Sudoku - University of Wisconsin–Whitewatermath.Uww.edu/sma/documents/Sudoku.pdfSudoku How to...
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Sudoku How to become a Sudoku Ninja: Tips, Tricks and Strategies 1
Sudoku - University of Wisconsin–Whitewatermath.Uww.edu/sma/documents/Sudoku.pdfSudoku How to become a Sudoku Ninja: Tips, Tricks and Strategies. 1. Be sure to have a personal introduction
Be sure to have a personal introduction at the beginning, including SMA. Any questions will be at the end of the presentation. Because of time constraints will not cover quads, addict strategies, and puzzle variations.
Benefits
• Fun• Exercises the Mind• Improves Memory• Improves Logical and Critical Reasoning • Helps to decline the effects of aging
– Can help halt the progress of Alzheimer’s disease
• Helps children learn• Helps develop Patience, Focus and Creativity
• Magic squares were known to Chinese mathematicians as early as 650 BCE and Arab mathematicians as early as the 7th century.
• In the late 18th century the Swiss mathematician Leonhard Euler developed the concept of Latin squares.
• The puzzle was designed by Howard Garns and published by Dell Magazines in 1979 as Number Place.
• The puzzle was introduced in Japan by Nikoli in 1984 as Suuji wadokushin ni kagiru, which can be translated as “the numbers must be single” and later abbreviated to Sudoku (“su = number, doku = single).
• Around 1997, Wayne Gould discovered Sudoku in a Japanese bookstore. He later publicized the puzzle in Britain, which resulted in a widespread interest all over the world today.
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For those interested in learning more about Magic Squares and Latin Squares, these can be easily found by searching the internet.
– 5,524,751,496,156,892,842,531,225,600 different Latin squares.
– 6,670,903,752,021,072,936,960 different Sudoku solution grids.
– 5,472,730,538 unique Sudoku solution grids.
– 9! = 362,880 versions of every grid available simply by rearranging the digits.
– There are 3,359,232 different ways to renumber the puzzle.
– Minimum 17 clues needed to guarantee a unique solution.
– Minimum 18 clues needed for a symmetric puzzle.
– Maximum 77 clues which does not give a unique solution.
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http://geometer.org/mathcircles/sudoku.pdf Generally, a valid Sudoku puzzle should have a unique solution. That is, there is only one way to fill in all the missing numbers. A minimal Sudoku puzzle is one such that if you delete any one given hint, the puzzle will result in more than one solution. Unique Sudoku solution grids takes into consideration different symmetries as well as the swapping of numbers. This gets the different patterns possible. 5) = [6) * 7) *8)] - # identical arrangements Different solution grids = (unique solution grids * geometric permutations * ways to renumber) – number identical arrangements
Difficulty of Sudoku
• Two things are obvious when you look at a Sudoku– Number of clues
– Placement of clues
• Two things are not as obvious when you look at a Sudoku– Time – How long it takes to complete the puzzle
– Logic – Which techniques are needed when solving the puzzle
• The placement of the clues and the logic needed to solve are what really determine the difficulty, not the number of clues.
• There are no standard rules to rate a Sudoku puzzle. – The same puzzle may have a different difficulty name based on where
it is published.
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“Published puzzles often are ranked in terms of difficulty. Perhaps surprisingly, the number of givens has little or no bearing on a puzzle's difficulty. A puzzle with a minimum number of givens may be very easy to solve, and a puzzle with more than the average number of givens can still be extremely difficult to solve. It is based on the relevance and the positioning of the numbers rather than the quantity of the numbers. Computer solvers can estimate the difficulty for a human to find the solution, based on the complexity of the solving techniques required. This estimation allows publishers to tailor their Sudoku puzzles to audiences of varied solving experience. Some online versions offer several difficulty levels.” “The primary factor which determines difficulty is the number of cells which can be solved at a given point. Suppose you have two puzzles, each with 10 cells left to fill in. On the first, you can immediately figure the value of 9 of those 10. On the second, you can only immediately figure the value of 4 of those 10. Then the second sudoku is the harder one at that point. Ultimately, of course, every correctly solved sudoku "converges" towards an equal difficulty. The last cell is always the easiest to fill out.” (sudokuplace.com) Newspapers tend to rate puzzles based on the number of given which is incorrect. This is why some harder puzzles (5 star) can seem easier than say a medium puzzle (3 stars).
