting Principals
If you already understand that each number can only be used one time in each row, column, and grid boxes, the next method in learning how to solve Sudoku is to use the process of elimination to try to fit all of the numbers into the squares. It is all right to jump around the board to try to fit all of the numbers into the squares. This is possibly one of the most common strategies for how to solve Sudoku puzzles.
Find the Definites
The definites are those numbers that are visible. Look for the definites in each puzzle and finish them first. This will improve your confidence and help you finish a few easy squares and eliminate the number possibilities. This is perhaps one of the easiest strategies on how to solve Sudoku puzzle and a good starting spot to start to solve the puzzle.
Check the Answers If You Get Stuck
When every strategies fail and you still cannot find the right number, then take a glance at the answers. At times, when you a few right numbers in place, the others will fall in more easily. Nevertheless, when you do look at the answers, only get one or two numbers to give you start then stop looking at the answers. When learning how to solve Sudoku puzzles, it is fine to glance at the answers.
Sudoku puzzles are the most famous game that all people all over the world play the puzzle. Sudoku puzzle is very delightful yet challenging and educational game. There is nothing wrong to attempt solving the Sudoku puzzles with several strategies that even generally available on the internet. With a little practice, you will learn how to solve Sudoku puzzles very soon.
So, are you now willing to know more about how to solve Sudoku ? Visit the links here, and these guidelines will make you smarter about it!
Where to Start
In Sudoku game, there is no literal area to start the game. Nevertheless, the best area to start is the center three squares. Within these three rows, find a number that come out twice and start with that number. Look through the columns and then try to fit the third number into that section of the grid. This will give you a starting spot on how to solve Sudoku puzzle.
Use Counting Principals
If you already understand that each number can only be used one time in each row, column, and grid boxes, the next method in learning how to solve Sudoku is to use the process of elimination to try to fit all of the numbers into the squares. It is all right to jump around the board to try to fit all of the numbers into the squares. This is possibly one of the most common strategies for how to solve Sudoku puzzles.
Find the Definites
The definites are those numbers that are visible. Look for the definites in each puzzle and finish them first. This will improve your confidence and help you finish a few easy squares and eliminate the number possibilities. This is perhaps one of the easiest strategies on how to solve Sudoku puzzle and a good starting spot to start to solve the puzzle.
Check the Answers If You Get Stuck
When every strategies fail and you still cannot find the right number, then take a glance at the answers. At times, when you a few right numbers in place, the others will fall in more easily. Nevertheless, when you do look at the answers, only get one or two numbers to give you start then stop looking at the answers. When learning how to solve Sudoku puzzles, it is fine to glance at the answers.
Sudoku puzzles are the most famous game that all people all over the world play the puzzle. Sudoku puzzle is very delightful yet challenging and educational game. There is nothing wrong to attempt solving the Sudoku puzzles with several strategies that even generally available on the internet. With a little practice, you will learn how to solve Sudoku puzzles very soon.
So, are you now willing to know more about how to solve Sudoku ? Visit the links here, and these guidelines will make you smarter about it!
Recent popularity
In 1997, retired Hong Kong judge Wayne Gould, 59, a New Zealander, saw a partly completed puzzle in a Japanese bookshop. Over six years he developed a computer program to produce puzzles quickly. Knowing that British newspapers have a long history of publishing crosswords and other puzzles, he promoted Sudoku to The Times in Britain, which launched it on 12 November 2004 (calling it Su Doku). The first letter to The Times regarding Su Doku was published the following day on 13 November from Ian Payn of Brentford, complaining that the puzzle had caused him to miss his stop on the tube.
The rapid rise of Sudoku in Britain from relative obscurity to a front-page feature in national newspapers attracted commentary in the media and parody (such as when The Guardian's G2 section advertised itself as the first newspaper supplement with a Sudoku grid on every page[16]). Recognizing the different psychological appeals of easy and difficult puzzles, The Times introduced both side by side on 20 June 2005. From July 2005, Channel 4 included a daily Sudoku game in their Teletext service. On 2 August, the BBC's programme guide Radio Times featured a weekly Super Sudoku which features a 16x16 grid.
