Sum sudoku

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Easy Medium Hard Expert. Auto-Check for Mistakes. Loading game. Easy Medium Hard Expert Evil. New Game Continue. Select Game Mode. Easy Medium Hard Expert Restart. Killer Sudoku by Sudoku. What is Killer Sudoku? How to play Killer Sudoku Fill all rows, columns, and 3x3 blocks with numbers exactly like in classic sudoku. Pay attention to the cages — groups of cells indicated by dotted lines. Make sure the sum of numbers in each cage is equal to the number in the upper left corner of the cage.

Numbers cannot repeat within cages, a single row, column, or 3x3 region. Are Killer Sudoku rules hard? Cookie preferences. Volume 1 1-per-page 4-per-page Volume 2 1-per-page 4-per-page Volume 3 1-per-page 4-per-page Volume 4 1-per-page 4-per-page Volume 5 1-per-page 4-per-page Volume 6 1-per-page 4-per-page Volume 7 1-per-page 4-per-page Volume 8 1-per-page 4-per-page Volume 9 1-per-page 4-per-page Volume 10 1-per-page 4-per-page Volume 11 1-per-page 4-per-page Volume 12 1-per-page 4-per-page Volume 13 1-per-page 4-per-page Volume 14 1-per-page 4-per-page Volume 15 1-per-page 4-per-page Volume 16 1-per-page 4-per-page Volume 17 1-per-page 4-per-page Volume 18 1-per-page 4-per-page Volume 19 1-per-page 4-per-page Volume 20 1-per-page 4-per-page Volume 21 1-per-page 4-per-page Volume 22 1-per-page 4-per-page Volume 23 1-per-page 4-per-page Volume 24 1-per-page 4-per-page How to use the hints My puzzles contain a page of hints in addition to a page of answers.

The hints are puzzles grids that contain numbers showing the order in which the squares were solved by the computer. Although this is not necessarily the order you would solve the puzzle, it's probably pretty close. Follow the numbered squares in order 1,2,3,4, This square or the one or two immediately after it is a good candidate to solve next.

Feel free to reproduce the puzzles for personal, church, school, hospital or institutional use. Please do not use my puzzles in for-profit publications without my permission. If you need free puzzles for your small circulation newspaper, or you would like to purchase puzzles for a book, periodical, app or website, contact me at dad krazydad.

Privacy Policy. Blog Publications Puzzles About. January 16, Volumes 23 and 24 of Printable Killer Sudoku are out! Happy puzzling! Prefer to donate by mail or Venmo? Here's how. How to use the hints My puzzles contain a page of hints in addition to a page of answers.

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If dropping the requirement for the uniqueness of the solution, clue minimal pseudo-puzzles are known to exist, but they can be completed to more than one solution grid. Removal of any clue increases the number of the completions and from this perspective none of the 41 clues is redundant. With slightly more than half the grid filled with givens 41 of 81 cells , the uniqueness of the solution constraint still dominates over the minimality constraint.

As for the most clues possible in a Sudoku while still not rendering a unique solution, it is four short of a full grid If two instances of two numbers each are missing and the cells they are to occupy are the corners of an orthogonal rectangle, and exactly two of these cells are within one region, there are two ways the last digits can be added two solutions. The number of minimal Sudokus Sudokus in which no clue can be deleted without losing uniqueness of the solution is not precisely known.

Other authors used faster methods and calculated additional precise distribution statistics. It has been conjectured that no proper Sudoku can have clues limited to the range of positions in the clear space of the first image above. The largest total number of empty groups rows, columns, and boxes in a Sudoku is believed to be nine.

One example is a Sudoku with 3 empty rows, 3 empty columns, and 3 empty boxes third image. A Sudoku grid is automorphic if it can be transformed in a way that leads back to the original grid, when that same transformation would not otherwise lead back to the original grid. One example of a grid which is automorphic would be a grid which can be rotated degrees resulting in a new grid where the new cell values are a permutation of the original grid.

