The New Sudoku Players' Forum

by **tarek** » Wed May 24, 2006 12:30 pm

ronk wrote:why isn't this simply a finned jellyfish? Is it one or more boxes in the "cover" that frankenizes it?

You have 6 columns (so 4 coulumns of vertices & 2 columns of fins)....Box 5 has only one column of fins....that is why it is not a finned Jellyfish......

Furthermore according to Ruud's brilliant formula, you can't have more than one line of fins for the finned Jellyfish.

tarek

by **ronk** » Wed May 24, 2006 3:07 pm

tarek wrote:Furthermore according to Ruud's brilliant formula, you can't have more than one line of fins for the finned Jellyfish.

That doesn't agree with Myth Jellies' illustrations IIRC. Do you have a link to "Ruud's brilliant formula"?

What is the below, and what is(are) the exclusion(s)?

Code: Select all: . . . | . . . | . . . . . . | . . . | . . . - - X | - - - | - - X ----------+----------+--------- - - X | X X X | X X X . . . | . . . | . . . - - X | X X X | X X X ----------+----------+--------- . . . | . . . | . . . - - X | - - - | X - X . . . | . . . | . . .

[edit: code block w/ question added]

by **tarek** » Wed May 24, 2006 4:02 pm

Ruud describing his brilliant formula wrote:Considering the remaining candidates of a digit, where P placements have been made for that digit (including the givens)

When we can find a set of A columns with candidates CA, and a set of B rows with candidates CB, where the size of A + B equals 9 - P, and the intersection on CA and CB is confined to a single box, we have found a finned fish of size (A or B whichever is larger) with the intersection as the fin.

Each candidate in the box containing the fin, that does not belong to CA or CB can be eliminated.

Extra feature: If there is NO intersection, every candidate that does not belong to CA or BC can be eliminated, as we have discovered the non-finned variety

the intersection cells are the Fins, if there are no intersection cells => No fins => Classic fish, one note on the description though [The fish size is the smaller of A & B not the larger]

so for a finned Jellyfish, the only possibilities are 4+4,4+5,5+4 you can't have a 6 (4+ 2 lines of fins)........ (that is my understanding, 1 example is enough to disprove it

)

I now have programmed it, it is slower than my previous finned algorithm but this one catches everything......

ronk wrote:What is the below, and what is(are) the exclusion(s)?
Code: Select all
. . . | . . . | . . . . . . | . . . | . . . - - X | - - - | - - X ----------+----------+--------- - - X | X X X | X X X . . . | . . . | . . . - - X | X X X | X X X ----------+----------+--------- . . . | . . . | . . . - - X | - - - | X - X . . . | . . . | . . .

you can eliminate candidates as per proof by contradiction, but I don't think there is enogh info to construct a fish pattern

I haven't tested it, but could it be possible that if the intersection of A+B is in 2or more boxes (of some form :?:

), that it could apply to these elusive Frankenfish (The last Franken Jellyfish I quickly found 4+4=9-1 pattern with intersections in boxes 4 & 5, which in theory eliminates any 8 in line 5 lying in boxes 4 & 5)???

I tried programming the franken algorithm, unfortunately it brought up this for the previous franken Jellyfish...

Code: Select all: *--------------------------------------------------------* |*258 -28 3 | 6 1458 14 |*48 9 7 | | 1 78 #4578 | 58 9 45 | 6 2 3 | | 6 9 #48 | 2 7 3 | 1 #48 5 | |------------------+------------------+------------------| | 3 128 6 | 1458 2458 245 | 7 145 9 | |*28 5 *278 | 147 3 9 |*248 6 1248 | | 4 1278 9 | 1578 258 6 | 3 15 128 | |------------------+------------------+------------------| | 9 3 1 | 45 245 8 | 245 7 6 | | 7 4 *258 | 3 6 125 | 9 *18 128 | |*258 6 *258 | 9 1245 7 |*2458 3 1248 | *--------------------------------------------------------* Eliminating 8 From r1c2 (Franken Jellyfish in Columns 1378 with 3 fin(s) in Boxes 13, or by constructing the Franken Jellyfish in rows 2346 with 3 fin(s) in Boxes 13)

Furthermore, extending the algorithm to X-wings & sworfishes brought up these strange creatures, I'm not sure if they're valid though..