Even the Greeks owe the publication of their first Sudoku magazine to British influence. It was at Heathrow airport in the middle of 2005 that a Greek computer magazine publisher first laid eyes on a British Sudoku magazine and - realizing the opportunity - proceeded to purchase the necessary software and quickly launch the first local Sudoku magazine, which became an instant success.
In the United States, the first newspaper to publish a Sudoku puzzle by Wayne Gould was The Conway Daily Sun (New Hampshire), in 2004.[17]
The world's first live TV Sudoku show, 1 July 2005, Sky One.The world's first live TV Sudoku show, Sudoku Live, was a puzzle contest first broadcast on 1 July 2005 on Sky One. It was presented by Carol Vorderman. Nine teams of nine players (with one celebrity in each team) representing geographical regions competed to solve a puzzle. Each player had a hand-held device for entering numbers corresponding to answers for four cells. Phil Kollin of Winchelsea, England was the series grand prize winner taking home over £23,000 over a series of games. The audience at home was in a separate interactive competition, which was won by Hannah Withey of Cheshire.
Later in 2005, the BBC launched SUDO-Q, a game show that combines Sudoku with general knowledge. However, it uses only 4x4 and 6x6 puzzles..
In 2006, a Sudoku website published songwriter Peter Levy's Sudoku tribute song,[18] but quickly had to take down the mp3 due to heavy traffic. British and Australian radio picked up the song, which is to feature in a British-made Sudoku documentary. The Japanese Embassy also nominated the song for an award, with Levy doing talks with Sony in Japan to release the song as a single.[19]
Sudoku software is very popular on PCs, websites, and mobile phones. It comes with many distributions of Linux. Software has also been released on video game consoles, such as the Nintendo DS, PlayStation Portable, the Game Boy Advance, Xbox Live Arcade, several iPod models, and the iPhone. In fact, just two weeks after Apple, Inc. debuted the online App Store within its iTunes store on July 11, 2008, there were already nearly 30 different Sudoku games, created by various software developers, specifically for the iPhone and iPod Touch. One of the most popular video games featuring Sudoku is Brain Age: Train Your Brain in Minutes a Day!. Critically and commercially well received, it generated particular praise for its Sudoku implementation[1][2][3] and sold more than 8 million copies worldwide.[4] Due to its popularity, Nintendo made a second Brain Age game titled Brain Age2, which has over 100 new sudoku puzzles and other activities.
In June 2008 an Australian drugs-related jury trial costing over AU$1 000 000 was aborted when it was discovered that five of the twelve jurors had been playing Sudoku instead of listening to evidence.[20]
The rapid rise of Sudoku in Britain from relative obscurity to a front-page feature in national newspapers attracted commentary in the media and parody (such as when The Guardian's G2 section advertised itself as the first newspaper supplement with a Sudoku grid on every page[16]). Recognizing the different psychological appeals of easy and difficult puzzles, The Times introduced both side by side on 20 June 2005. From July 2005, Channel 4 included a daily Sudoku game in their Teletext service. On 2 August, the BBC's programme guide Radio Times featured a weekly Super Sudoku which features a 16x16 grid.
Even the Greeks owe the publication of their first Sudoku magazine to British influence. It was at Heathrow airport in the middle of 2005 that a Greek computer magazine publisher first laid eyes on a British Sudoku magazine and - realizing the opportunity - proceeded to purchase the necessary software and quickly launch the first local Sudoku magazine, which became an instant success.
In the United States, the first newspaper to publish a Sudoku puzzle by Wayne Gould was The Conway Daily Sun (New Hampshire), in 2004.[17]
The world's first live TV Sudoku show, 1 July 2005, Sky One.The world's first live TV Sudoku show, Sudoku Live, was a puzzle contest first broadcast on 1 July 2005 on Sky One. It was presented by Carol Vorderman. Nine teams of nine players (with one celebrity in each team) representing geographical regions competed to solve a puzzle. Each player had a hand-held device for entering numbers corresponding to answers for four cells. Phil Kollin of Winchelsea, England was the series grand prize winner taking home over £23,000 over a series of games. The audience at home was in a separate interactive competition, which was won by Hannah Withey of Cheshire.