Terms to know
Groups
Rows
Columns
Boxes
Cells
Givens
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Groups are also called Houses. Boxes also called Regions or Nonets. Givens also called Clues. Each cell has 20 buddy cells (8 in same row, 8 in same column, and 4 others in same box not in column or row)
The One Rule
Fill in all blank cells so that each group contains the numbers 1 through 9.
AND
Each group can not have any duplicate numbers.
AND
There is only one solution per puzzle.
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Numbers can not repeat within a group. Note: 3 part rule.
Basic Skills
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One and Only Choice
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When a cell is the only one in a group that can receive a number, it must contain that number. Top left is missing a 8. Bottom left is missing a 7. Right is missing a 6 in the cell. The examples are also called a Full House or Counting.
Scanning
Squeezing Cross Hatching
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Left (To a part of a group) can have an 8 in the middle box middle column. Middle (To a cell within a group) can have an 8 in the middle box middle column middle cell. The value is squeezed into the cell and can only be placed there. This is also called Forced Digit. Right (To a specific cell from different directions) uses the squeeze method but in both rows and columns, with the 1 only able to go in the cell with the X. You can also use this method with candidates, especially locked candidates. You could also call these techniques “Slicing and Dicing” respectively.
Candidates
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Candidates (pencilmarks) help you to determine possible values for the cells you have not found out yet. Many times candidate techniques help to eliminate other candidates which helps you to find where a digit goes. What I do is go in order by a group at a time and get all my candidates filled in. I do this once my other basic techniques do not work out. I also place the digits in a grid system so that it is easier for me to see patterns/techniques. Next I will show you how to remove those candidates. There are a lot of techniques that use candidates and the process of elimination to help solve the puzzle. Strategy: Use pencil, not pen, in case you make a mistake.
Naked Singles
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Naked single means there are no other candidates showing for a particular cell. This means it is the only number that can go into that cell, thus allowing you to remove that candidate from every other cell in the group, as well as any buddy cells. When a cell has a lone candidate, then that candidate must go there. Big chain of good things come from this one (9,7,6,8,4).
Hidden Single
Where is the other hidden single?
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Hidden Singles are where a candidate can only go in one place, but is “hidden” by the other candidates in the group. Even if there are other numbers as candidates, but there is a number that appears only once in a group then that cell must contain that number. Left: After the 4 and 7, you have a 5 hidden single. Right: After the 8 in the left box, you also have a 4 in the right box.
Intermediate Skills
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Locked Candidates
Restricted to a row or column in a box
Restricted to a row or column 15
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Top: 7s have to be in bottom row of right box, so you can remove any 7 candidates from other boxes and same row, also results in naked 8 and 1,4 naked pair. Left is also called pointing pairs/ triples. (Pointing) Bottom: Row 4 only has 9s in box 3. Therefore A or B must be a 9, then 9s can be removed from C, D, and E. Results in 6,7 naked pair in box 3 column 8. (Claiming)
Locked Candidates 2
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Here is another examples with many pointing pairs/triples. http://www.sudokuwiki.org/Intersection_Removal Top: Shows the many different locked candidates with the yellow candidates being removed.
Locked Candidates 34 2 8 3
8 1 4 2
7 2 6 8 5 1 4
8 6
5 6 1 8
4
8
8
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Sometimes it helps to combine different methods, such as locked candidates and scanning. This could be called Advanced Scanning (Squeezing/Crosshatching). Top: Look at 1, forced into box 3 row 3 because 1 candidates locked into box 1 row 1. Bottom: Look at 8, forced into box 3 row 2 column 3 because 8 candidates locked into box 2 row 3.