Code: Select all: *--------------------------------------------------------* | 258 *28 3 | 6 *1458 14 | 48 9 7 | | 1 78 4578 | 58 9 45 | 6 2 3 | | 6 9 48 | 2 7 3 | 1 48 5 | |------------------+------------------+------------------| | 3 #128 6 | 1458 2458 245 | 7 145 9 | | 28 5 278 | 147 3 9 | 248 6 1248 | | 4 #1278 9 | 1578 258 6 | 3 15 128 | |------------------+------------------+------------------| | 9 3 1 | 45 245 8 | 245 7 6 | | 7 *4 258 | 3 *6 -125 | 9 18 128 | | 258 6 258 | 9 #1245 7 | 2458 3 1248 | *--------------------------------------------------------* Eliminating 1 From r8c6 (Franken XWing in Columns 25 with 3 fin(s) in Boxes 48) *--------------------------------------------------------* | 258 28 3 | 6 458 1 | 48 9 7 | | 1 78 4578 | 58 9 45 | 6 2 3 | | 6 9 48 | 2 7 3 | 1 48 5 | |------------------+------------------+------------------| | 3 #128 6 | 158 258 245 | 7 #145 9 | | 28 5 278 | 147 3 9 | 248 6 1248 | | 4 *1278 9 | 1578 258 6 | 3 *15 -128 | |------------------+------------------+------------------| | 9 3 1 | 45 245 8 | 25 7 6 | | 7 *4 258 | 3 6 25 | 9 *18 128 | | 258 6 258 | 9 1 7 | 2458 3 248 | *--------------------------------------------------------* Eliminating 1 From r6c9 (Franken XWing in Columns 28 with 2 fin(s) in Boxes 46, or by constructing the Franken XWing in rows 45 with 2 fin(s) in Boxes 46) *--------------------------------------------------------* | 258 28 3 | 6 458 1 | 48 9 7 | | 1 78 4578 | 58 9 45 | 6 2 3 | | 6 9 48 | 2 7 3 | 1 48 5 | |------------------+------------------+------------------| | 3 #128 6 | 158 258 245 | 7 145 9 | | 28 5 278 | 147 3 9 | 248 6 #1248 | | 4 *1278 9 | 1578 258 6 | 3 -15 *28 | |------------------+------------------+------------------| | 9 3 1 | 45 245 8 | 25 7 6 | | 7 *4 258 | 3 6 25 | 9 18 *128 | | 258 6 258 | 9 1 7 | 2458 3 24 | *--------------------------------------------------------* Eliminating 1 From r6c8 (Franken XWing in Columns 29 with 2 fin(s) in Boxes 46, or by constructing the Franken XWing in rows 45 with 2 fin(s) in Boxes 46)

tarek

by **Mike Barker** » Thu May 25, 2006 4:48 am

Ron, an interesting test. Row 3 and 8 form a classic finned X-wing. This allows elimination in r79c9. This looks to be an extension of the Franken Jellyfish since it has two big fins and one little one. The almost constrained subset is A={r3,r4,r6, r8} and B={c3,c7,c9,b5} with the fin=r46c8. This allows eliminations in r5c79. With B={c3,c9,b5,b6} and the fin=r8c7, the X-wing eliminations are also possible. I guess the lesson in this is that applying almost constrained subsets can possibly yield more eliminations than just N*N fish alone, whether they be basic fish, finned fish, big finned fish, or Frankenfish as currently defined. I think it makes sense to further investigate almost constraint sets independently or as an aid to identifying other useful patterns which even I might be able to detect in a puzzle. The other thing I learned was that ACS can have multiple "B" sets for a given "A" set.

Code: Select all: . . . | . . . | . . . . . . | . . . | . . . - - X | - - - | - - X ----------+----------+--------- - - X | X X X | X X X . . . | . . . | . . . - - X | X X X | X X X ----------+----------+--------- . . . | . . . | . . . - - X | - - - | X - X . . . | . . . | . . .

