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.
Saturday, April 4, 2009
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.
SUKODU HISTORY
Number puzzles first appeared in newspapers in the late 19th century, when French puzzle setters began experimenting with removing numbers from magic squares. Le Siècle, a Paris-based daily, published a partially completed 9×9 magic square with 3×3 sub-squares in 1892.[8] It was not a Sudoku because it contained double-digit numbers and required arithmetic rather than logic to solve, but it shared key characteristics: each row, column and sub-square added up to the same number.
Within three years Le Siècle's rival, La France, refined the puzzle so that it was almost a modern Sudoku. It simplified the 9×9 magic square puzzle so that each row and column contained only the numbers 1–9, but did not mark the sub-squares. Although they are unmarked, each 3×3 sub-square does indeed comprise the numbers 1–9. However, the puzzle cannot be considered the first Sudoku because, under modern rules, it has two solutions. The puzzle setter ensured a unique solution by requiring 1–9 to appear in both diagonals.
These weekly puzzles were a feature of newspaper titles including L'Echo de Paris for about a decade but disappeared about the time of the First World War.[9]
According to Will Shortz, the modern Sudoku was most likely designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor from Indiana, and first published in 1979 by Dell Magazines as Number Place (the earliest known examples of modern Sudoku). Garns's name was always present on the list of contributors in issues of Dell Pencil Puzzles and Word Games that included Number Place, and was always absent from issues that did not.[10] He died in 1989 before getting a chance to see his creation as a worldwide phenomenon.[10] It is unclear if Garns was familiar with any of the French newspapers listed above.
The puzzle was introduced in Japan by Nikoli in the paper Monthly Nikolist in April 1984[10] as Suuji wa dokushin ni kagiru (数字は独身に限る ?), which can be translated as "the digits must be single" or "the digits are limited to one occurrence." At a later date, the name was abbreviated to Sudoku by Maki Kaji (鍜治 真起 ,Kaji Maki?), taking only the first kanji of compound words to form a shorter version.[10] In 1986, Nikoli introduced two innovations: the number of givens was restricted to no more than 32, and puzzles became "symmetrical" (meaning the givens were distributed in rotationally symmetric cells). It is now published in mainstream Japanese periodicals, such as the Asahi Shimbun.
Within three years Le Siècle's rival, La France, refined the puzzle so that it was almost a modern Sudoku. It simplified the 9×9 magic square puzzle so that each row and column contained only the numbers 1–9, but did not mark the sub-squares. Although they are unmarked, each 3×3 sub-square does indeed comprise the numbers 1–9. However, the puzzle cannot be considered the first Sudoku because, under modern rules, it has two solutions. The puzzle setter ensured a unique solution by requiring 1–9 to appear in both diagonals.
These weekly puzzles were a feature of newspaper titles including L'Echo de Paris for about a decade but disappeared about the time of the First World War.[9]
According to Will Shortz, the modern Sudoku was most likely designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor from Indiana, and first published in 1979 by Dell Magazines as Number Place (the earliest known examples of modern Sudoku). Garns's name was always present on the list of contributors in issues of Dell Pencil Puzzles and Word Games that included Number Place, and was always absent from issues that did not.[10] He died in 1989 before getting a chance to see his creation as a worldwide phenomenon.[10] It is unclear if Garns was familiar with any of the French newspapers listed above.
The puzzle was introduced in Japan by Nikoli in the paper Monthly Nikolist in April 1984[10] as Suuji wa dokushin ni kagiru (数字は独身に限る ?), which can be translated as "the digits must be single" or "the digits are limited to one occurrence." At a later date, the name was abbreviated to Sudoku by Maki Kaji (鍜治 真起 ,Kaji Maki?), taking only the first kanji of compound words to form a shorter version.[10] In 1986, Nikoli introduced two innovations: the number of givens was restricted to no more than 32, and puzzles became "symmetrical" (meaning the givens were distributed in rotationally symmetric cells). It is now published in mainstream Japanese periodicals, such as the Asahi Shimbun.
