Locking Cages and it's inverse,
Blocking Cages, both use cage combinations to either lock candidates or block permutations/combinations. This technique is called
Locked Cages in sudopedia but I feel that Locking Cages is a better name since the inverse Blocking Cages makes more sense than "Blocked Cages". But, take your pick.
Locking Cages is when two cages contain the only candidate for a house which then "locks" a second candidate
if that second candidate has a direct link to the first in both cages. It is easiest to see when two cages in the same house have the same cage total.
Two-cell cages In this example above from A167, 1 in c1 must be in one of the 10(2) cages -> one of those cages must also have 9 -> 9 locked for c1.
For some reason, it is much more common that one of the cages is hidden.
In MessyOne#1 above, r56c3 (in yellow) is a h9(2) cage. Candidate 8 in this nonet must be in this hidden 9(2)
or in the 9(2) cage -> 1 locked for the nonet. These become easy to find since the cage totals are the same.
Much more difficult is when the cage totals are different, ie, when at least one of the cages is more than 2 cells.
One two-cell and one three-cell cages.
This pic above is an alternative way to do Andrew's step 18 in A174
here, using Locking Cages. 5 in n8 must be in the 11(2) = {56} or the hidden 19(3) at r7c45+r8c4 = {289/379/469/478/568}. The only combination with 5 is {568}, so, if it has 5 it must also have 6 (and of course 8 but that is not relevent here). In both cages, 5 forces candidate 6 -> 6 locked for n8.
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Extension: locking-out combinations/candidatesBut another extention of this move is possible, to lock-out combinations and possibly candidates within the locking cages. In the final example above,
Other combinations in the h19(3) with the second candidate (6) can be eliminated since they don't also have the forcing candidate (5). So, {469} is not possible since it would leave no 5 available for n8! This second part is essential to Andrew's step 18.
The table in the previous post may make it easier to find Locking Cages when the two cages do NOT have the same total. Andrew notes that his move was available at the beginning of the puzzle. As the table shows, a 19(3) cage is one of those special cages that has this direct-link feature.
This pic above is from Texas Jigsaw Killer 39 and is from
manu's walkthrough
here.
Quote:
3)a) 1 locked for n6 at r3c9+5(2)={14/23} : r3c9 <> 4 !
This move above involves just one cage (and one cell) but works by the locking-out principle to eliminate 4 from r3c9.
Locking-out also works for
two 2-cell cages that have different cage totals. For example, if 6 is only in a 15(2) and 10(2) (in the same house) -> {19} is blocked from the 10(2)
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Now, a look at the inverse,
Blocking Cages.
This is from A171 ALT and is an alternative way to do manu's step 5b
here. There is a hidden 12(2) at r23c3 which overlaps a 13(2) & a 10(3) -> [84] permutation is blocked from the h12(2) since 5 is also required in both cages ie, 5 is forced into both the 13(2)=[58] and 10(3)={15}[4]. So, the 13(2) and 10(3) work as Blocking Cages to h12(2) permutations. This leaves the h12(2) = {57} and cracks the puzzle.
Further applications of these techniques are available but those can wait for a separate post sometime. For example, almost Locking Cages can form killer subsets with other cages or almost Blocking Cages will eliminate combinations in other cages. The interesting sudopedia article shown in the link above includes an example of innies/outies as another way to use this locking technique.
Many thanks to Andrew and Afmob for encouraging this post onto this technques forum.
Cheers
Ed