Naked Pairs
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A naked pair is where only two identical candidates are in two different cells within a group. If two cells in a group contain an identical pair of candidates and only those two candidates, then no other cells in that group could be those values. Top: 4,6 pair which results in a 3,8 naked pair Bottom: 1,4 pair
Hidden Pairs
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If two cells in a group contain a pair of candidates (hidden amongst other candidates) that are not found in any other cells in that group, then other candidates in those two cells can be excluded safely.��In the example on the right, the candidates 1 & 9 are only located in two highlighted cells of a box, and therefore form a 'hidden' pair. All candidates except 1 & 9 can safely be excluded from these two cells as one cell must be the 1 while the other must be the 9. Top: 2,9 pair, results in locked 1s Bottom: 2,6 pair. http://www.learn-sudoku.com/hidden-pairs.html
Advanced Skills
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Naked Triples
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A Naked Triple occurs when three cells in a group contain no candidates other than the same three candidates. The cells which make up a Naked Triple don't have to contain every candidate of the Triple, but it will contain at least 2 candidates otherwise if it is just one it is a naked single. If these candidates are found in other cells in the group they can be excluded.��Top: Naked Triple of 5,6,9, resulting in a hidden 2 and 1,4 naked pair. Bottom: A Naked Triple is formed by the top left, bottom left & bottom right cells of a box since they only contain the candidates 1, 4 & 6. Therefore the candidates 1 & 4 in the highlighted cells can be excluded safely. (Results in a naked single 3, then making a 5,7,8 naked triple)
Hidden Triples
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A Hidden Triple occurs when three cells in a group contain no candidates other than the same three candidates. The cells which make up a Hidden Triple don't have to contain every candidate of the Triple, but it will contain at least 2 candidates otherwise if it is just one it is a hidden single. The other candidates in these three cells can be excluded safely. �Hidden triples are much harder to spot. They will occur in harder puzzles. Hidden triples like naked triples are restricted to three cells in a row, column, or region. Hidden triples like hidden pairs have additional digits that camouflage the three candidates. Top: Hidden triple of 4, 8, and 9. Bottom: Hidden triple of 3, 6 and 7 are found only in column four, six and seven. Therefore, all other candidates can be excluded from those three cells. Remove the extra numbers from the cells circled in red. Do you think hidden triples are tough to find? Try quads.
Naked Quads
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A Naked Quad occurs when four cells in a group contain no candidates other than the same four candidates. The cells which make up a Naked Quad don't have to contain every candidate of the Quad, but it will contain at least 2 candidates otherwise if it is just one it is a naked single. These candidates can be safely excluded from the other cells in the group. Top: Naked Quad of 3, 5, 6, and 8. Remove any instance of these four numbers from the other cells in this row. Hidden 4,9 is easier to see. �Bottom: Naked Quad of 2, 5, 7, and 9 in the 3 left most cells and bottom middle cell of the box. Therefore candidates 5 and 7 that are in the highlighted cells can be excluded.�
Hidden Quads
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A Hidden Quad occurs when four cells in a group contain no candidates other than the same four candidates. The cells which make up a Hidden Quad don't have to contain every candidate of the Quad, but it will contain at least 2 candidates otherwise if it is just one it is a hidden single. The other candidates in these four cells can be excluded safely. These four numbers are hidden by additional candidates. Hidden quads are very difficult to find. The good news is I have rarely seen them. (Maybe because they are hidden so well!) They occur only in a few of the more difficult puzzles. Hidden Quads are very rare, which is fortunate since they're almost impossible to spot even when you know they're there. Left: Hidden Quad of 1, 5, 6, and 8. It is safe to remove the extra digits (3,4,7,9) from these four cells. Right: Hidden Quad of 5,6,7,8. �
For the Addict
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X Wing
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For every Sudoku, a value can exist only once in each row, column and box. If a value has only 2 possible locations in a given row (ie it has a candidate in only 2 cells in that row), then it must be assigned to one of these 2 cells. Extra Proof: Given a particular puzzle that has two rows where a given candidate 'C' is restricted to exactly the same two columns (and no more than 2 columns), and since� 1) candidate C must be assigned once in each of these two rows, and� 2) no column can contain more than one of candidate C�then candidate C must be assigned exactly once in each of these two columns within these two rows. Therefore, it's not possible for any other cells in these two columns to contain candidate C. This same logic applies when a puzzle that has two columns where candidate C is restricted to exactly the same two rows. Look for a candidate that appears twice in only two rows and if they share the same two columns they form a vertical X wing. You can also have it where a candidate appears twice in two columns and share the same two rows, forming a horizontal X wing. �Left: The cells marked with a blue highlight form an "X-Wing" since rows one and nine both have only two cells with candidate 6's and these two cells share the same two columns. Therefore, other candidate 6's in columns six and nine (highlighted yellow) can safely be removed.� Right: This example uses two columns that share the same two rows. Therefore the candidate 3 has to be in a diagonal pair of the yellow cells. This means the candidate can not be in any of the cells colored green. � Note: It is called the X Wing because it obviously makes an X between the candidates in the rectangle. http://www.sudokuwiki.org/X_Wing_Strategy