Later in 2005, the BBC launched SUDO-Q, a game show that combines Sudoku with general knowledge. However, it uses only 4x4 and 6x6 puzzles..
In 2006, a Sudoku website published songwriter Peter Levy's Sudoku tribute song,[18] but quickly had to take down the mp3 due to heavy traffic. British and Australian radio picked up the song, which is to feature in a British-made Sudoku documentary. The Japanese Embassy also nominated the song for an award, with Levy doing talks with Sony in Japan to release the song as a single.[19]
Sudoku software is very popular on PCs, websites, and mobile phones. It comes with many distributions of Linux. Software has also been released on video game consoles, such as the Nintendo DS, PlayStation Portable, the Game Boy Advance, Xbox Live Arcade, several iPod models, and the iPhone. In fact, just two weeks after Apple, Inc. debuted the online App Store within its iTunes store on July 11, 2008, there were already nearly 30 different Sudoku games, created by various software developers, specifically for the iPhone and iPod Touch. One of the most popular video games featuring Sudoku is Brain Age: Train Your Brain in Minutes a Day!. Critically and commercially well received, it generated particular praise for its Sudoku implementation[1][2][3] and sold more than 8 million copies worldwide.[4] Due to its popularity, Nintendo made a second Brain Age game titled Brain Age2, which has over 100 new sudoku puzzles and other activities.
In June 2008 an Australian drugs-related jury trial costing over AU$1 000 000 was aborted when it was discovered that five of the twelve jurors had been playing Sudoku instead of listening to evidence.[20]
Mathematics of Sudoku
A completed Sudoku grid is a special type of Latin square with the additional property of no repeated values in any partition of the 9×9 block into contiguous 3×3 blocks. The relationship between the two theories is now completely known, after Denis Berthier proved in his recent book, "The Hidden Logic of Sudoku", that a first order formula that does not mention blocks (also called boxes or regions) is valid for Sudoku if and only if it is valid for Latin Squares (this property is trivially true for the axioms and it can be extended to any formula).
The first known calculation of the number of classic 9×9 Sudoku solution grids was posted on the USENET newsgroup rec.puzzles in September 2003[11] and is 6,670,903,752,021,072,936,960 (sequence A107739 in OEIS). This is roughly 1.2×10−6 times the number of 9×9 Latin squares. A detailed calculation of this figure was provided by Bertram Felgenhauer and Frazer Jarvis in 2005.[12] Various other grid sizes have also been enumerated—see the main article for details. The number of essentially different solutions, when symmetries such as rotation, reflection and relabelling are taken into account, was shown by Ed Russell and Frazer Jarvis to be just 5,472,730,538[13] (sequence A109741 in OEIS).
The maximum number of givens provided while still not rendering a unique solution is four short of a full grid; if two instances of two numbers each are missing and the cells they are to occupy form the corners of an orthogonal rectangle, and exactly two of these cells are within one region, there are two ways the numbers can be assigned. Since this applies to Latin squares in general, most variants of Sudoku have the same maximum. The inverse problem—the fewest givens that render a solution unique—is unsolved, although the lowest number yet found for the standard variation without a symmetry constraint is 17, a number of which have been found by Japanese puzzle enthusiasts,[14][15] and 18 with the givens in rotationally symmetric cells. Over 47,000 examples of Sudokus with 17 givens resulting in a unique solution are known.
The first known calculation of the number of classic 9×9 Sudoku solution grids was posted on the USENET newsgroup rec.puzzles in September 2003[11] and is 6,670,903,752,021,072,936,960 (sequence A107739 in OEIS). This is roughly 1.2×10−6 times the number of 9×9 Latin squares. A detailed calculation of this figure was provided by Bertram Felgenhauer and Frazer Jarvis in 2005.[12] Various other grid sizes have also been enumerated—see the main article for details. The number of essentially different solutions, when symmetries such as rotation, reflection and relabelling are taken into account, was shown by Ed Russell and Frazer Jarvis to be just 5,472,730,538[13] (sequence A109741 in OEIS).
