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Extended Innies and Outies

In the previous section, we saw how Innies and Outies can be used when we can work out the sum of either 8 cells in a group or 10 cells with one outside the group. We can, however, extend this rule to cover cases when there is more than one cell under or over.

For the Extended Innies technique, the rule is as follows: if we can work out the sum of a subset of the cells in a group, we can use Combination Culling to remove possibilities from the remainder. Here is an example of this:

In this example, we can remove the candidate values "6" and "8" from the cells highlighted in red. The cages highlighted in orange contain 7 cells between them, the contents of which must add up to 29 in total. We know that the entire box has a sum of 45, meaning that the cells highlighted in red must have a sum of 16. Thus, they must contain a "7" and a "9" (which is the only way to make 16). So we can eliminate "6" and "8" from the candidates.

The Extended Outie rule is very similar: if we can work out the sum of a superset of a group (all the cells in the group plus some outside it), and if the cells outside the group in this set are all shared by another group or contained within one cage, we can use Combination Culling to remove possibilities from them. This sounds more complicated than it is, so here is an example:

In this example, we can remove all candidates except for "8" and "9" from the cells highlighted in red. The highlighted cells exactly fill three cages, with sums of 19 + 17 + 26 = 62. The cells highlighted in orange, however, form a complete box, and their contents must add up to 45. So the two cells left over (highlighted in red) must add up to 17, meaning they must be an "8" and a "9".

The important extra bit in the outies rule is that all the "outie" cells must share either a group or a cage. If they do not, then repeated digits might be possible, and the normal combination rules don't apply. You can sometimes still work out what the cells must contain (for instance, if they sum to 2, they both need to be "1", or if they sum to 17, they need to be "8" or "9"), but the number of combinations is vastly greater for sums between the extremes. Puzzle Tiger ignores this case altogether, so it won't be necessary for you to use it to solve puzzles. On the other hand, if you spot a useful case, you might as well use it!

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