Automorphic Sudokus are Sudoku puzzles which solve to an automorphic grid. Two examples of automorphic Sudokus, and an automorphic grid are shown below. Since these Sudokus are automorphic, so too their solutions grids are automorphic. Notice that in the second example, the Sudoku also exhibits translational or repetition symmetry; clues are clustered in groups, with the clues in each group ordered sequentially i. The third image is the Most Canonical solution grid. In these examples the automorphisms are easy to identify, but in general automorphism is not always obvious.

The table at right shows the number of the essentially different Sudoku solution grids for all existing automorphisms. An enumeration technique based on band generation was developed that is significantly less computationally intensive. The strategy begins by analyzing the permutations of the top band used in valid solutions. Once the Band1 symmetries and equivalence class for the partial solutions are identified, the completions of the lower two bands are constructed and counted for each equivalence class.

Pettersen [31]. The permutations for Band1 are 9! The permutations of B1 are the number of ways to relabel the 9 digits, 9! Counting the permutations for B2 is more complicated, because the choices for B2 depend on the values in B1. This is a visual representation of the expression given above.

The conditional calculation needs a branch sub-calculation for each alternative. Fortunately, there are just 4 cases for the top B2 triplet r21 : it contains either 0, 1, 2, or 3 of the digits from the B1 middle row triplet r Once this B2 top row choice is made, the rest of the B2 combinations are fixed.

The Band1 row triplet labels are shown on the right. Note: Conditional combinations becomes an increasingly difficult as the computation progresses through the grid. At this point the impact is minimal. The Case 0 diagram shows this configuration, where the pink cells are triplet values that can be arranged in any order within the triplet.

Each triplet has 3! The 6 triplets contribute 6 6 permutations. The same logic as case 0 applies, but with a different triplet usage. In the Case 1 diagram, B1 cells show canonical values, which are color-coded to show their row-wise distribution in B2 triplets. Colors reflect distribution but not location or values. For this case: the B2 top row triplet r21 has 1 value from B1 middle triplet, the other colorings can now be deduced. Fill in the number of B2 options for each color, The B3 color-coding is omitted since the B3 choices are row-wise determined by B1, B2.

B3 always contributes 3! For B2, the triplet values can appear in any position, so a 3! However, since some of the values were paired relative to their origin, using the raw option counts would overcount the number of permutations, due to interchangeability within the pairing. The option counts need to be divided by the permuted size of their grouping 2 , here 2!

Case 2: 2 Matches for r21 from r The same logic as case 1 applies, but with the B2 option count column groupings reversed. Totaling the 4 cases for Band1 B B3 gives 9! Symmetries are used to reduce the computational effort to enumerate the Band1 permutations. A symmetry is an operation that preserves a quality of an object. For a Sudoku grid, a symmetry is a transformation whose result is also a valid grid.

The following symmetries apply independently for the top band:. Combined, the symmetries give 9! A symmetry defines an equivalence relation , here, between the solutions, and partitions the solutions into a set of equivalence classes.

Since the solution for any member of an equivalence class can be generated from the solution of any other member, we only need to enumerate the solutions for a single member in order to enumerate all solutions over all classes. The Band1 symmetries above are solution permutation symmetries defined so that a permuted solution is also a solution.

For the purpose of enumerating solutions, a counting symmetry for grid completion can be used to define band equivalence classes that yield a minimal number of classes. Counting symmetry partitions valid Band1 permutations into classes that place the same completion constraints on lower bands; all members of a band counting symmetry equivalence class must have the same number of grid completions since the completion constraints are equivalent. Counting symmetry constraints are identified by the Band1 column triplets a column value set, no implied element order.

Using band counting symmetry, a minimal generating set of 44 equivalence classes [56] was established. The following sequence demonstrates mapping a band configuration to a counting symmetry equivalence class. Begin with a valid band configuration 1. Build column triplets by ordering the column values within each column. This is not a valid Sudoku band, but does place the same constraints on the lower bands as the example 2.