The maximum number of givens provided while still not rendering a unique solution is four short of a full grid; if two instances of two numbers each are missing and the cells they are to occupy form the corners of an orthogonal rectangle, and exactly two of these cells are within one region, there are two ways the numbers can be assigned. Since this applies to Latin squares in general, most variants of Sudoku have the same maximum. The inverse problem—the fewest givens that render a solution unique—is unsolved, although the lowest number yet found for the standard variation without a symmetry constraint is 17, a number of which have been found by Japanese puzzle enthusiasts,[14][15] and 18 with the givens in rotationally symmetric cells. Over 47,000 examples of Sudokus with 17 givens resulting in a unique solution are known.
Although the 9×9 grid with 3×3 regions is by far the most common, variations abound. Sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with pentomino regions have been published under the name Logi-5; the World Puzzle Championship has featured a 6×6 grid with 2×3 regions and a 7×7 grid with six heptomino regions and a disjoint region. Larger grids are also possible. The Times offers a 12×12-grid Dodeka sudoku with 12 regions of 4×3 squares. Dell regularly publishes 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal). Nikoli offers 25×25 Sudoku the Giant behemoths.
Another common variant is to add limits on the placement of numbers beyond the usual row, column, and box requirements. Often the limit takes the form of an extra "dimension"; the most common is to require the numbers in the main diagonals of the grid also to be unique. The aforementioned Number Place Challenger puzzles are all of this variant, as are the Sudoku X puzzles in the Daily Mail, which use 6×6 grids.
A variant named "Mini Sudoku" appears in the American newspaper USA Today, which is played on a 6x6 grid with 3x2 regions. The object is the same as standard Sudoku, but the puzzle only uses the numbers 1 through 6.
Another variant is the combination of Sudoku with Kakuro on a 9 x 9 grid, called Cross Sums Sudoku, in which clues are given in terms of cross sums. The clues can also be given by cryptic alphametics in which each letter represents a single digit from 0 to 9. An excellent example is NUMBER+NUMBER=KAKURO which has a unique solution 186925+186925=373850. Another example is SUDOKU=IS*FUNNY whose solution is 426972=34*12558.
Killer Sudoku combines elements of Sudoku with Kakuro - usually no initial numbers are given, but the 9*9 grid is divided into regions, each with a number that the sum of all numbers in the region must add up to, with no repeated numerals. These must be filled in while obeying the standard rules of Sudoku.
Hypersudoku puzzle. As seen in The Age.
Solution to Hypersudoku puzzle.
Hypersudoku is one of the most popular variants. It is published by news papers and magazines around the world and is also known as "NRC Sudoku", "Windoku", "Hyper-Sudoku" and "4 Square Sudoku". The layout is identical to a normal Sudoku, but with additional interior areas defined in which the numbers 1 to 9 must appear. The solving algorithm is slightly different from the normal Sudoku puzzles because of the leverage on the overlapping squares. This overlap gives the player more information to logically reduce the possibilities in the remaining squares. The approach to playing is still similar to Sudoku but with possibly more emphasis on scanning the squares and overlap rather than columns and rows.
Puzzles constructed from multiple Sudoku grids are common. Five 9×9 grids which overlap at the corner regions in the shape of a quincunx is known in Japan as Gattai 5 (five merged) Sudoku. In The Times, The Age and The Sydney Morning Herald this form of puzzle is known as Samurai SuDoku. The Baltimore Sun and the Toronto Star publish a puzzle of this variant (titled High Five) in their Sunday edition. Often, no givens are to be found in overlapping regions. Sequential grids, as opposed to overlapping, are also published, with values in specific locations in grids needing to be transferred to others.
