These form a family of techniques specific to Killer Sudoku. JSudoku will search and solve them. When only one cell remains, its value is immediatly set. When several cells remains, a split cage is created. It will be displayed with a transparent colored background unless you turn off Options > Show Split Cages.
You may also use them for manual solving. JSudoku will make most of the tedious tasks for you: computing the sum, checking for combinations, taking care of numbers which may repeat or not... Leaving you the fun part of thinking, finding your way into the puzzle. You should first select with the mouse all the cells in the house(s): rows, columns, nonets... Then choose the appropriate command from the Solve menu.
In this example, the solver will compute the value of the innies of N3 -> R3C7 = 45 - (17+17+9) = 2. Since only one cell remains, its value is immediatly set.
In this example, the solver will compute the value of the innies of two adjacent nonets: Innies of N23 -> R3C4 = (2*45) - (6+9+12+17+17+11+9) = 9. Again only one cell remains, so its value is set to 9. As a result, it updates the combination and restrict R4C4 to 6. Here the option Solve all Naked Singles after Deduce was off. If it were on, it would also have set R4C4 to 6.
Here we compute the innies of two columns : C12. Notice that the two cells at R89C3 are now restricted to {234678}. Indeed, they add up to 19-9 = 10 which can't contain a 5.
In this example, the innies of N2 generates a split cage. The log will list : Innies of N2 -> R2C4+R3C4 = 3. The possibilities of these two cells are then restricted to the possible combination(s) for that sum, in this case {12}.
In this example, the innies of N23 generates a split cage over 3 cells. Although the cells are from differents cages, they may not hold any repeated numbers since they are all within the same nonet.
In this example, the innies of N12 generates a complex cage which sum is 10. Since the two resulting cells are in different rows, columns, nonets and cages, they may also hold repeated numbers. 5+5 is a valid combination here.
Outies are the counter part of Innies. The solver will compute the value of the outies of N1 -> R4C2 = (10+15+9+12) - 45 = 1. Since only one cell remains, its value is set.
As for Innies, we may select several adjacent groups. Here the two rows R12.
Or three adjacent nonets: N124.
Note : it would have been simpler and more efficient to compute innies and outies step by step :
Previouly defined split cages are also used to compute innies-outies when appropriate, as are the cells already solved and naked subsets.
At step 3, computing the innies of N1 will make use of the split cage 16/2 in R23C3 created at step 2.
At step 4, computing the outies of N1 will make use of the split cage 3/2 in R23C4 created at step 1.
At step 5, computing the outies of N4 will make use of the split cage 5/2 in R3C12 created at step 3.
As for Innies, the outies may result in complex cages with possible repeated numbers. Here the two cells add up to 12, which can be 6+6 since the cells are in different rows, columns, nonets and cages.
This solver will compute the other half of a split cage. In this example, splitting the cage 19/4 gives: R23C3 = 19-3 = 16
This is a mix of innies and outies which may result in a difference of cells. For each cage crossing the house(s), the solver will keep the innies or outies with the least number of unsolved cells.
In this example, the outies minus innies of N8 give us the difference of two cells: (15+18+11) - 45 = -1 (a negative number). This tell us that R9C7 (outies) - R7C6 (innies) = -1. Which is equivalent to R9C7 = R7C6 - 1 and also R7C6 = R9C7 + 1.
The Odd Combinations solver will later deduce that R9C7={1..8}, R7C6={2..9}. Here the difference is quite small, hence not "much" useful.
In this example, the outies minus innies of R6789 give us the difference of three cells: R45C6 - R6C2 = -5. Which is equivalent to R45C6 = R6C2 - 5 and also R6C2 = R45C6 + 5.
The Odd Combinations solver will later deduce that R45C6={123}, R6C2={89}.
This can also be deduced by computing the valid ranges of the two parts by propagating their minimum and maximum:
The maximum of one cell is 9, so R6C2 is at maximum 9. Therefore, the maximum of R45C6 = 9-5 = 4 = {13}.
The minimum of R45C6 is {12} = 3 (numbers cannot repeat here). Therefore, the minimum of R6C2 = 3+5 = 8.
This kind of deduction is not implemented as such, but the Odd Combinations solver will give the same result.
When a crossing cage has an equal number of cells in and out, the solver will always choose the outies. It will not consider other cases involing some of the innies. In this example, computing the outies minus innies of R12 will choose the outies R34C9 since the cage 23/4 in C9 has two cells in and two out. It will not consider the other case with the innies R12C9. Outies minus innies of R12 gives: R3C1+R34C9-R2C2 = 0, not really useful here.
However, it is sometimes possible to compute the other case, by seleting the complementary group of houses. Outies minus innies of R3..9 gives: R2C2+R12C9-R3C1 = 23. Much more useful !
This is a more general technique based on the "45" rule where houses of various kind may be mixed. The principle uses addition and subtraction of cells in groups and cages.
In this example there are 3 overlapping cells at the intersection of the top row and top center nonet. Select with the mouse all cells in R1. Then, while pressing the alt key, extend the selection to include also all cells in N2. Then choose the Overlaps form the Solve or Highlight menu.
We can easilly compute the sum of these 3 cells:
R1C456 = 90 - (17+23+27) = 90 - 67 = 23.
Therefore, these 3 cells in green must be {689}
Here is how it works:
So the Overlaps gives the formula: R1C4+R1C5+R1C6 = 23.
In this example, there are three overlapping groups: R1,C1 and N1. The Overlaps will give us a formula where R1C1 becomes counted twice and R1C23+R23C1 are counted once.
Here is how it works:
So the Overlaps gives the formula: 2*R1C1+R1C2+R1C3+R2C1+R3C1 = 43.
Overlaps will do its best trying to keep the number of resulting cells as low as possible, using the same principle as Innies minus Outies. Using previously defined split cages, solved cells and naked subsets if appropriate. The result may also mix some innies and/or outies.
In this example, we have a mix of Overlaps and Innies. We first computed the 4 split cages in blue:
Now we may compute the Overlaps of R6 and C6 to get the sum of the three cells in yellow:
So the Overlaps gives the formula: R2C6+R6C6+R6C2 = 25
An even more complex use of Overlaps may gives a formula with both innies and outies, like in Killer #9 by Nate Dorward.
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