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[UP: Killer Solving Techniques]
[PREV: Extended Innies and Outies]

Multi Group Innies and Outies

So far, we've only looked at innies and outies resulting from a single group. An obvious extension of this is to put multiple groups together, to form the basis for an innie or outie. If you take any two neighbouring groups of the same type (two rows, two columns or two boxes) and add up all the cells in them, the sum must be 90. If you take three, the sum must be 135 and so on. If you can find one cell left over from this area, then you have an innie or an outie.

Here is an example of a Multi Group Innie:

In this example, the digit "1" can be placed in the cell highlighted in red. The highlighted area is formed from two boxes, so all the highlighted cells together must add up to 90. When we add up the empty cages and the filled cells highlighted in orange, we find that they sum to 89. So the left over cell, highlighted in red, must contain a "1".

Obviously, the extended form of innies and outies can be used too. In this example, there is an extended multi group outie:

Here, we can remove the candidates 2, 4 and 7 from the cells highlighted in red. The cells highlighted in orange form two complete rows, so they must add up to 90. If we add up the sums of all the highlighted cages, the result is 104. This means that the two cells in red must add up to 14. Neither cell can contain a "9" or a "5", because the already share a row with both, so the only combination for the two cells is "6" and "8". All the other candidates can be removed from both cells.

Finally, it is sometimes possible to use a Superset Cull on the cells of the group containing the outie. Here is an example:

Here, we can remove the candidates "6" and "8" from the cells highlighted in red. This is the superset cull of the previous example. We know (from the logic of the previous example), that the empty, white cells on the third row form an outie of the rows highlighted in orange, and must contain the digits "6" and "8". We can therefore eliminated these candidates from the remainder of the row.

In this case, the same result could have been achieved using an extended innie of the first three rows, but this isn't always the case. If, for example, the outie cells all belonged to a different type of group (a box or a column), then the superset cull would be the only way of making that reduction.

The Extended, Multi Group Innies and Outies techniques are the most advanced techniques used by Sudoku Tiger. This means that if you can spot and use all the techniques discussed so far, you will be able to solve all of the Killer puzzles generated by Puzzle Tiger. There are other, more advanced Killer techniques that you could use. For instance, you can merge different types of group together to form mixed group innies and outies, provided that you know the sum of the intersection between them. It is also possible to make reductions by listing all of the permutations that can go in a cage, and culling options which don't fit the list. Puzzle Tiger doesn't use either of these techniques (or anything more advanced), because it quickly becomes very difficult indeed for a person to spot the exact place where such a technique needs to be used.

Copyright © Adam A. Brown, 2006, All Rights Reserved. www.sudokutiger.com