Construct an equivalence class ID from the B2, B3 column triplet values. Use column and box swaps to achieve the lowest lexicographical ID. The last figure shows the column and box ordering for the ID: All Band1 permutations with this counting symmetry ID will have the same number of grid completions as the original example. An extension of this process can be used to build the largest possible band counting symmetry equivalence classes 3.

Note, while column triplets are used to construct and identify the equivalence classes, the class members themselves are the valid Band1 permutations: class size Sb. Counting symmetry is a completion property and applies only to a partial grid band or stack. Solution symmetry for preserving solutions can be applied to either partial grids bands, stacks or full grid solutions.

Lastly note, counting symmetry is more restrictive than simple numeric completion count equality: two distinct bands belong to the same counting symmetry equivalence class only if they impose equivalent completion constraints. Symmetries group similar object into equivalence classes. Two numbers need to be distinguished for equivalence classes, and band symmetries as used here, a third:.

The not less than and up to caveats are necessary, since some combinations of the transformations may not produce distinct results, when relabeling is required see below. Consequently, some equivalence classes may contain less than 6 5 distinct permutations and the theoretical minimum number of classes may not be achieved. Each of the valid Band1 permutations can be expanded completed into a specific number of solutions with the Band2,3 permutations. By virtue of their similarity, each member of an equivalence class will have the same number of completions.

Consequently, we only need to construct the solutions for one member of each equivalence class and then multiply the number of solutions by the size of the equivalence class. We are still left with the task of identifying and calculating the size of each equivalence class. Further progress requires the dexterous application of computational techniques to catalogue classify and count the permutations into equivalence classes.

Block 1 uses a canonical digit assignment and is not needed for a unique ID. Equivalence class identification and linkage uses the lowest ID within the class. Since the size is fixed, the computation only needs to find the equivalence class IDs. Note: in this case, for any Band1 permutation, applying these permutations to achieve the lowest ID provides an index to the associated equivalence class.

Application of the rest of the block, column and row symmetries provided further reduction, i. When the B1 canonical labeling is lost through a transformation, the result is relabeled to the canonical B1 usage and then catalogued under this ID. This approach generated equivalence classes, somewhat less effective than the theoretical minimum limit for a full reduction.

Application of counting symmetry patterns for duplicate paired digits achieved reduction to and then to 71 equivalence classes. Each of the 44 equivalence classes can be expanded to millions of distinct full solutions, but the entire solution space has a common origin in these The 44 equivalence classes play a central role in other enumeration approaches as well, and speculation will return to the characteristics of the 44 classes when puzzle properties are explored later.

Enumerating the Sudoku solutions breaks into an initial setup stage and then into two nested loops. Initially all the valid Band1 permutations are grouped into equivalence classes, who each impose a common constraint on the Band2,3 completions.

For each of the Band1 equivalence classes, all possible Band2,3 solutions need to be enumerated. An outer Band1 loop iterates over the 44 equivalence classes. In the inner loop, all lower band completions for each of the Band1 equivalence class are found and counted. The computation required for the lower band solution search can be minimised by the same type of symmetry application used for Band1.

There are 6! At this point, completing 10 sets of solutions for the remaining 48 cells with a recursive descent, backtracking algorithm is feasible with 2 GHz class PC so further simplification is not required to carry out the enumeration. Using this approach, the number of ways of filling in a blank Sudoku grid has been shown to be 6,,,,,,, 6. The result, as confirmed by Russell, [56] also contains the distribution of solution counts for the 44 equivalence classes.

The listed values are before application of the 9! The number of completions for each class is consistently on the order of ,,, while the number of Band1 permutations covered by each class however varies from 4 — Within this wide size range, there are clearly two clusters.

The disparity in consistency between the distributions for size and number of completions or the separation into two clusters by size is yet to be examined. From Wikipedia, the free encyclopedia. Mathematical investigation of Sudoku. This article is about the mathematical analysis of Sudoku puzzles. For solving and generating algorithms, see Sudoku solving algorithms.

This article may contain excessive or inappropriate references to self-published sources. Please help improve it by removing references to unreliable sources where they are used inappropriately. September Learn how and when to remove this template message. Retrieved 20 October The Hidden Logic of Sudoku Second, revised and extended ed.

