So we have a third grid which doesnt have a 19 - despite the MCN as low as 13 !

I think [simplistically ] that a grid will by represented with a minimal number of clues if all the unavoidables can be covered by 17,18,19 or 20 clues.

If a grid needs 20 clues it is because there is no combination of 19 clues which hits all the unavoidable sets.

There are two processes here

1. A high MCN - eg dukuso16 grid - it doesnt have a 19 because there is no combination of 16 plus 3 clues which hits all the [high number of] 4 sets in this grid.

2.Reduced number of 2-perms - I think the reduced number of 2-perms inherently means there are more small 2-clue unavoidables - which you get with higher rates of 4,8 and 16 perms. This means that the RW grid doesnt happen to have any combination of 13 plus 6 clues which hits all the unavoidables.

Similar grids to RWs must be rare - and I would be impressed if we get one with less than 9 2-perms.

Incidently - how are we working out the 36 2-rookeries ?

I am running {checker128 19} on rededmax and entwiner2 [somewhat optimstically]

I have thought about the 170 essentially different 2-rookerys - if it helps in understanding constructing the grids.

2-perm

- Code: Select all
`AAA`

AAA

AAA one 18 clue unavoidable sets between 2 clues - fully entwined

+---+---+---+

|1..|...|..2|

|.2.|1..|...|

|...|.2.|1..|

+---+---+---+

|.1.|..2|...|

|...|.1.|2..|

|..2|...|.1.|

+---+---+---+

|..1|2..|...|

|...|..1|.2.|

|2..|...|..1|

+---+---+---+

+---+---+---+

|12.|...|...|

|...|1..|.2.|

|...|.2.|1..|

+---+---+---+

|.1.|..2|...|

|..2|.1.|...|

|...|...|.12|

+---+---+---+

|..1|2..|...|

|2..|..1|...|

|...|...|2.1|

+---+---+---+ these two cannot be morphed by row/box swapping

4 perm

- Code: Select all
`ABB `

ABB

BBB =

+---+---+---+

|12.|...|...|

|...|1..|.2.|

|...|..2|1..|

+---+---+---+

|21.|...|...|

|...|.1.|..2|

|...|2..|.1.|

+---+---+---+

|..1|...|2..|

|..2|..1|...|

|...|.2.|..1|

+---+---+---+ but there will be different ways to complete a 14 clue B unavoidable set.

here are more

4+14 6+12 6+12 8+10 8+10

ABB ABB AAB AAB AAB

ABB ABB ABB ABB AAB

BBB ABB BBB ABB BBB

8 perm

- Code: Select all
`ABC ACC ABB ABC ABB ACC ABC ABC ABB `

ABC ABC ABC ABC ABB ACC ABC ABC ABC

CCC CBC CCC BBC CCC BBC BBB ABC ACC

16 perm

- Code: Select all
`ABC ABB ABB `

ABC ACC ABC

DDD DDD DDC

These ones are impossible

- Code: Select all
`ACC ABC ABC`

ACC ABC ACB

?BB ?DD CBB

There is only one way to have the 4-unavoidable and the two types of 6-unavoidable between 2 clues.

I am not sure about the different ways to have 8,10,12 and 14-unavoidable sets between 2 clues - this would then presumably add up to 170 from dukuso.

I am up to 33 clues in the RW grid.

C