How To Solve Sudoku Puzzles
I divide the solving techniques into two categories: standard and advanced. Many published puzzles can be solved using just the standard technique, including even the puzzles labelled "Suicide" in the Sunday Toronto Star.
The Standard Technique
Here are the basic steps:
0) First, for each blank cell on the grid, determine all the possible values the cell can have. This can be done by looking at the seed values in all other cells of the associated groups (row, column, and 3x3 square).
1) Whenever the value of a cell is known, that value can be eliminated from the list of possible values for all other cells of the associated groups.
2) Look for an unsolved cell that has just one possible value. That becomes the value of the cell. Then apply rule 1.
3) Look through all the unsolved cells of a group looking for a possible value that exists only in one cell. That value now becomes the value of that cell. Then apply rule 1.
Repeat steps 1, 2, and 3 until either all cells are solved or you reach an impasse. If the puzzle is not yet completely solved, then you need to use the advanced techniques.
With a bit of practice, for most puzzles, you don't actually have to list out all possibilities for the unsolved cells. When starting to solve a puzzle, you can often easily spot several cells where the value is obvious. For some puzzles then, the challenge is to write down the solution in pen without jotting down any notes. But for many puzzles, at least some annotation is needed to keep track of possibilities.
The Advanced Techniques
In the standard technique, you look at just just one cell or one possible value at a time. In the advanced techniques, you look at several cells or possible values. Using an advanced technique, you don't actually determine a cells value - you can only eliminate possibilities. You can then continue solving using the standard technique.
Single Group Analysis
Double or triple cells
In any group (row, column, or square), look for two cells with the same two possible values, and only those two values. The two values must exist in either of those two cells, and can therefore be eliminated from the other cells of the group.
Likewise, you can look for three cells in a group with the same three possible values, and no others. These values must exist within these three cells, and so can be eliminated from the other cells of the group. Note that not all three values have to exist in all three cells for this technique to apply. For example, you could have three cells with the possibilities {1,2,3}, {2,3}, and {1,3}.
Double or triple possibilities
Look for two possibilities that exist only in two cells of a group. Since these possibilities must exist within these two cells, all other possibilities can be eliminated from the two cells.
You can also look for three possibilities that exist in only three cells of a group, and eliminate all other possible values from the three cells.
Partitioning
Actually, the above two techniques are simply flip sides of the same general technique. For example, say you have a group with eight unknown cells. Looking for six cells with the same six possibilities is the same as looking for two possibilities that exist in only two cells. Either way, possibilities may get eliminated from the two cells.
Therefore, to generalize the technique: Partition the group into two groups of n and m cells. If there are exactly n possibilities present in the group of n cells, then these possibilities can be eliminated from the cells of the other group.
This is just another way to express the technique. Use whichever way makes sense to you.
Intersection of two groups
Look at the intersection of a 3x3 square (call it set A) and either a row or a column (call it set B). There are three cells in the intersection, which we'll call set C. (That is, C=A∩B.) For this method to work, there must be at least two unknown cells in the intersection.
Now then, if some possibility x exists in set C but not in A-C, then that possibility also cannot exist in B-C. Thus, possibility x can then be eliminated from the cells of B not part of the intersection.
Intersection of four groups
This technique is the most complicated of all, and is best illustrated with an example. In the following puzzle, we reach an impasse:
The Standard Technique
Here are the basic steps:
0) First, for each blank cell on the grid, determine all the possible values the cell can have. This can be done by looking at the seed values in all other cells of the associated groups (row, column, and 3x3 square).
1) Whenever the value of a cell is known, that value can be eliminated from the list of possible values for all other cells of the associated groups.
2) Look for an unsolved cell that has just one possible value. That becomes the value of the cell. Then apply rule 1.
3) Look through all the unsolved cells of a group looking for a possible value that exists only in one cell. That value now becomes the value of that cell. Then apply rule 1.