Alphabetical variations have emerged; there is no functional difference in the puzzle unless the letters spell something. Some variants, such as in the TV Guide, include a word reading along a main diagonal, row, or column once solved; determining the word in advance can be viewed as a solving aid.
A tabletop version of Sudoku can be played with a standard 81-card Set deck (see Set game). A three-dimensional Sudoku puzzle was invented by Dion Church and published in the Daily Telegraph in May 2005. There is a Sudoku version of the Rubik's Cube named Sudoku Cube.
The 2005 U.S. Puzzle Championship included a variant called Digital Number Place: rather than givens, most cells contain a partial given—a segment of a number, with the numbers drawn as if part of a seven-segment display. This version has also appeared in GAMES magazine.
One more variant of Sudoku is Greater Than Sudoku (GT Sudoku). In this a 3x3 grid of the Sudoku is given with 12 symbols of Greater Than (>) or Less Than (<) on the common line of the two adjacent numbers. Depending on difficulty this type of Sudoku may or may not be given with numbers.
Another common variant is to add limits on the placement of numbers beyond the usual row, column, and box requirements. Often the limit takes the form of an extra "dimension"; the most common is to require the numbers in the main diagonals of the grid also to be unique. The aforementioned Number Place Challenger puzzles are all of this variant, as are the Sudoku X puzzles in the Daily Mail, which use 6×6 grids.
A variant named "Mini Sudoku" appears in the American newspaper USA Today, which is played on a 6x6 grid with 3x2 regions. The object is the same as standard Sudoku, but the puzzle only uses the numbers 1 through 6.
Another variant is the combination of Sudoku with Kakuro on a 9 x 9 grid, called Cross Sums Sudoku, in which clues are given in terms of cross sums. The clues can also be given by cryptic alphametics in which each letter represents a single digit from 0 to 9. An excellent example is NUMBER+NUMBER=KAKURO which has a unique solution 186925+186925=373850. Another example is SUDOKU=IS*FUNNY whose solution is 426972=34*12558.
Killer Sudoku combines elements of Sudoku with Kakuro - usually no initial numbers are given, but the 9*9 grid is divided into regions, each with a number that the sum of all numbers in the region must add up to, with no repeated numerals. These must be filled in while obeying the standard rules of Sudoku.
Hypersudoku puzzle. As seen in The Age.
Solution to Hypersudoku puzzle.
Hypersudoku is one of the most popular variants. It is published by news papers and magazines around the world and is also known as "NRC Sudoku", "Windoku", "Hyper-Sudoku" and "4 Square Sudoku". The layout is identical to a normal Sudoku, but with additional interior areas defined in which the numbers 1 to 9 must appear. The solving algorithm is slightly different from the normal Sudoku puzzles because of the leverage on the overlapping squares. This overlap gives the player more information to logically reduce the possibilities in the remaining squares. The approach to playing is still similar to Sudoku but with possibly more emphasis on scanning the squares and overlap rather than columns and rows.
Puzzles constructed from multiple Sudoku grids are common. Five 9×9 grids which overlap at the corner regions in the shape of a quincunx is known in Japan as Gattai 5 (five merged) Sudoku. In The Times, The Age and The Sydney Morning Herald this form of puzzle is known as Samurai SuDoku. The Baltimore Sun and the Toronto Star publish a puzzle of this variant (titled High Five) in their Sunday edition. Often, no givens are to be found in overlapping regions. Sequential grids, as opposed to overlapping, are also published, with values in specific locations in grids needing to be transferred to others.
Alphabetical variations have emerged; there is no functional difference in the puzzle unless the letters spell something. Some variants, such as in the TV Guide, include a word reading along a main diagonal, row, or column once solved; determining the word in advance can be viewed as a solving aid.
A tabletop version of Sudoku can be played with a standard 81-card Set deck (see Set game). A three-dimensional Sudoku puzzle was invented by Dion Church and published in the Daily Telegraph in May 2005. There is a Sudoku version of the Rubik's Cube named Sudoku Cube.
