Yet, 5 cells within the same cage totaling 25 has twelve possible combinations. In the early stages of the game, the most common way to begin filling in numbers is to look at such low-sum or high-sum cages that form a 'straight line'.

As the solver can infer from these that certain numbers are in a certain row or column, they can begin 'cross-hatching' across from them. A further technique can be derived from the knowledge that the numbers in all houses rows, columns and nonets add up to By adding up the cages and single numbers in a particular house, the user can deduce the result of a single cell.

If the cell calculated is within the house itself, it is referred to as an 'innie'; conversely if the cell is outside it, it is called an 'outie'. Even if this is not possible, advanced players may find it useful to derive the sum of two or three cells, then use other elimination techniques see below for an example of this.

A short-cut to calculating or checking the value of a single 'innie' or 'outie' on a large number of cages is to add up the cages using 'clock' arithmetic correctly, Modular Arithmetic modulo 10 , in which all digits other than the last in any number are ignored.

When two numbers are added together, the last digit of the total is not affected by anything other than the last digits of the two original numbers.

Adding together a number ending in 7 and a number ending in 8 always results in a number ending in 5, for example. The biggest number an 'innie' or 'outie' can hold is 9, so adding or subtracting that value will change the last digit of the total in a way that no other value would - allowing the 'innie' or 'outie' to be directly calculated.

Example: A set of cages form a complete nonet with an 'outie'. Clock arithmetic has the additional bonus that, when the final digits of two cage totals add up to 10 1 3 and 2 7 , for example , the pair will make no difference to the overall clock total, and can simply be skipped.

Clock arithmetic should at most be used with caution for houses with more than one 'innie' or 'outie', when more than one set of values may result in the same final number, but may still be useful as a quick arithmetic check.

Even though some cages can have multiple combinations of numbers available, there can often be one or more numbers that are consistent within all available solutions.

For example: a 4 cell cage totaling 13 has the possible combinations of 1, 2, 3, 7 , 1, 2, 4, 6 , or 1, 3, 4, 5. Even though, initially, there is no way to tell which combination of numbers is correct, every solution available has a 1 in it.

The player then knows for certain that one of the numbers within that cage is 1 no matter which is the final solution. This can be useful if, for example, they have already deduced another cell within a nonet the cage resides in as having the number 1 as its solution.

They then know that the 1 can only reside in cells that are outside of this nonet. If there is only one cell available, it is a 1.

The 1 cannot be in the top line as that conflicts with our first 2 cells therefore the top cell of this pair is 3 and the lower cell 1.

The four cells in the top right cage totaling 15 can only include one of 1, 3, 7, or 9 if at all because of the presence of 1, 3, 7, and 9 in the top right hand nonet.

If any one of 1, 3, 7, or 9 is present then this must be the lone square in the nonet below. Looking at the nonet on the left hand side in the middle, we can see that there are three cages which do not cross over into another nonet; these add up to 33, meaning that the sum of the remaining two cells must be This does not seem particularly useful, but consider that the cell in the bottom right of the nonet is part of a 3-cage of 6; it can therefore only contain 1, 2 or 3.

If it contained 1 or 2, the other cell would have to contain 11 or 10 respectively; this is impossible. It must, therefore, contain 3, and the other cell 9.

With 6-cell, 7-cell or 8-cell cages, correlating the combinations with their 3-cell, 2-cell, or 1-cell complements usually simplifies things. The table for 6 cell cages is the complement of the 3 cell table adding up to 45 minus the listed value; similarly, the 7 cell table complements the 2 cell table.

The other 3 cages are completely located inside the nonet. On the left, only the 3 cages are shown that are fully inside the nonet. The missing cell is the innie.

The cell sticking out of the nonet is the outie. In this particular situation, it does not really matter if you decide to calculate the innie or the outie, because there is one of each.

This is not always the case. You must be able to check both possibilities. There is a single cage that partially overlaps this row. It has 3 cells, 2 outside the row, and 1 inside the row.