Swordfish
Jellyfish – variation of the X Wing on four rows and columnsSquirmbag – variation of the X Wing on five rows and columns
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The Swordfish pattern is a variation on the "X-Wing" pattern above.��Extra Proof:�Given a particular puzzle that has three rows where a given candidate 'C' is restricted to the same three columns, and since candidate C must be assigned once in each of these three rows no column can contain more than one of candidate C then candidate C must be assigned exactly once in each of these three columns within these three rows. Therefore, it's not possible for any other cells in these three columns to contain candidate C.�This same logic applies when a puzzle that has three columns where candidate C is restricted to exactly the same three rows. �Example: Three rows (3, 5, & 7) have candidate 5 in no more than three cells (only two cells each in this example), and these cells all share the same three columns (3, 4 & 7). A "Swordfish" pattern is established. Other candidate 5's in these three columns (highlighted yellow) can be excluded safely.��Note: As shown in this example, there does not have to be exactly 3 cells in each row (or column), there may be less. However, there must not be more that three candidates in the pattern defining rows. Another interesting fact is that the Swordfish does not have to have the candidate in one of the column cells or it can already have a number in place. You could also have a variation of a Horizontal Swordfish where 3 columns have the same candidate in no more than 3 cell sharing the same rows, then that candidate in the other row cells could be eliminated. http://www.sudokuwiki.org/Sword_Fish_Strategy
XY Wing: Version 1
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Not to be confused with X-Wing (also called Y wing). Formal definition:�Given 3 cells where - all cells have exactly 2 candidates and they share the same 3 candidates in the form - xy, yz, xz. One cell (the Y 'stem' with canidates xy) shares a group with the other 2 cells (Y 'branches' with candidates xz & yz) then any other cell which shares a group with both 'branch' cells can have excluded the 'z' candidate that is common to these 'branch' cells. Extra - Proof: If a cell sharing a group with both branch cells is assigned the 'z' candidate, then neither branch can be assigned the 'z' candidate. Consequently, one branch would be the 'x' and the other the 'y' leaving the 'stem' without a candidate, an invalid state. http://sudokuandpuzzles.com/ http://www.brainbashers.com/sudokuxywing.asp� How to pick the cells for a XY wing: Choose two cells in a box with each cell containing only two candidates (preferably different rows and different columns). Next take that cell and remove the candidate in common. For example in the top example, you pick 1,2 and 1,3. Therefore this leaves you with looking for a 2,3. You want to look for the 2,3 cell that is in the same row or column as on of the two cells first chosen. If you find a cell that fits what you are looking for, then the cell that appears to be in the middle is your stem cell. The branch cells should contain a candidate in common. This common candidate is the candidate that can be removed from the cells that can see both branch cells. �Note: If all 3 cells in a xy-wing pattern shared the same group, then they would be called a 'naked triple'. Top: 1,2,3 candidates. R2C1 can not be a 2, it must be a 3. Huge chain happens. Bottom: 1,5,8 candidates. The R1C2, R3C2, R6C3 cannot be an 8. � Note: up to 5 cells can have a value eliminated using XY wing. http://www.sudokuwiki.org/Y_Wing_Strategy http://www.dagblastit.com/blog/2010/04/finding-y-wing-aka-xy-wing/�
XY Wing: Version 2
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Not to be confused with X-Wing (also called Y wing). Formal definition:�Given 3 cells where - all cells have exactly 2 candidates and they share the same 3 candidates in the form - xy, yz, xz. One cell (the Y 'stem' with canidates xy) shares a group with the other 2 cells (Y 'branches' with candidates xz & yz) then any other cell which shares a group with both 'branch' cells can have excluded the 'z' candidate that is common to these 'branch' cells. Extra - Proof: If a cell sharing a group with both branch cells is assigned the 'z' candidate, then neither branch can be assigned the 'z' candidate. Consequently, one branch would be the 'x' and the other the 'y' leaving the 'stem' without a candidate, an invalid state. http://sudokuandpuzzles.com/ http://www.brainbashers.com/sudokuxywing.asp� How to pick the cells for a XY wing: Choose two cells in a box with each cell containing only two candidates (preferably different