The 2005 U.S. Puzzle Championship included a variant called Digital Number Place: rather than givens, most cells contain a partial given—a segment of a number, with the numbers drawn as if part of a seven-segment display. This version has also appeared in GAMES magazine.
One more variant of Sudoku is Greater Than Sudoku (GT Sudoku). In this a 3x3 grid of the Sudoku is given with 12 symbols of Greater Than (>) or Less Than (<) on the common line of the two adjacent numbers. Depending on difficulty this type of Sudoku may or may not be given with numbers.
How to solve Sudoku puzzles: methods, hints and tips
The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analysing.
Scanning
Scanning is performed at the outset and periodically throughout the solution. Scans may have to be performed several times in between analysis periods. Scanning comprises two basic techniques, which may be used alternately:
Cross-hatching: the scanning of rows (or columns) to identify which line in a particular region may contain a certain number by a process of elimination. This process is then repeated with the columns (or rows). For fastest results, the numbers are scanned in order of their frequency. It is important to perform this process systematically, checking all of the digits 1-9.
Counting 1-9 in regions, rows, and columns to identify missing numbers. Counting based upon the last number discovered may speed up the search. It also can be the case (typically in tougher puzzles) that the value of an individual cell can be determined by counting in reverse - that is, scanning its region, row, and column for values it cannot be to see which is left.
Advanced solvers look for "contingencies" while scanning - that is, narrowing a number's location within a row, column, or region to two or three cells. When those cells all lie within the same row (or column) and region, they can be used for elimination purposes during cross-hatching and counting. Particularly challenging puzzles may require multiple contingencies to be recognized, perhaps in multiple directions or even intersecting - relegating most solvers to marking up (as described below). Puzzles which can be solved by scanning alone without requiring the detection of contingencies are classified as "easy" puzzles; more difficult puzzles, by definition, cannot be solved by basic scanning alone.
Marking up
Scanning comes to a halt when no further numbers can be discovered. From this point, it is necessary to engage in some logical analysis. Many find it useful to guide this analysis by marking candidate numbers in the blank cells. There are two popular notations: subscripts and dots. In the subscript notation the candidate numbers are written in subscript in the cells. The drawback to this is that original puzzles printed in a newspaper usually are too small to accommodate more than a few digits of normal handwriting. If using the subscript notation, solvers often create a larger copy of the puzzle or employ a sharp or mechanical pencil. The second notation is a pattern of dots with a dot in the top left hand corner representing a 1 and a dot in the bottom right hand corner representing a 9. The dot notation has the advantage that it can be used on the original puzzle. Dexterity is required in placing the dots, since misplaced dots or inadvertent marks inevitably lead to confusion and may not be easy to erase without adding to the confusion.
Analysing
There are two main analysis approaches - elimination and what-if:
In elimination, progress is made by successively eliminating candidate numbers from one or more cells to leave just one choice. After each answer has been achieved, another scan may be performed - usually checking to see the effect of the latest number. There are a number of elimination tactics. One of the most common is "unmatched candidate deletion". Cells with identical sets of candidate numbers are said to be matched if the quantity of candidate numbers in each is equal to the number of cells containing them. For example, cells are said to be matched within a particular row, column, or region if two cells contain the same pair of candidate numbers (p,q) and no others, or if three cells contain the same triple of candidate numbers (p,q,r) and no others. These are essentially coincident contingencies. These numbers (p,q,r) appearing as candidates elsewhere in the same row, column, or region in unmatched cells can be deleted.
In the what-if approach, a cell with only two candidate numbers is selected and a guess is made. The steps above are repeated unless a duplication is found, in which case the alternative candidate is the solution. In logical terms this is known as reductio ad absurdum. Nishio is a limited form of this approach: for each candidate for a cell, the question is posed: will entering a particular number prevent completion of the other placements of that number? If the answer if yes, then that candidate can be eliminated. The what-if approach requires a pencil and eraser. This approach may be frowned on by logical purists as too much trial and error but it can arrive at solutions fairly rapidly.
