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Killer Sudoku Solving Strategies

Solving puzzles involves using the normal Sudoku rules plus additional strategies designed specifically for Killer Sudoku puzzles. Some of the strategies are listed below.

Single Combinations

To start solving a Killer Sudoku puzzle it is easiest to identify any cages that have a single combination. This is because, even though each cell value may not be known, the values may still be used to remove options from other cages that are in scope. For example, in the following diagram the 2 cell cage, with a sum of 4, only has a single combination of 1, 3. The three-cell cage, with the sum of 11, has 5 possible combinations, which encompass the cell values 1 through to 7. Since only one solution excludes the values 1, 3, then it can be deduced that the cell values in the sum 11 cage are 2, 4 and 5. Note the 6 and 7 values could be removed, since these were parts of combinations including 1 or 3. Furthermore the 4 cell part of the 5 cell cage, with a sum of 36, must contain the remaining numbers i.e. 6,7,8 and 9.

Innies & Outies

"Innies" are partial cage cells contained within a row, cage or block. Whereas "Outies" are partial cage cells outside of a row, column or block.

A useful fact in solving Killer Sudoku puzzles is that each row, column, or block must add up to 45. This knowledge can be exploited to calculate sums of some cell combinations that are not given. For example the "Innie" and "Outie" sums in the above diagram can be calculated as follows:

  • "Innie" Sum = 45 - (27 + 13) = 5
  • "Outie" Sum = 27 + 13 + 14 - 45 = 9

Although this example uses a block, the same principle can be used for rows and columns too.

The "Innie" and "Outie" calculations, described above for a single block, can be extended to include multiple rows, columns or blocks. The "Innie" and "Outie" calculations can be generalised as:

  • "Innie" Sum = 45 x N - sum1 ... where sum 1 equals the sum of all complete cages contained within the row, column, or block and N equals the number or rows, columns or blocks
  • "Outie" Sum = sum 2 - 45 x N ... where sum 2 equals sum of all intersected cages and N equals the number or rows, columns or blocks

Innie & Outie Pairs

The diagram below shows an "Innie" and "Outie" pair (highlighted A and B respectively). Using the 45 rule the following must be true:

  • 27 + A + 24 - B = 45

Rearranging this equation gives:

  • B - A = 27 + 24 - 45 = 6

In general the following must be true:

  • sum1 = sum of cages before the "Innie"
  • sum2 = sum of cages after the "Innie"
  • B - A = sum1 + sum2 - 45 ... where sum1 + sum2 > 45
  • A - B = 45 - (sum1 + sum2) ... where sum1 + sum2 < 45

This rule can also be applied to multiple rows or columns. In this case 45 must be replaced with 45 x N in the above equations (where N equals the number of rows or columns).

Rule of Necessity

The partial Killer Sudoku puzzle illustrated shows a mix of some cells being filled in, some marked up with possible values and others blank.

The sum 24 cage in the first column of the bottom block contains a 7, 8 and 9. Since a column can contain values only once, then this implies that these values can't be present in the first column of the two blocks above. Also since there is an 8 in the third column of the middle block, it means that this value can't be present in the third column of the block above and below it. This means that an 8 must exist only in the middle column of the top block. However none of the combinations in the sum 12 cage in the first block include 8 and so it follows that an 8 must exist in the second column of the sum 21 cage. There are three combinations that sum to 21, but only two include the value 8. Further 7, 8 and 9 can be excluded from the first cell of this cage. The diagram is annotated with the resulting possible cell values for the sum 21 cage.

Using Include OR

The following partial Killer Sudoku puzzle includes a 4 cell cage with a sum of 19.

At first this would appear to have 11 possible combinations. However the "Innie" in the second block is equal to a sum of 5 (i.e. 45 - (35 + 5)), which only has two combinations. Using this knowledge, the 19 cage combinations can be reduced from 11 down to 4, since these are the only combinations that include either (1,4) or (2,3). Furthermore it can be deduced that the two cells of the 19 cage in the first block must contain either (5,9) or (6,8).

Another example is shown below. Here the top two cells of the 23 cage is known to contain either 3, 5 or 9.

There are 3 combinations of numbers that could satisfy this condition, namely (3,5), (5,9) or (3,9). Therefore the 9 combinations can be reduced down to 5, since these are the only combinations that include one of these sets.

Combinations with Duplicates

Cells C4, D2 and D3 represent the "outie" cells of the first block. Their sum can be calculated as follows:

  • Sum = 19 + 6 + 16 + 28 - 45 = 24

The combinations without duplicates that add up to this sum are 768. However cell C4 can't see cells  D2 and D3 and therefore duplicates are possible. The combinations with duplicates that add up to  24 are 699, 789 and 888. Clearly 888 can't be the solution as this would result in both D2 and D3 being the same value. However 699 is a legal solution, which would result in D2 = 9 (can't be  6 as the only legal values are 8 or 9), D3 = 6 and C4 = 9.

Therefore there are two legal combinations for C4, D2 and D3 namely:

  • 789 or
  • 699




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