You can now calculate the innie, but only the sum of the 2 outies. This row has a single outie. By now, you must be able to do these calculations yourself.

By themselves, these two rows have little to offer in a 45 test. Too many innies and outies lie between them. However, when we combine the two rows, a different picture emerges.

There is now only a single innie. Both rows must contain digits 1 through 9, so the sum of all cells in these 2 rows equals Sometimes the innies and outies do not present themselves so easily.

You have to search a little more and try a few combinations of nonets. This outie can only be seen when these 3 nonets are combined into this L shape. Not only are they difficult to see, but because you have to add all the cage sums for these 3 nonets, the chance of making a calculation error is highly increased.

If solving the killer would depend on this outie, the puzzle would have a higher difficulty rating for it. To do these calculations yourself, add all cage sums together and subtract from the result.

This gives you the digit to place in the outie. You can perform these 45 tests right at the beginning. In a later stage, you should watch out for new 45 test that may become available, because the number of innies and outies are reduced by placements.

Here is an example that shows how this happens:. The 8 placed inside this nonet allows us to do a new 45 test to determine the placements in the 9[2] cage. When there are multiple innies or outies, there may still be an opportunity to do some placements.

For this, you need to be on the lookout for pairs of innies or outies that have a minimum or maximum difference. Here is an example:. The 19[3] cage no longer contributes outies, because of the 3 placed inside the nonet.

This leaves 2 outies. A sum of 2 can only mean that both outies contain digit 1. Because they do not share a row, column or nonet, this is allowed for these two outies.

The innie for the 7[2] cage will be 6 and the innie in the 10[2] cage will receive a 9. Here is a practice puzzle for innies and outies. It starts with the outer columns, and works its way to the interior.

Do not forget to check the nonet boundaries. The sum value on the cages is not only useful for adding and subtracting, it can sometimes tell us immediately what digits the cage contains.

A cage with sum 3 and size 2 written as 3[2] in this manual can contain digits 1 and 2. Nothing else fits. This is very similar to naked subset reductions. When the entire cage lies within a row, column or nonet, the cage can also be seen as a naked subset within that house.

This is a term that is often used in sudoku solving guides. Two cells that belong to the same row , column or nonet cannot both have the same value. In killer sudoku, according to the killer convention, two cells that belong to the same cage can also see each other.

Alternatively, these cells are called buddies or peers. There are some cage sums that leave a choice of digits, but some of the digits are always part of the configuration. These cages allow us to eliminate candidates for those digits outside the cage.

Here are a few examples. Unfortunately, there are no almost minimum or maximum cages of size 2. It is rare to find larger cages inside a single house, making it unlikely that you can perform reductions for these larger cages, but in harder puzzles, these may be the key to solving it.

Killer sudoku with many cages of size 2, or with little elbow cages of size 3 offer an opportunity to reduce a large part of the puzzle to pairs.

You start with a minimum or maximum cage and perform the reductions, which, in turn, will reduce the number of possible configurations in other cages, having a ripple effect throughout the puzzle.

Start with the two 17[2] cages. They eliminate candidates 8 and 9 from the remainder of the first two rows. Then look at the 12[2] cage. Eliminate 5 and 7 in the third nonet.

This cage is neatly aligned with the third row, so you can eliminate these 3 digits from the remainder of that row. At this stage, you need input from other parts of the puzzle to continue.

In easy or gentle killers, working the pairs in combination with a few 45 placements is all you need to solve the puzzle. Because this can be done so quickly, some players consider Killer Sudokus easier to solve than regular Sudokus.

Here is the complete puzzle that we used in our little demonstration. You should now be able to solve this puzzle working through the pairs, and using an occasional 45 test. When you are working the pairs, it is good practice to check for opportunities using regular sudoku solving techniques.

Hidden and naked singles often emerge after a few reductions, but line-box interactions can also become available as more and more candidates are eliminated. Because so many cells are reduced to pairs and triples, you can also find naked pairs and triples, which are not confined to a single cage.