Tarek, although the eliminations you've identified are valid, I'm not sure your Franken X-wings are. These normally require that there be a big fin (two lines of the fish in the same chute). What you have are vanilla X-wings with two fins which are not legitimate based on Ruud's formula. For example in your last example, given CA={c2,c9}, there is no way to find 2 rows such that CB intersects CA in only one box. If there was a big fin (or multiple big fins based on Ron's example) and eliminations may be possible, for example,

Code: Select all: . . . | . . . | X X - . . . | . . . | X X - . . . | . . . | X X - ------+-------+-|-|-- . . . | . . . | | | - * * * | * * * | X X - . . . | . . . | . . - ------+-------+------ . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

where the '-' cells have candidates eliminated either by the fish or locked candidates.

by **ronk** » Thu May 25, 2006 12:09 pm

tarek, thanks for posting Ruud's formula.

Unfortunately, none of the eliminations in your post are valid IMO. It's because, in all instances, the elimination cell does not see all the [edit: fin cells].

tarek wrote:Eliminating 8 From r1c2 (Franken Jellyfish in Columns 1378 with 3 fin(s) in Boxes 13, or by constructing the Franken Jellyfish in rows 2346 with 3 fin(s) in Boxes 13)

Based on 8s alone, this outcome is possible.

Code: Select all: . 8 . | . . . | . . . . . . | 8 . . | . . . . . . | . . . | . 8 . -------+-------+------ . . . | . 8 . | . . . 8 . . | . . . | . . . . . . | . . . | . . 8 -------+-------+------ . . . | . . 8 | . . . . . 8 | . . . | . . . . . . | . . . | 8 . .

tarek wrote:Eliminating 1 From r8c6 (Franken XWing in Columns 25 with 3 fin(s) in Boxes 48)
(...)
Eliminating 1 From r6c8 (Franken XWing in Columns 29 with 2 fin(s) in Boxes 46, or by constructing the Franken XWing in rows 45 with 2 fin(s) in Boxes 46)

Based on 1s alone, this outcome is possible.

Code: Select all: . . .| . 1 .| . . . 1 . .| . . .| . . . . . .| . . .| 1 . . ------+------+------- . 1 .| . . .| . . . . . .| 1 . .| . . . . . .| . . .| . 1 . ------+------+------- . . 1| . . .| . . . . . .| . . 1| . . . . . .| . . .| . . 1

tarek wrote:Eliminating 1 From r6c9 (Franken XWing in Columns 28 with 2 fin(s) in Boxes 46, or by constructing the Franken XWing in rows 45 with 2 fin(s) in Boxes 46)

Based on 1s alone, this outcome is possible.

Code: Select all: . . . | . . 1 | . . . 1 . . | . . . | . . . . . . | . . . | 1 . . -------+-------+------- . 1 . | . . . | . . . . . . | 1 . . | . . . . . . | . . . | . . 1 -------+-------+------- . . 1 | . . . | . . . . . . | . . . | . 1 . . . . | . 1 . | . . .

by **tarek** » Thu May 25, 2006 12:19 pm

Thanx Ron & mike,

I'll try fiddling with the formula & see if a proof by contradiction can be obtained consistanly, I think it's time again for deep sea fishing

[Edit: I tried reasoning it out, I couldn't find how would an elimination cell SEE a fin cell which is in another box, that is why you can have a miximum of 4 fin cells in one box that see the elimination cell, No multiple box fins, The only way for one eliminatin cell to see fins in another box is indirectly via strong links]

tarek

by **ronk** » Thu May 25, 2006 8:34 pm

Mike Barker wrote:The almost constrained subset is A={r3,r4,r6, r8} and B={c3,c7,c9,b5} with the fin=r46c8. This allows eliminations in r5c79. With B={c3,c9,b5,b6} and the fin=r8c7, the X-wing eliminations [edit: r79c9] are also possible.

