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 Post subject: Combined Cages
PostPosted: Tue May 12, 2009 11:08 pm 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Combined Cages can be useful to reduce the number of possible combinations in two or more cages when all the cells in each cage "see" all the cells in each of the other cages. In the following discussion I'll simplify things by only talking about two cages; more would be possible.

Combined cages most commonly occur within one row, column or nonet; more complicated examples have wider coverage, see the diagram and example below.

The simplest examples, ignoring single cell cages, are two 2-cell cages in the same row, column or nonet. A 3(2) cage and a 7(2) cage which "see" each other can be combined to become a 10(4) cage = {1234} from which one can see that the 3(2) cage must be {12} and the 7(2) cage {34}; of course one can obtain the same result from 3(2) cage = {12}, 7(2) cage = {34} (cannot be {16/25} which clash with the 3(2) cage).

Combining two 2-cell cages is most productive when the total for the combined cage is close to 10 or 30, the minimum and maximum totals for 4-cell cages.

Three more complicated examples occur in my walkthrough for A153. They are available from the starting position, as in the following diagram.

Image

All the cells of the 27(6) cage can "see" both cells of the 15(2) cage.

a) 27(6) cage at R1C2 + R23C2 = 42 = {12456789}, no 3
b) 3 in N1 locked in R1C13, locked for R1
c) 45 rule on N1 2 innies R1C13 = 1 outie R4C2 + 3
d) R4C2 = {124568} -> R1C13 {13/23/34/35/36/38} (because R1C13 must contain 3), no 9 in R1C1

Here step a) is the combined 42(8) cage. I've included the following steps to show what resulted immediately after using it. 9 would already have been eliminated from R1C3 in the prelims because it's part of a 9(2) cage.

An advantage with a 8-cell cage is that it can only have one combination so one candidate can be eliminated from each cell of the cage.

If you look at the the full diagram for A153 here you should be able to find two more examples of 6-cell and 2-cell cages forming combined 8-cell cages.

This cage pattern was particularly interesting because it also allowed another technique to be used. Ed used the fact that the 6-cell and 2-cell cages "see" each other for IOE here.

Thanks to Ed for providing the diagram!


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