The 45 test is a versatile tool that can be used in many ways. You've already learned how to use this tool to find single innies and outies. Here are a few other ways to use the 45 test.

This is not something that you find in your everyday Killer Sudoku. A construction like this is likely to have been placed here on purpose by the maker of the puzzle, requiring you to find and use it.

Here is a group of cages that covers exactly 1 row and 1 column. All digits of the row add up to 45, and so do all digits in the column.

However, we will have counted the cell in the intersection of the row and column twice. This is the digit that goes into the cell at the intersection. In theory, it is possible to extend this to multiple rows and columns.

For practical purposes, 1 row and 2 columns or 2 rows and 1 column are still useful, but more complex combinations have a too large intersection to yield any useful information. Here we have a nonet with 3 innies.

This can be the result when you're working the pairs for a while. Because they are a naked pair, we know what sum these 2 cells will have together, even if we don't know their individual values.

When larger cages cross the boundary of a 45 test area, it can be useful to split these cages in two parts. Both parts can be treated as separate cages, but the constraint no repeats of the original cage also applies.

This gives you 3 times the solving power of the original cage. The 10[4] cage lies exactly in the middle of the 2 nonets. Thus, the useles quad has been split into two very productive pairs.

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The biggest number an 'innie' or 'outie' can hold is 9, so adding or subtracting that value will change the last digit of the total in a way that no other value would - allowing the 'innie' or 'outie' to be directly calculated.

Example: A set of cages form a complete nonet with an 'outie'. Clock arithmetic has the additional bonus that, when the final digits of two cage totals add up to 10 1 3 and 2 7 , for example , the pair will make no difference to the overall clock total, and can simply be skipped.

Clock arithmetic should at most be used with caution for houses with more than one 'innie' or 'outie', when more than one set of values may result in the same final number, but may still be useful as a quick arithmetic check.

Even though some cages can have multiple combinations of numbers available, there can often be one or more numbers that are consistent within all available solutions.

For example: a 4 cell cage totaling 13 has the possible combinations of 1, 2, 3, 7 , 1, 2, 4, 6 , or 1, 3, 4, 5. Even though, initially, there is no way to tell which combination of numbers is correct, every solution available has a 1 in it.

The player then knows for certain that one of the numbers within that cage is 1 no matter which is the final solution. This can be useful if, for example, they have already deduced another cell within a nonet the cage resides in as having the number 1 as its solution.

They then know that the 1 can only reside in cells that are outside of this nonet. If there is only one cell available, it is a 1.

The 1 cannot be in the top line as that conflicts with our first 2 cells therefore the top cell of this pair is 3 and the lower cell 1. The four cells in the top right cage totaling 15 can only include one of 1, 3, 7, or 9 if at all because of the presence of 1, 3, 7, and 9 in the top right hand nonet.

If any one of 1, 3, 7, or 9 is present then this must be the lone square in the nonet below. Looking at the nonet on the left hand side in the middle, we can see that there are three cages which do not cross over into another nonet; these add up to 33, meaning that the sum of the remaining two cells must be This does not seem particularly useful, but consider that the cell in the bottom right of the nonet is part of a 3-cage of 6; it can therefore only contain 1, 2 or 3.

If it contained 1 or 2, the other cell would have to contain 11 or 10 respectively; this is impossible. It must, therefore, contain 3, and the other cell 9. With 6-cell, 7-cell or 8-cell cages, correlating the combinations with their 3-cell, 2-cell, or 1-cell complements usually simplifies things.

The table for 6 cell cages is the complement of the 3 cell table adding up to 45 minus the listed value; similarly, the 7 cell table complements the 2 cell table. An 8-cell cage is of course missing only one digit 45 minus the sum of the cage.

As a 2-cell cage totalling 4 can contain only 1 and 3, we deduce that a 7-cell cage totalling 41 contains neither 1 nor 3.