Good show! I set up the illustration for eliminations at r5c7 and r79c9 -- since these non-finned ACS eliminations each see the fin at r8c7 -- and didn't even notice your first "B={c3,c7,c9,b5} with the fin=r46c8".

by **Mike Barker** » Fri May 26, 2006 4:00 am

In looking at Ron's test a little more, I've realized that what is happening is a result of siamese fish (two fish combined into one). In the example (rotated for comparison to other big fin fish), c2 and c5 form a finned X-wing with the elimination in r6c46. The cells, c2, c5, c7, c8 form a standard Franken Jellyfish with the fin in b6 (r4c5 becomes part of the body and the eliminations in r46c9)

Code: Select all: . . . | . . . | X X . . . . | . . . | X X . . . . | . . . | X X . ------+-------+-|-|-- . . . | . # . | X X * . . . | . | . | # # . . X . | - X - | X X * --|---+---|---+-|-|-- . X . | . X . | X X . . . . | . . . | . . . . . . | . . . | . . .

The eliminations in a skinny Franken Jellyfish occur because it consists of a bigfin swordfish (no fins) in c2, c7, and c8 (with the elimination in r2c9) as well as a skinny Franken Jellyfish with the fin in box 2 and the elimination in r2c46.

Code: Select all: . . . | . # . | X X . . X . | * X * | X X - . | . | . # . | # # . ---|---+---|---+-|-|--- . | . | . | . | | | . . X . | . X . | X X . . | . | . | . | | | . ---|---+---|---+-|-|--- . | . | . | . | | | . . | . | . | . | | | . . X . | . X . | X X .

I guess I had already encountered a siamese fish here which consists of two different Franken Swordfish {r4, r9, and b3} and {r5, r9, b3}:

Code: Select all: . . . | . | . | # # . . . . | . | . | # # . . . . | . | . | # # . ------+---|----+-|--|--- . . . | . X# . | | | - . . . | * | * | X# X# . . . . | . | . | | | . ------+---|----+-|--|--- . . . | . | . | | | . . . . | . | . | | | . . . . | . X . | X X .

What this means is that there are not really any additional eliminations occurring in these cases. The cells shown as "-" are already eliminated by the existance of the equal or smaller half of the siamese fish. More importantly I can pull out a bunch of code I needed to identify the r2c9 elimination in the second example. I have to correct my identification of Ron's test - its a siamese finned X-wing and Franken Jellyfish. I guess this shouldn't be a surpise since two covering ACS were identified.

by **ronk** » Fri May 26, 2006 8:19 pm

Mike Barker wrote:I guess I had already encountered a siamese fish here which consists of two different Franken Swordfish {r4, r9, and b3} and {r5, r9, b3}:
Code: Select all
. . . | . | . | # # . . . . | . | . | # # . . . . | . | . | # # . ------+---|----+-|--|--- . . . | . X# . | | | - . . . | * | * | X# X# . . . . | . | . | | | . ------+---|----+-|--|--- . . . | . | . | | | . . . . | . | . | | | . . . . | . X . | X X .

That's the only one I've seen where one of the fin cells must be true, so that makes it unique IMO. (I don't count the candidates in box 3 as fin cells.)

Mike Barker wrote:What this means is that there are not really any additional eliminations occurring in these cases. The cells shown as "-" are already eliminated by the existance of the equal or smaller half of the siamese fish.

This thread teems (pun intended) with examples where deductions can be made with simpler techniques. Have you forgotten all the frankenfish with ERs?

by **Steve R** » Sun May 28, 2006 7:46 pm

Before trying to understand headless, franken and so on I thought it might help me to define a finned fish. This is done in two stages and, for brevity, based on columns:

An m-fin base for a candidate, X, is a set of m (> 0) columns (the base columns), a non-empty set of boxes (the target boxes) and a non- empty set of rows (the target rows) subject to the following constraints:
(a) X remains to be placed in each base column. The cells in the base columns which may admit X are referred to as the base cells.
(b) Each of the target rows contains a base cell which is not in a target box.
(c) Each of the target boxes contains a base cell which is not in a target row.

Condition (c) ensures that the boxes contain at least one fin.

A finned m-fish is an m-fin base with tb target boxes and tr target rows such that tb + tr = m or tb + tr = m + 1.

(1) If F is a finned m-fish for which tb + tr = m, X may be eliminated from its target boxes and rows except for the base cells.

The base column must hold m Xs, each in a base cell. There are precisely m targets so each receives one X, again in a base cell. No other cell in a target may contain X.