ISBN EA 5 : — Springer International Publishers, Google Discussiegroepen. The New Sudoku Players' Forum. Frazer Jarvis's home page. University of Sheffield. Retrieved 29 April McGuire, B. Tugemann, G. Retrieved 13 July Retrieved 19 October Discrete Mathematics. Archived from the original on 6 February Retrieved 3 October Journal of Combinatorial Mathematics and Combinatorial Computing.

April: 94— Retrieved 28 February Retrieved 2 October Retrieved 5 October Retrieved 22 October Retrieved 11 September Retrieved 14 September Retrieved 8 November Archived from the original on 12 October Archived from the original on 26 November Retrieved 6 January Lin, I-C. Retrieved 30 November See in particular Figure 7, p. Retrieved 14 August Archived from the original on 11 August Retrieved 25 January Retrieved 2 February Clueless Sudoku".

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She demonstrates an example with only eight relations. It is not known whether this is the best possible. The most clues for a minimal Sudoku is believed to be 40, of which only two are known. If any clue is removed from either of these Sudokus, the puzzle would have more than one solution and thus not be a proper Sudoku.

In the work to find these Sudokus, other high-clue puzzles were catalogued, including more than 6,,, minimal puzzles with 36 clues. About minimal Sudokus with 39 clues were also found. If dropping the requirement for the uniqueness of the solution, clue minimal pseudo-puzzles are known to exist, but they can be completed to more than one solution grid.

Removal of any clue increases the number of the completions and from this perspective none of the 41 clues is redundant. With slightly more than half the grid filled with givens 41 of 81 cells , the uniqueness of the solution constraint still dominates over the minimality constraint.

As for the most clues possible in a Sudoku while still not rendering a unique solution, it is four short of a full grid If two instances of two numbers each are missing and the cells they are to occupy are the corners of an orthogonal rectangle, and exactly two of these cells are within one region, there are two ways the last digits can be added two solutions.

The number of minimal Sudokus Sudokus in which no clue can be deleted without losing uniqueness of the solution is not precisely known. Other authors used faster methods and calculated additional precise distribution statistics.

It has been conjectured that no proper Sudoku can have clues limited to the range of positions in the clear space of the first image above. The largest total number of empty groups rows, columns, and boxes in a Sudoku is believed to be nine. One example is a Sudoku with 3 empty rows, 3 empty columns, and 3 empty boxes third image. A Sudoku grid is automorphic if it can be transformed in a way that leads back to the original grid, when that same transformation would not otherwise lead back to the original grid.

One example of a grid which is automorphic would be a grid which can be rotated degrees resulting in a new grid where the new cell values are a permutation of the original grid. Automorphic Sudokus are Sudoku puzzles which solve to an automorphic grid. Two examples of automorphic Sudokus, and an automorphic grid are shown below.

Since these Sudokus are automorphic, so too their solutions grids are automorphic. Notice that in the second example, the Sudoku also exhibits translational or repetition symmetry; clues are clustered in groups, with the clues in each group ordered sequentially i.

The third image is the Most Canonical solution grid. In these examples the automorphisms are easy to identify, but in general automorphism is not always obvious. The table at right shows the number of the essentially different Sudoku solution grids for all existing automorphisms.

An enumeration technique based on band generation was developed that is significantly less computationally intensive. The strategy begins by analyzing the permutations of the top band used in valid solutions. Once the Band1 symmetries and equivalence class for the partial solutions are identified, the completions of the lower two bands are constructed and counted for each equivalence class.

Pettersen [31]. The permutations for Band1 are 9! The permutations of B1 are the number of ways to relabel the 9 digits, 9! Counting the permutations for B2 is more complicated, because the choices for B2 depend on the values in B1. This is a visual representation of the expression given above. The conditional calculation needs a branch sub-calculation for each alternative. Fortunately, there are just 4 cases for the top B2 triplet r21 : it contains either 0, 1, 2, or 3 of the digits from the B1 middle row triplet r Once this B2 top row choice is made, the rest of the B2 combinations are fixed.