From Wikipedia, the free encyclopedia. Arithmetical puzzle game. Not to be confused with "killer"-level i. This article does not cite any sources. Please help improve this article by adding citations to reliable sources.

Unsourced material may be challenged and removed. December Learn how and when to remove this template message. Categories : Sudoku Logic puzzles. You will learn a variety of new solving techniques, with many opportunities to practice the new techniques you've learned.

There is also an overall strategy that you will learn, but as you gain experience, you may develop a better approach to solving Killer Sudokus. A Killer Sudoku is a puzzle with 9 rows of 9 cells each.

Vertically, there are 9 columns. This grid of 81 cells is also divided in 9 nonets of 3x3 cells. These groups are also called regions , blocks , or boxes.

Rows are numbered from top to bottom, columns from left to right and nonets 1,2,3 for the top layer, 4,5,6 for the middle layer and 7,8,9 for the bottom layer.

You can recognize the nonets by the thicker or darker borders separating them. Together, we call rows, columns and nonets houses , as they all have the same constraint of requiring digits 1 though 9.

The terms unit or group are sometimes used in stead of house. The dotted shapes with a number in the left top corner we call cages. A cage encloses between 1 and 9 cells.

Sometimes a cage of size 1 is not shown as a cage, because we already know what digits goes into the cell. This cell then contains a given digit.

The number in the left top is the cage sum , or simply the sum. Cages are identified by their sum, size and location. Sometimes the size is omitted, when there is only a single cage with that sum.

The mission in solving a killer is to place a digit in each of the cells in such a way that each house contains all digits 1 though 9, and the sum of the digits within each cage equals the cage sum.

Once you have accomplished that feat, you have found the solution. In this guide, there is a clear distinction between placed digits and candidates. The candidates are the remaining possible digits for a cell.

When a placement or a solving technique causes certain candidates to become invalid, we say that these candidates are eliminated or removed. The process of eliminating candidates is also called reduction.

This does not tell us which digit goes into which cell, but it limits the number of candidates for each cell within the cage.

A digit that cannot be used in a valid combination is an obsolete digit or candidate. A digit that is found in every possible combination is a mandatory digit.

This killer solving guide is written with the killer convention in mind. This convention is used by most killer publishers, after it was originally introduced by the Times newspaper. Each digit is unique within a cage, even when repeats would be allowed by normal sudoku rules.

This convention allows us to treat all cages in the same way. It also narrows down the number of digit combinations. Further more, the maximum size for a cage is thus limited to 9, including all digits from 1 through 9.

Unless you have a photographic memory, you should use pencilmarks to write down which values go into each cell. More than with regular sudokus, solving a killer is often achieved by long series of candidate eliminations.

In regular sudoku, a few of those steps appear in the 2 or 3 bottlenecks that a difficult sudoku has, but killers seem to require candidate eliminations all the way to the end.

Each variation of sudoku has this line: All the techniques that you would use for a regular sudoku do also apply. This being true, do not expect many advanced regular sudoku techniques in a killer.

It is just too difficult to create a killer with advanced techniques like swordfish or coloring. There are a few techniques that can be used in killers on a regular basis:.

The use of uniqueness test requires a warning. You can only perform them if the 4 cells of the unique rectangle are located in 2 rows, 2 columns, 2 nonets and 2 cages.

This happens a lot in killers, making this technique very useful to learn and apply. X-Wings can occur when two aligned size 2 cages are reduced to a single pair, sharing a digit.

Naked subsets can also occur inside a cage. When this happens, the remainder of the cage can be cleared of those candidates.

In odd-shaped cages, a naked pair inside a cage can cause eliminations outside the cage, when there are cells that share a house with all cells of the subset. You need to perform a lot of little calculations when solving a killer, making this type of puzzle hated by one group of people and loved by others.

Use any tool you want when doing these calculations, but beware: errors in calculations are catastrophic when it comes to solving a killer.

