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PostPosted: Fri Mar 20, 2009 2:38 pm 
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Expert
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Joined: Sat Jan 17, 2009 8:30 am
Posts: 118
Location: france
Hi,

I would like to understand clearly the difference between a cage blocker and a (naked) killer subset and use the accurate terminology ; I think that some of the others may wonder the same thing and could be interested in that question

Could anyone tell me if I make a mistake : let' see two exemples :

1) Suppose for instance that a row contains both cages 13(2) and 14(2) : could I say that {69} is a killer pair of cages 13(2)&14(2) since because both 6 and 9 are contained in the combined cages 13(2)&14(2) ?

2) Suppose for instance that a row contains both cages 8(3) and 15(3). Of course, 1 is locked at cage 8(3) which enables to remove combinations {159/168}. But (most of killer sudoku solvers know it !), combinations
{249/357/456} must be also removed since cage 8(3) must contain one of (24), (35) and (45). Could I say that (24) (35) and (45) are cage blockers ? And, more precisely, what is the terminology used for saying it? ( (24) is a cage blocker (of 8(2)) for combination {249} etc...)

In fact, I have seen that Richard SSolver uses the word '' cage blockers'' but I was not able to find references about that.

Cheers,

manu


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PostPosted: Fri Mar 20, 2009 4:38 pm 
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From my point of view, both of your examples are Killer pairs. Especially in your second one, I think it's more precise that say that (24), (35) and (45) are Killer pairs of 8(3) because a cage blocker could be anything from a Killer pair to a Killer quad.

For me a cage blocker would be a set of cells that blocks certain combination(s) of a cage only because of its candidates.

Examples:
1) Let R1C12 be a 12(2) cage and R1C6 = (57), then R1C6 is a cage blocker for 12(2) = {57}
2) Let R12C7 be a 11(2) cage and R579C7 = (239), then R579C7 is a cage blocker for 11(2) = {29}

Of course, in both examples those blocked combos are (Implied?) Killer pairs of these cells, but I think it's just easier to look at it with the candidates.

By the way, in your first example (19), (29) , ..., (89) are also Killer pairs of the combined cage since 13(2) + 14(2) = 27(4) = 9{378/468/567}, so it must have a 9.


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PostPosted: Sat Dec 26, 2009 5:42 am 
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Joined: Wed Apr 23, 2008 6:04 pm
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Thanks manu and Afmob for interesting discussion about killer subsets.

There are two main ways to use killer subsets, the traditional way and the wider way used by Afmob and some others.

The traditional way, the one that udosuk refers to as naked killers, uses killer values in two or more cages/cells in a row, column or nonet (also diagonal if the puzzle is a Killer-X) to eliminate those values from all other cells in that row, ...

In addition to using the traditional way, Afmob also uses another version where killer subsets in one cage are used to eliminate combination(s) from another cage in the same row, ... This use seems to me to be a form of hidden killer subset.

Both ways can be correctly described as killers although I stick to the traditional way, preferring to describe the other type as clashes with, or blocked by, without using the phrase "killer".

A186 provides examples of both ways and also prompted me to make this post.

Image

First, here are some examples from Afmob's walkthrough for A186.

The traditional type of killer is used in step 4d

d) ! Killer pair (49) locked in Outies R789 + R6C4 for R6

and the blocking type of killer in step 1c

c) Innies D/ = 13(3) <> 9 since {139} blocked by Killer pair (13) of 8(2) @ D/


Afmob suggested to me that step 8 of my walkthrough for A186 used a killer triple.

8. 45 rule on N7 3 innies R7C23 + R8C3 = 15 = {249/258/348} (cannot be {159/357} which clash with 8(2) cage, cannot be {168/267} which clash with R78C1, cannot be {456} which clashes with R9C23), no 1,7

While this step can be seen as a blocking type of killer, I don't see it that way. To me it's a simple clash, the sort of thing that someone solving a newspaper killer sudoku would easily spot.

In this case R9C23 contains 123456. Any combination in a 3 cell cage in N7 that clashes with either 123456 or 123456 is a simple clash.

At one stage, I was looking at a similar but slightly harder clash for 45 rule on N1, 3 innies R2C3 + R3C23 = 14 where I could eliminate {257} because it clashed with R1C23. In this case any combination in a 3 cell cage in N1 that clashes with either 123567 or 123567 can be eliminated. In each of these cases the clash is with alternating candidates in a cage. However by the time I was ready to use these innies in step 17, I was able to use a simpler clash with R23C1.

Both of the above are, in my opinion, much simpler when seen as clashes rather than as killer subsets.


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