Ideally one needs to find a combination of techniques which avoids some of the drawbacks of the above elements. The counting of regions, rows, and columns can feel boring. Writing candidate numbers into empty cells can be time-consuming. The what-if approach can be confusing unless you are well organised. The Holy Grail is to find a technique which minimises counting, marking up, and rubbing out.
Scanning
Scanning is performed at the outset and periodically throughout the solution. Scans may have to be performed several times in between analysis periods. Scanning comprises two basic techniques, which may be used alternately:
Cross-hatching: the scanning of rows (or columns) to identify which line in a particular region may contain a certain number by a process of elimination. This process is then repeated with the columns (or rows). For fastest results, the numbers are scanned in order of their frequency. It is important to perform this process systematically, checking all of the digits 1-9.
Counting 1-9 in regions, rows, and columns to identify missing numbers. Counting based upon the last number discovered may speed up the search. It also can be the case (typically in tougher puzzles) that the value of an individual cell can be determined by counting in reverse - that is, scanning its region, row, and column for values it cannot be to see which is left.
Advanced solvers look for "contingencies" while scanning - that is, narrowing a number's location within a row, column, or region to two or three cells. When those cells all lie within the same row (or column) and region, they can be used for elimination purposes during cross-hatching and counting. Particularly challenging puzzles may require multiple contingencies to be recognized, perhaps in multiple directions or even intersecting - relegating most solvers to marking up (as described below). Puzzles which can be solved by scanning alone without requiring the detection of contingencies are classified as "easy" puzzles; more difficult puzzles, by definition, cannot be solved by basic scanning alone.
Marking up
Scanning comes to a halt when no further numbers can be discovered. From this point, it is necessary to engage in some logical analysis. Many find it useful to guide this analysis by marking candidate numbers in the blank cells. There are two popular notations: subscripts and dots. In the subscript notation the candidate numbers are written in subscript in the cells. The drawback to this is that original puzzles printed in a newspaper usually are too small to accommodate more than a few digits of normal handwriting. If using the subscript notation, solvers often create a larger copy of the puzzle or employ a sharp or mechanical pencil. The second notation is a pattern of dots with a dot in the top left hand corner representing a 1 and a dot in the bottom right hand corner representing a 9. The dot notation has the advantage that it can be used on the original puzzle. Dexterity is required in placing the dots, since misplaced dots or inadvertent marks inevitably lead to confusion and may not be easy to erase without adding to the confusion.
Analysing
There are two main analysis approaches - elimination and what-if:
In elimination, progress is made by successively eliminating candidate numbers from one or more cells to leave just one choice. After each answer has been achieved, another scan may be performed - usually checking to see the effect of the latest number. There are a number of elimination tactics. One of the most common is "unmatched candidate deletion". Cells with identical sets of candidate numbers are said to be matched if the quantity of candidate numbers in each is equal to the number of cells containing them. For example, cells are said to be matched within a particular row, column, or region if two cells contain the same pair of candidate numbers (p,q) and no others, or if three cells contain the same triple of candidate numbers (p,q,r) and no others. These are essentially coincident contingencies. These numbers (p,q,r) appearing as candidates elsewhere in the same row, column, or region in unmatched cells can be deleted.
In the what-if approach, a cell with only two candidate numbers is selected and a guess is made. The steps above are repeated unless a duplication is found, in which case the alternative candidate is the solution. In logical terms this is known as reductio ad absurdum. Nishio is a limited form of this approach: for each candidate for a cell, the question is posed: will entering a particular number prevent completion of the other placements of that number? If the answer if yes, then that candidate can be eliminated. The what-if approach requires a pencil and eraser. This approach may be frowned on by logical purists as too much trial and error but it can arrive at solutions fairly rapidly.
Ideally one needs to find a combination of techniques which avoids some of the drawbacks of the above elements. The counting of regions, rows, and columns can feel boring. Writing candidate numbers into empty cells can be time-consuming. The what-if approach can be confusing unless you are well organised. The Holy Grail is to find a technique which minimises counting, marking up, and rubbing out.
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