A single digit off will send you in a completely wrong direction, with no means to trace back to the error point. Some players measure the difficulty of a puzzle by the number of times they had to start over.

When you're not playing against the clock, it may be a good idea to recheck your calculations before you act upon them. When you gain more experience, you will learn which calculations require a recheck.

The numbers 1 through 9, when added together, sum up to You know that each row, column and nonet requires digits 1 though 9, so the sum of each house is The sum of 2 adjacent rows is You also need to know 3 x 45 and 4 x 45 This is all you need.

For 5 rows, you can also test the remaining 4 rows. There are only 9 rows, columns and nonets, after all. When you look at the cages located inside a house, you often find some cages that are only partially located inside that house.

A part of the cage sticks out. As killer solvers, we are very interested in innies and outies, because they are the most important tools to solve the puzzle.

This is a nonet with a single cage that has a partial overlap. The other 3 cages are completely located inside the nonet. On the left, only the 3 cages are shown that are fully inside the nonet.

The missing cell is the innie. The cell sticking out of the nonet is the outie. In this particular situation, it does not really matter if you decide to calculate the innie or the outie, because there is one of each.

This is not always the case. You must be able to check both possibilities. There is a single cage that partially overlaps this row.

It has 3 cells, 2 outside the row, and 1 inside the row. You can now calculate the innie, but only the sum of the 2 outies. This row has a single outie. By now, you must be able to do these calculations yourself.

By themselves, these two rows have little to offer in a 45 test. Too many innies and outies lie between them. However, when we combine the two rows, a different picture emerges. There is now only a single innie.

Both rows must contain digits 1 through 9, so the sum of all cells in these 2 rows equals Sometimes the innies and outies do not present themselves so easily. You have to search a little more and try a few combinations of nonets.

This outie can only be seen when these 3 nonets are combined into this L shape. Not only are they difficult to see, but because you have to add all the cage sums for these 3 nonets, the chance of making a calculation error is highly increased.

If solving the killer would depend on this outie, the puzzle would have a higher difficulty rating for it. To do these calculations yourself, add all cage sums together and subtract from the result.

This gives you the digit to place in the outie. You can perform these 45 tests right at the beginning. In a later stage, you should watch out for new 45 test that may become available, because the number of innies and outies are reduced by placements.

Here is an example that shows how this happens:. The 8 placed inside this nonet allows us to do a new 45 test to determine the placements in the 9[2] cage.

When there are multiple innies or outies, there may still be an opportunity to do some placements. For this, you need to be on the lookout for pairs of innies or outies that have a minimum or maximum difference.

Here is an example:.

Relax with ten fresh categories every day! To complete a puzzle you must fill the board with numbers that obey traditional sudoku rules AND make each outlined region sum to a specific value. However, for those of you who want something harder to work on during the week, we also have a harder killer board here. A partially solved puzzle. When applying the above rule to rows, columns or regions, also remember that you can extend it to cover two or three complete regions, with sums totaling 90 or etc. Add to Wishlist. Account Options Sign in.

New feature: cross out items how to play killer sudoku a combination iiller by tapping on them. How to play killer sudoku logic game Sudoku. Beware, these puzzles can be click hard! An unsolved killer sudoku puzzle. The more extreme high or low a sum is, the fewer possible combinations of numbers there will be. Solutions to the weekly puzzle are posted a week later, also in the archives. First try to find cages with only one cell. You can also play Sudoku on Facebook.

Conceptis SumSudoku. You can how to play killer sudoku make combination lists that float above the hoq how to play killer sudoku read article be rearranged on the screen! When applying the above rule to rows, columns or regions, also remember that you can extend it to cover two or three complete regions, with sums totaling 90 or etc. Account Options Sign in. There are also more than three thousand puzzles available to purchase, but the ten free puzzles might just keep you busy for a long time : All our killer sudoku puzzles have a unique solution. Simple and addictive! Happy puzzling!

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