The Band1 row triplet labels are shown on the right. Note: Conditional combinations becomes an increasingly difficult as the computation progresses through the grid. At this point the impact is minimal. The Case 0 diagram shows this configuration, where the pink cells are triplet values that can be arranged in any order within the triplet. Each triplet has 3! The 6 triplets contribute 6 6 permutations.

The same logic as case 0 applies, but with a different triplet usage. In the Case 1 diagram, B1 cells show canonical values, which are color-coded to show their row-wise distribution in B2 triplets. Colors reflect distribution but not location or values. For this case: the B2 top row triplet r21 has 1 value from B1 middle triplet, the other colorings can now be deduced. Fill in the number of B2 options for each color, The B3 color-coding is omitted since the B3 choices are row-wise determined by B1, B2.

B3 always contributes 3! For B2, the triplet values can appear in any position, so a 3! However, since some of the values were paired relative to their origin, using the raw option counts would overcount the number of permutations, due to interchangeability within the pairing. The option counts need to be divided by the permuted size of their grouping 2 , here 2! Case 2: 2 Matches for r21 from r The same logic as case 1 applies, but with the B2 option count column groupings reversed.

Totaling the 4 cases for Band1 B B3 gives 9! Symmetries are used to reduce the computational effort to enumerate the Band1 permutations. A symmetry is an operation that preserves a quality of an object. For a Sudoku grid, a symmetry is a transformation whose result is also a valid grid.

The following symmetries apply independently for the top band:. Combined, the symmetries give 9! A symmetry defines an equivalence relation , here, between the solutions, and partitions the solutions into a set of equivalence classes.

Since the solution for any member of an equivalence class can be generated from the solution of any other member, we only need to enumerate the solutions for a single member in order to enumerate all solutions over all classes.

The Band1 symmetries above are solution permutation symmetries defined so that a permuted solution is also a solution. For the purpose of enumerating solutions, a counting symmetry for grid completion can be used to define band equivalence classes that yield a minimal number of classes.

Counting symmetry partitions valid Band1 permutations into classes that place the same completion constraints on lower bands; all members of a band counting symmetry equivalence class must have the same number of grid completions since the completion constraints are equivalent. Counting symmetry constraints are identified by the Band1 column triplets a column value set, no implied element order.

Using band counting symmetry, a minimal generating set of 44 equivalence classes [56] was established. The following sequence demonstrates mapping a band configuration to a counting symmetry equivalence class. Begin with a valid band configuration 1. Build column triplets by ordering the column values within each column. This is not a valid Sudoku band, but does place the same constraints on the lower bands as the example 2.

Construct an equivalence class ID from the B2, B3 column triplet values. Use column and box swaps to achieve the lowest lexicographical ID. The last figure shows the column and box ordering for the ID: All Band1 permutations with this counting symmetry ID will have the same number of grid completions as the original example. An extension of this process can be used to build the largest possible band counting symmetry equivalence classes 3. Note, while column triplets are used to construct and identify the equivalence classes, the class members themselves are the valid Band1 permutations: class size Sb.

Counting symmetry is a completion property and applies only to a partial grid band or stack. Solution symmetry for preserving solutions can be applied to either partial grids bands, stacks or full grid solutions. Lastly note, counting symmetry is more restrictive than simple numeric completion count equality: two distinct bands belong to the same counting symmetry equivalence class only if they impose equivalent completion constraints.

Symmetries group similar object into equivalence classes. Two numbers need to be distinguished for equivalence classes, and band symmetries as used here, a third:. The not less than and up to caveats are necessary, since some combinations of the transformations may not produce distinct results, when relabeling is required see below.

Consequently, some equivalence classes may contain less than 6 5 distinct permutations and the theoretical minimum number of classes may not be achieved. Each of the valid Band1 permutations can be expanded completed into a specific number of solutions with the Band2,3 permutations. By virtue of their similarity, each member of an equivalence class will have the same number of completions.