Repeat steps 1, 2, and 3 until either all cells are solved or you reach an impasse. If the puzzle is not yet completely solved, then you need to use the advanced techniques.
With a bit of practice, for most puzzles, you don't actually have to list out all possibilities for the unsolved cells. When starting to solve a puzzle, you can often easily spot several cells where the value is obvious. For some puzzles then, the challenge is to write down the solution in pen without jotting down any notes. But for many puzzles, at least some annotation is needed to keep track of possibilities.
The Advanced Techniques
In the standard technique, you look at just just one cell or one possible value at a time. In the advanced techniques, you look at several cells or possible values. Using an advanced technique, you don't actually determine a cells value - you can only eliminate possibilities. You can then continue solving using the standard technique.
Single Group Analysis
Double or triple cells
In any group (row, column, or square), look for two cells with the same two possible values, and only those two values. The two values must exist in either of those two cells, and can therefore be eliminated from the other cells of the group.
Likewise, you can look for three cells in a group with the same three possible values, and no others. These values must exist within these three cells, and so can be eliminated from the other cells of the group. Note that not all three values have to exist in all three cells for this technique to apply. For example, you could have three cells with the possibilities {1,2,3}, {2,3}, and {1,3}.
Double or triple possibilities
Look for two possibilities that exist only in two cells of a group. Since these possibilities must exist within these two cells, all other possibilities can be eliminated from the two cells.
You can also look for three possibilities that exist in only three cells of a group, and eliminate all other possible values from the three cells.
Partitioning
Actually, the above two techniques are simply flip sides of the same general technique. For example, say you have a group with eight unknown cells. Looking for six cells with the same six possibilities is the same as looking for two possibilities that exist in only two cells. Either way, possibilities may get eliminated from the two cells.
Therefore, to generalize the technique: Partition the group into two groups of n and m cells. If there are exactly n possibilities present in the group of n cells, then these possibilities can be eliminated from the cells of the other group.
This is just another way to express the technique. Use whichever way makes sense to you.
Intersection of two groups
Look at the intersection of a 3x3 square (call it set A) and either a row or a column (call it set B). There are three cells in the intersection, which we'll call set C. (That is, C=A∩B.) For this method to work, there must be at least two unknown cells in the intersection.
Now then, if some possibility x exists in set C but not in A-C, then that possibility also cannot exist in B-C. Thus, possibility x can then be eliminated from the cells of B not part of the intersection.
Intersection of four groups
This technique is the most complicated of all, and is best illustrated with an example. In the following puzzle, we reach an impasse:
To break the impasse, we need to look at two rows and two columns at the same time. In this diagram, look at the four cells where the marked rows and columns intersect. Note that the possibility "6" occurs in all four cells. Note also that in the marked rows, possibility "6" occurs only in the intersecting cells. Thus, the value "6" must appear either in the top right and bottom left cells, or in the top left and bottom right cells. Either way, "6" cannot appear in any other cell in the two marked columns, and can therefore be eliminated from those other cells. In this case, the three underlined sixes can be eliminated.
Note that this technique doesn't apply to just intersecting rows and columns. You also have to consider other combinations of four intersecting rows, columns, and squares:
2 rows and 2 columns
2 rows, 1 square, and 1 column
2 rows and 2 squares
1 row, 1 square, and 2 columns
2 columns and 2 squares
You could extend this technique to three rows intersecting with three columns. But again, I doubt you'll ever have to resort to that.
SUDOKU
Su Doku is usually played on a 9 by 9 board, divided into 3 by 3 cells. You can generalize this into playing on an M×M square board broken into non-overlapping rectangular cells, each containing M squares (an obvious question is to how define a puzzle where M is a prime number). The solution of the puzzle is to place M symbols on the board such that each row, column or cell contains each symbol exactly once, without moving the initial clues. Puzzles with M<9 are called Sub Dokus, those for M>9 are called Super Dokus.
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