So far, the

arcilla lists have been used first to convert large Fish into smaller ones having the same eventual-elimination (EE) cell, and then more recently to help verify the Fluke (see previous post). But, how can the lists be applied in a top-down, logical approach to

find that EE? Let's look again at

DAJ's 7s grid, except this time without any foreknowledge of the EE cell. I'll proceed slowly here since this is new ground for me.

The 7s grid has the following two

arcilla lists:

RPC: (

29)(167)(457)(3589)(124679)(

269)(2367)(45689)(13568)

CPR: (259)(1567)(479)(358)(3489)(256789)(2357)(489)(14568)

Per the standard Rx, we check the lists' cells for Almost-Locked Sets (ALS), where the Almost list entry is the spoiler/obstacle, and the LS is the Locked Set. The spoiler translates to a Fin, and the LS translates to a basic Fish. Inspection of the two bolded entries in the RPC list above immediately suggests a finned X-Wing. In the grid below, (*) marks a covered base cell, (p) marks a potential elimination (PE), and (#) marks the Fin.

- Code: Select all
`2-Fish c16\r29 + fr6c6. `

. 7 . | . 7 . | . . 7

*7 . . | . p7 *7 | p7 . .

. . . | 7 . . | 7 . 7

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

. . 7 | . 7 . | . 7 .

. . 7 | 7 . . | . 7 7

. 7 . | . 7 #7 | 7 7 7

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

. 7 7 | . 7 . | 7 . .

. . . | 7 . . | . 7 7

*7 . . | p7 p7 *7 | . p7 .

We already know from basic finned-Fish strategy that the Fin, if true, must be able to see (either directly or remotely) one or more of the PEs for any valid exclusions. However, the Fin above cannot

directly see any of the PEs, so does that mean we have to look for five separate remote paths to the PEs, starting at the fin? Well, we

could, but that would certainly feel like T&E!

Or, can we just use the Fish=Fin strong-link rule and begin to develop an implication stream? In what follows, we'll denote the “definned” X-Wing as XW(c16), and for clarity let's also refer to the term Fish as being definned, and that Fin in this case refers to a

single fin cell. We now begin the implication stream by treating the Fish as false:

XW(c16)=r6c6...

This forces the Fin to be true, so we set r6c6=7, omit all of its peers (r6c25789|r29c6|r4c5|r5c4), and adjust the grid accordingly. We denote the deleted peer cells in the implication stream as:

XW(c16)=r6c6-[r6c6 peers]...

The modified grid appears as:

- Code: Select all
` . 7 . | . 7 . | . . 7`

7 . . | . 7 . | 7 . .

. . . | 7 . . | 7 . 7

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

. . 7 | . . . | . 7 .

. . 7 | . . . | . 7 7

. . . | . . 7 | . . .

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

. 7 7 | . 7 . | 7 . .

. . . | 7 . . | . 7 7

7 . . | 7 7 . | . 7 .

We next examine a

new pair of

arcilla lists for the modified grid:

RPC: (

29)(

17)(457)(389)(

1279)(6)(

237)(4589)(1358)

CPR: (259)(157)(479)(38)(389)(6)(2357)(489)(1458)

We'll stick with the RPC list for the moment and look for suitable ALS. The first three bolded entries form a nice 3-cell ALS, but all four row numbers occur twice, meaning that any Fin selected will comprise

two different fin cells, which we'd like to avoid if at all possible. So, we add the fourth bolded entry, which has a single occurrence of row number 3 and therefore implies only a single Fin cell. The 1279 row entries define the cover sectors, while the cell positions, 1257, define the base sectors. This translates to a finned Jellyfish as shown below in the modified grid:

- Code: Select all
`4-Fish c1257\r1279 + fr3c7.`

. *7 . | . *7 . | . . p7

*7 . . | . *7 . | *7 . .

. . . | 7 . . | #7 . 7

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

. . 7 | . . . | . 7 .

. . 7 | . . . | . 7 7

. . . | . . 7 | . . .

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

. *7 p7 | . *7 . | *7 . .

. . . | 7 . . | . 7 7

*7 . . | p7 *7 . | . p7 .

So, we next add the entire

finned Jellyfish (denoted FJF(c1257)) to the implication stream:

XW(c16)=r6c6-[r6c6 peers]=FJF(c1257)...

But now, a true Fin at r3c7 can

directly see the PE at r1c9, which implies r1c9<>7. One can then immediately extend the stream from the false r1c9 with a Kite (ala

JC Van Hay) to form:

XW(c16)=r6c6-[r6c6 peers]=FJF(c1257)-r1c9=Kite(r1c1) => r9c5<>7,

where Kite(r1c1) is the bidirectional AIC, r1c5=r1c2-r2c1=r9c1 => r9c5<>7.

The implication stream actually defines a

strong inference between XW(c16) and Kite(r1c1), meaning that at least one of them must be true. Either way, the exclusion at r9c5 applies.

One should note that the CPR list of the modified grid also generates a suitable ALS from the entries in row positions 3458 that translates to:

4-Fish r3458\c3489 + fr3c7.

This finned Jellyfish can be substituted directly for FJF(c1257) to produce the same EE.

These “hybrid” implication streams are not as compact as a single NxM Fish (with possible duplicate sectors), but they do seem to offer understandable alternatives to the manual solver.

EDIT: The basic approach described above is fundamentally sound, but the implication stream is technically flawed. See subsequent posts and

DAJ's final resolution

here.