rows and different columns). Next take that cell and remove the candidate in common. For example in the top example, you pick 1,2 and 1,3. Therefore this leaves you with looking for a 2,3. You want to look for the 2,3 cell that is in the same row or column as on of the two cells first chosen. If you find a cell that fits what you are looking for, then the cell that appears to be in the middle is your stem cell. The branch cells should contain a candidate in common. This common candidate is the candidate that can be removed from the cells that can see both branch cells. �Note: If all 3 cells in a xy-wing pattern shared the same group, then they would be called a 'naked triple'. Top: 1,2,3 candidates. R2C1 can not be a 2, it must be a 3. Huge chain happens. Bottom: 1,5,8 candidates. The R1C2, R3C2, R6C3 cannot be an 8. � Note: up to 5 cells can have a value eliminated using XY wing. http://www.sudokuwiki.org/Y_Wing_Strategy http://www.dagblastit.com/blog/2010/04/finding-y-wing-aka-xy-wing/�
XYZ Wing
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This is an extension of the Y Wing with the main difference being the pivot cell having XYZ values and the pincers having XZ and YZ values. This means the cell (up to 2 cells) can see the trio and the common number can be eliminated. The candidate that appears all three cells of the XYZ wing can be eliminated in the cell which can see all three cells. What I mean by see is shares a group with each cell of the XYZ wing. http://www.sudokuwiki.org/XYZ_Wing
Other Techniques
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http://www.sudopedia.org/wiki/Solving_Technique http://www.sudokuwiki.org/Strategy_Families ALS (Almost Locked Sets)
Special Fish (Shape and/or Quality)
Coloring
Chains/Loops
Uniqueness/BUG
ALS
Nishio (Trial and Error)
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http://www.sudopedia.org/wiki/Solving_Technique http://www.sudokuwiki.org/Strategy_Families ALS (Almost Locked Sets)
Review
Naked• N cells contain N
numbers• Remove the N numbers
from other cells in the group
Hidden• N numbers are
contained in N cells• Remove other numbers
from the N cells in the group
Hidden becomes Naked33
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Naked means you can remove those candidates from other cells in the respective group. With this you will find the candidates you are looking at in other cells in the respective group. Hidden means you can remove the other candidates from the cells in the respective group. This turns the hidden into a naked. With this you will not find the candidates you are looking at in other cells in the respective group, but you will find other candidates hiding the candidates you are looking at, thus why they are called hidden.
Strategy
• If a puzzle is Easy, you should be able to use basic techniques without candidate lists.
• If the puzzle is Hard, use basic techniques to start, but be prepared to use candidate lists.
• Start with a difficulty level you are comfortable with. Once you feel you have mastered that level, then you may want to move up in difficulty. You may need some practice to get good at higher level puzzles.
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Strategy
• Good idea to start with the number/group with the most givens, then cycle through the other numbers/groups.
• Once you fill in a cell, adjust all other cells (candidate lists) that are affected. Many times one technique can result in a chain of good things happening.
• Use elimination. It is easier to know where a number can’t go, then where it can go.
• Look for the Obvious! • Don’t Guess! Use Logic Learned!
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3rd points especially true for naked and hidden singles, as well as pairs or triples that are part of multiple groups. Such as a naked pair in a box and same column. This will effect the box as well as other cells in same column outside the box. Harder puzzles tend to “bottleneck”. This means that the more you solve the number of possibilities to remove gets smaller “like a bottle”. Eventually you will get stuck until you figure out the one technique to get you past the “bottleneck” or single move and then it will open up. Many times I have had this happen and once I got past the “bottleneck” I literally was able to finish the puzzle with ease. Sometimes puzzles can have multiple “bottlenecks” in them.
Strategy
• If you make a mistake, you have 2 choices:– Try to find and correct your mistake
• You can see where you have duplicate numbers and by looking at other numbers determine which one is incorrect.
• You can assume one of them is correct and hope you picked the right one. If you chose the one that didn’t work you know it must be the other choice.
– Start over• Redo the current puzzle.• Recycle your puzzle and start a different one.
http://www.sudoku.4thewww.com/9x9.php http://www.sudokuwiki.org/X_Wing_Strategy http://www.sudokuwiki.org/Y_Wing_Strategy http://www.dagblastit.com/blog/2010/04/finding-y-wing-aka-xy-wing/ http://www.sudokuessentials.com/sudoku-strategy.html http://www.thepuzzleclub.com http://www.sudopedia.org *May want to add different sudoku programs that I have found/enjoyed for the computer, android, and ipod touch.*
In Diagonal Sudoku, fill in the grid so that every row, column, 3x3 box, and main diagonals contains the digits 1 through 9.
It is also called Sudoku–X.42
Hyper Sudoku
In Hyper Sudoku, fill in the grid so that every row, column, 3x3 box, contains the digits 1 through 9. Instead of 9 boxes, there are now 13.
