KiMo Sudoku
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DG KiMo 7

Double Killers

MikeJapan and H3lix have been putting up Double Killers (two layouts with very large cages which have the same solution, neither of which is solvable alone {perhaps without trial and error} but which can de solved jointly) on DJape’s Forum.

I have really enjoyed them and intend to put a few up in KiMo form over the next few weeks.

Disjoint Killer Explanation

The Row-Column to Nonet-Spot Transform

Consider the following DG solution (from Menneske of course)
In each Row we have 1..9.
In each Column we have 1..9.
In each Nonet we have 1..9.
Also as it is a Disjoint Group Solution we also have 1..9 in each Spot (where the spots are identified by their position in each Nonet (and colour coded to help).

The Rows and Columns map the whole puzzle exactly, i.e. every cell position can be described uniquely as RxCy where x and y both belong to the set (1..9).

This is not the only way the puzzle can be described completely and uniquely. We can also describe each cell position in terms of the Nonet it is in and the Spot in that Nonet e.g. N6S8 (the eight Spot in the sixth Nonet) = 9.

Note that we cannot do this with the other combinations: N5C6 describes three possible cells, whereas N5C7 has no meaning. The same happens with NR, CS and RS.

Consider the above puzzle solution laid out with its cells listed in terms of their Nonet and Spot. Please check a few of the values in the following table with the Nonet and Spot positions in the above table to confirm their identity.

Consider the top left nine numbers "block":
N1s1 = r1c1 = 1
N1s2 = r1s2 = 5
N1s3 = r1c3 = 4
N2s1 = r1c4 = 8
N2s2 = r1c5 = 3
N2s3 = r1c6 = 6
N3s1 = r1c7 = 2
N3s2 = r1c8 = 7
N3s3 = r1c9 = 9
This is Row 1 and the other "blocks" are the other rows in order 2 to 9.

If we check the "spots" in this layout we find they are the columns in the normal layout.

We now have two complete-unique descriptions of the set of puzzle cells with an associated transform between the two descriptions

Disjoint Group Killers using both Layouts

When we create a killer (or KiMo) based on a Disjoint Group solution, we can make it exactly like a normal killer but with an extra constraint, see:
http://www.diceboard.co.uk/kimo27.html

In a standard killer (barring some of MM’s) each cell is included in one and only cage. There is nothing to stop us having overlapping cages or cells that are not in any cage. However if we do this we have two problems. The first is that having the cages exactly cover the puzzle gives an extra bit of information. The second one is that the overlapping cages could become very confusing (and would be difficult to draw).

By using both the DG layouts we can create interesting cage combinations while avoiding too much confusion. Because of the first point above we do end up using more cages.

Consider the top part of a puzzle based on the above numbers, RC form:

NS form:

We can redo this on one puzzle as follows with the cells marked in one colour totalling to the value on the right:

Using the two formats has (to me) the advantage of being less confusing, but more importantly I can see the Spot interactions directly.

If you find any part of this explanation unclear or wish to make any comments please contact me.

HATMAN (Maurice Smith)