Consequently, we only need to construct the solutions for one member of each equivalence class and then multiply the number of solutions by the size of the equivalence class. We are still left with the task of identifying and calculating the size of each equivalence class. Further progress requires the dexterous application of computational techniques to catalogue classify and count the permutations into equivalence classes.

Block 1 uses a canonical digit assignment and is not needed for a unique ID. Equivalence class identification and linkage uses the lowest ID within the class. Since the size is fixed, the computation only needs to find the equivalence class IDs. Note: in this case, for any Band1 permutation, applying these permutations to achieve the lowest ID provides an index to the associated equivalence class.

Application of the rest of the block, column and row symmetries provided further reduction, i. When the B1 canonical labeling is lost through a transformation, the result is relabeled to the canonical B1 usage and then catalogued under this ID. This approach generated equivalence classes, somewhat less effective than the theoretical minimum limit for a full reduction. Application of counting symmetry patterns for duplicate paired digits achieved reduction to and then to 71 equivalence classes.

Each of the 44 equivalence classes can be expanded to millions of distinct full solutions, but the entire solution space has a common origin in these The 44 equivalence classes play a central role in other enumeration approaches as well, and speculation will return to the characteristics of the 44 classes when puzzle properties are explored later.

Enumerating the Sudoku solutions breaks into an initial setup stage and then into two nested loops. Initially all the valid Band1 permutations are grouped into equivalence classes, who each impose a common constraint on the Band2,3 completions. For each of the Band1 equivalence classes, all possible Band2,3 solutions need to be enumerated. An outer Band1 loop iterates over the 44 equivalence classes. In the inner loop, all lower band completions for each of the Band1 equivalence class are found and counted.

The computation required for the lower band solution search can be minimised by the same type of symmetry application used for Band1. There are 6! At this point, completing 10 sets of solutions for the remaining 48 cells with a recursive descent, backtracking algorithm is feasible with 2 GHz class PC so further simplification is not required to carry out the enumeration.

Using this approach, the number of ways of filling in a blank Sudoku grid has been shown to be 6,,,,,,, 6. The result, as confirmed by Russell, [56] also contains the distribution of solution counts for the 44 equivalence classes. The listed values are before application of the 9! The number of completions for each class is consistently on the order of ,,, while the number of Band1 permutations covered by each class however varies from 4 — Within this wide size range, there are clearly two clusters.

The disparity in consistency between the distributions for size and number of completions or the separation into two clusters by size is yet to be examined. From Wikipedia, the free encyclopedia. Mathematical investigation of Sudoku. This article is about the mathematical analysis of Sudoku puzzles. For solving and generating algorithms, see Sudoku solving algorithms. This article may contain excessive or inappropriate references to self-published sources. Please help improve it by removing references to unreliable sources where they are used inappropriately.

September Learn how and when to remove this template message. Retrieved 20 October The Hidden Logic of Sudoku Second, revised and extended ed. ISBN EA 5 : — Springer International Publishers, Google Discussiegroepen. The New Sudoku Players' Forum. Frazer Jarvis's home page. University of Sheffield. Retrieved 29 April McGuire, B.

Tugemann, G. Retrieved 13 July Retrieved 19 October Discrete Mathematics. Archived from the original on 6 February Retrieved 3 October Journal of Combinatorial Mathematics and Combinatorial Computing. April: 94— Retrieved 28 February Retrieved 2 October Retrieved 5 October Retrieved 22 October Retrieved 11 September Retrieved 14 September Retrieved 8 November Archived from the original on 12 October Archived from the original on 26 November Retrieved 6 January Lin, I-C.

Retrieved 30 November See in particular Figure 7, p. Retrieved 14 August Archived from the original on 11 August City Racing Games. All Racing Games. Cooking Games. All Simulation Games. For you. Join for free. No games found. View more results. Learn More. We have other games that don't require Flash. Here's a few of them. Hey, don't go yet! Check out these awesome games! Add to Favorites. Login or Join now to add this game to your faves. Full screen. A challenging twist injects extra complexity to classic sudoku.

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