Killer Solving Guide

Last update: June 22, 2006

Introduction

With the introduction of Killer Sudoku, you can train both your logic skills and your mental arithetics. This guide introduces you to the strange world of Killer Sudokus. 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.

But first you need an introduction in the language of Killer Sudoku.

Terminology

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. Nonet is a fancy word for “group of 9”. 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 “cage 9[2] in N5” is the cage with sum 5, has 2 cells and lies within nonet 5.

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.

The possible digit combinations within a cage are written in curly brackets, like {1,2,4}. 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.

The Killer Convention

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.

Use pencilmarks

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.

Regular sudoku techniques

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:

• Hidden singles
• Naked singles
• Line-box interactions (a.k.a locked candidates or pointing pairs)
• Naked subsets (often aligned with a cage)
• Hidden subsets
• X-Wing
• Uniqueness test

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.

Little Arithmetics

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.

Practice the table of 45

The numbers 1 through 9, when added together, sum up to 45. You know that each row, column and nonet requires digits 1 though 9, so the sum of each house is 45. The sum of 2 adjacent rows is 90. You also need to know 3 x 45 (135) and 4 x 45 (180). 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.

Innies and outies

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. Now when you omit such a cage, the cells from that cage inside the house are called “innies”. When on the other hand you include that cage in your calculations, the cells sticking out of the house are called “outies”. 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 sum of these 3 cages equals (13+14+16)=43. The missing cell is the innie. It must contain (45-43)=2 On the right, you can see all cages that are fully or partially inside the nonet. The sum of all cages now equals (14+13+16+5)=48. The cell sticking out of the nonet is the outie. It must contain (48-45)=3. 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.

Here is an example of an innie in a row.

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. Innie calculation: 45-(16+10+14) = 5, leaving 15 for the 2 outies. As a result, the outies can only be {69} or {78}.

And also an example of a row with an outie.

This row has a single outie. By now, you must be able to do these calculations yourself. This is what you should have found: (23+18+8)-45=4.

Using multiple rows.

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 90. The sum of all cages fully inside these two rows equals (9+11+16+10+9+8+15+7)=85, so the innie must be (90-85)=5.

Using multiple nonets.

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 135 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. We can now add (10+19+6+8)=43. The innie of the 9[2] cage will be (45-43)=2, the outie will be 7.

Innie or Outie pairs

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. When adding the cages sums, we get (9+3+11+7+10+7)=47. (47-45)=2 for both 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.

Innie & Outie Practice

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 following code can be copied and pasted in SumoCue:

SumoCueV1=3=18+1=23+3=7+5=21+7+0=11+1=12+3=7+5=9+7=16+10=13+12=5+14=13+16=5+18=9+20=10+22=7+24=15+26=10+28=7+30=9+32=10+34=9+36=5+38=10+40=13+42=11+44=13+46=13+48=9+50=10+52=15+54=16+56=6+58=24+60=11+62+64+64+66+66+68+68+70+70+62

Minimum and Maximum Cages

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. A cage 4[2] can only contain digits 1 and 3, because 2+2 is not a valid combination.

Here are all cage sums upto size 4 that only have a single configuration:

• 3[2] = {1,2}
• 4[2] = {1,3}
• 16[2] = {7,9}
• 17[2] = {8,9}
  
• 6[3] = {1,2,3}
• 7[3] = {1,2,4}
• 23[3] = {6,8,9}
• 24[3] = {7,8,9}
  
• 10[4] = {1,2,3,4}
• 11[4] = {1,2,3,5}
• 29[4] = {5,7,8,9}
• 30[4] = {6,7,8,9}

You can eliminate these candidates from all cells that can be ‘seen’ by all members of the cage. 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.

Almost Minimum and Maximum Cages

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.

• 8[3] = {1,2,5} or {1,3,4}
• 22[3] = {5,8,9} or {6,7,9}
  
• 12[4] = {1,2,3,6} or {1,2,4,5}
• 13[4] = {1,2,3,7} or {1,2,4,6} or {1,3,4,5}
• 27[4] = {3,7,8,9} or {4,6,8,9} or {5,6,7,9}
• 28[4] = {4,7,8,9} or {5,6,8,9}

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.

Work the Pairs

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. This stage of killer solving is commonly known as “working the 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.

Here is an example, so you can have a taste for this technique.

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. Because 8 and 9 are out of the running, it can only contain {5,7}. Eliminate 5 and 7 in the third nonet. Now the 6[2] cage can no longer contain {1,5}, so it must be {2,4}. The 10[3] cage is reduced to {1,3,6}. This cage is neatly aligned with the third row, so you can eliminate these 3 digits from the remainder of that row.

After a few more reductions, this is the result:

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. That is certainly true for these easier types, but you haven’t finished reading yet!

Working the Pairs Practice

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.

The following code can be copied and pasted in SumoCue:

SumoCueV1=8+0=17+2=14+4=12=6+7=10=8=5+11=15+4+6=17+16+9+10=13+20+13+13=10+24+24=14=16+28=10+30=13+32=11+34+27+27+28=17+39=13+41=5+43=15+45+45+39=9+49=11+51=17=11+54=5+56=10+58=14+53+53=11+63=7=13=6=12+60+60=12=15+72+65+66+67+68=3+78+71

Regular Sudoku Techniques (reprise)

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.

Advanced 45 Tests

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.

Intersection of a single row and column.

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. Now when we add all sums of these cages together: (21+13+22+18+15)=89, the difference 90-89=1. 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.

Using naked subsets in a 45 test.

Here we have a nonet with 3 innies. Normally, we would ignore this situation, but 2 of these innies form a naked pair {17}. 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. Here are the calculations for the third innie: 45-(12+10+7+(1+7))=8.

Cage Splitting

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.

Here is an example:

The 10[4] cage lies exactly in the middle of the 2 nonets. 22+9+8=39, so the left nonet receives a 6[2] part and the right nonet receives the 4[2] part. Since the 10[4] cage can only contain {1,2,3,4}, there is only one way to split these candidates: {2,4} and {1,3}. Thus, the useles quad has been split into two very productive pairs.

To be continued...