It is also called Windoku, NRC-Sudoku, and Four-Square Sudoku.43
Killer Sudoku
The puzzle has no givens. The cells are placed into cages, where the number in the cage represents the sum of the digits.
It is also called Sum Sudoku.44
Killer Sudoku
Example Example Solution
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Another example of Killer Sudoku, but where the cages are colored.
Samurai Sudoku
The puzzle has 5 grids, with the center grid overlapping exactly one corner box with each of the remaining grids. It is also called Gattai-5. 46
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Each Sudoku does not affect the other Sudoku unless the cells are overlapping. This is part of a bigger group called overlapping or gattai sudokus. Some other examples are Shogun (11), Sumo (13), Shaolin (25), plus many others.
Jigsaw Sudoku
This puzzle is the same as Normal Sudoku, except in a Jigsaw Sudoku the boxes are irregular shapes.
It is also called Irregular or Geometry Sudoku.47
Other Variations
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Greater Than Sudoku
The puzzle has no givens. Instead, there are greater-than (>) signs between some adjacent cells, which signify that the digit in one cell
should be greater than another. All other ordinary Sudoku rules apply. 49
Odd/Even Sudoku
This puzzle is also called Tanto Sudoku.The grey cells are even, then the white cells are odd.
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(Smaller than Normal) Sudoku
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Same as Normal Sudoku, but instead of 9x9, it usually is a 4x4 or 6x6. Used primarily for kids or beginners.
(Bigger than Normal) Sudoku
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Same as Normal Sudoku, but instead of 9x9, it can be 16x16 or 25x25.
Center Dot Sudoku
The center cell of each box makes an additional group, meaning the blue cells must contain digits 1-9.
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Clueless Sudoku
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This puzzle has 9 constituent Sudokus. The central 3x3 boxes in each of these 9 Sudokus (colored blue), in the configuration of the constituent puzzles, form a 10th Sudoku, which is completely clueless. None of the constituent puzzles can be solved independently, you need to feed information back and forth using the 10th puzzle to solve the Clueless Special as a whole. Clueless Center Dot Sudoku: http://www.sudocue.net/explosion.php
This is also called Picross, Paint by numbers, and Griddlers.
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KenKen (Calcudoku)
Each bold-outlined group of cells is a cage containing digits which achieve the specified result using the specified mathematical operation: addition (+), subtraction (−), multiplication (×), and
division (÷). (Unlike Killer Sudoku, digits may repeat within a cage). 58
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Presentation Notes
For example, 26 is the same as 62 when referring to 3 division.
Kakuro (Cross Sums)
Referred to a mathematical transliteration of the crossword.
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Nurikabe (Islands in the Stream)
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Numbers represent island size and the water must be continuous/unbroken (no ponds/lakes).
Akari (Light Up)
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Light up all the white cells by using light bulbs (white circles) which light up horizontally/vertically until they hit black. Number in black represents how many light bulbs are connected to that box. No number boxes can vary.
Hitori
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Remove duplicate numbers in rows and columns. However it is ok if a number is not in a row or column.
Fillimino
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Make block pieces with numbers given.
Futoshiki (Unequal)
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Fill in the grid by using the inequality signs with rows/columns like Sudoku/Latin Square.
Tents
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Find where all the tents go. There can not be any adjacent tents and a tent must be by a tree. The numbers on the side tell you how many tents are in that row or column.
Hidoku (Hidato)
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Presentation Notes
Make a path connecting all the values from 1 to N. The next cell in a path can be in any direction.
Logic Puzzle
NAME MONTH DAY
Abigail February MondayBrenda December WednesdayMary June SundayPaula March FridayTara July Saturday
FINAL SOLUTION:
THE PUZZLE:Five sisters all have their birthday in a different month and each on a different day of the week. Using the clues below, determine the month and day of the week each sister's birthday falls.1.Paula was born in March but not on Saturday. Abigail's birthday was not on Friday or Wednesday. 2.The girl whose birthday is on Monday was born earlier in the year than Brenda and Mary. 3.Tara wasn't born in February and her birthday was on the weekend. 4.Mary was not born in December nor was her birthday on a weekday. The girl whose birthday was in June was born on Sunday. 5.Tara was born before Brenda, whose birthday wasn't on Friday. Mary wasn't born in July.
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Presentation Notes
Using the clues given and the assistance of the grid, solve the question and fill out the table.