(2) If F is a finned m-fish for which tb + tr = m + 1, X may be eliminated from the intersection of any target box and any target row except for the base cells. (It is not asserted that a non-empty intersection is bound to be present.)

Suppose, if possible, that X does occupy such a cell. m Xs remain to be placed because the new insertion is not in a base cell. However, the number of boxes available is reduced from tb to tb – 1 and the number of rows from tr to tr – 1. The m Xs have m – 1 recipients and the contradiction establishes (2).

It is noteworthy that the propositions represent nothing more than counting in finite subsets of the grid. There is no requirement that the boxes are of a particular shape or size. As a matter of fact there is no requirement for columns and rows to run in straight lines.

Duality is much easier to understand in square n x n grids with square boxes, however. Suppose p Xs have been placed when a finned m-fish, F, is found. Then
- the n – p – tr rows which are void of X and are not target rows of F
- the tb boxes of F and
- the n – p – m columns which are void of X and are not base columns of F
form a finned (n – p – tr)-fish, F*, based on rows. This is the fish dual to F.
From the construction:
(3) F and F* make precisely the same eliminations.
(4) The dual of F* is F.

It is interesting to inspect Havard’s splendid discovery against this background:

Code: Select all: Cannibalistic Franken-Whale in columns: 3 4 5 6 7 9 19 159 37 | 1368X 1578X 4 | 2 356789 1358X 8 125 37 | 1236X 1257X 9 | 1346X 3567 1345X 129 6 4 | 1238X 1578X 178X | 1389X 35789 1358X ---------------------+----------------------+--------------------- 3 4 128X | 7 1268X 5 | 168X 268 9 5 7 128X | 9 4 16X- | 1368X 2368 138X 129 289 6 | 128X 128X 3 | 5 4 7 ---------------------+----------------------+--------------------- 4 18 9 | 5 368 168X | 7 38 2 6 3 128X | 18X 1789X 278 | 489 589 458 7 28 5 | 4 39 28 | 39 1 6

I am not convinced it is a finned 6-fish because it seems to me as that the elimination follows if column 6 is omitted from the recipe.

Code: Select all: ------------------------- | . . . | X X . | . . X | | . . . | X X . | X . X | | . . . | X X . | X . X | ------------------------- | . . X | . X . | X . . | | . . X | . . - | X . X | | . . . | X X . | . . . | ------------------------- | . . . | . . . | . . . | | . . X | X X . | . . . | | . . . | . . . | . . . | -------------------------

This has the look of a finned 5-fish
- base columns 3, 4, 5, 7 and 9 (m = 5)
- target boxes 2, 3, and 5 (tb = 3)
- target rows 4, 5 and 8 (tr = 3)

According to the theory above, its dual should also be a finned 5-fish because p = 1 and m – p – tr = 9 – 1 – 3 = 5.

Code: Select all: ------------------------- | X X . | X X . | . . X | | . X . | X X . | X . X | | X . . | X X X | X . X | ------------------------- | . . . | . . . | . . . | | . . . | . . - | . . . | | X . . | X X . | . . . | ------------------------- | . X . | . . X | . . . | | . . . | . . . | . . . | | . . . | . . . | . . . | -------------------------

Finned 5-fish
- base rows 1, 2, 3, 6 and 7
- target boxes 2, 3, and 5
- target columns 1, 2 and 6

If there is a smaller fish there I cannot find it.

Of course there will be other ways of solving the puzzle, perhaps easier ways. At the same time, unless the eliminations are false, I find it difficult to deny Havard his catch.

Steve

by **Mike Barker** » Sat Jun 10, 2006 1:00 am

This just came up in a puzzle in the help section. Starting from

Code: Select all: *-----------* |.68|.52|.7.| |...|6.9|.5.| |...|..3|2.6| |---+---+---| |4.6|...|389| |859|346|127| |73.|...|564| |---+---+---| |.73|...|...| |.8.|237|...| |.9.|86.|73.| *-----------*

then after a few elimination there is what appears to be a Franken Squirmbag. I'm not sure I believe it. To the best of my knowledge no apparant sighting of this creature has ever withstood closer examination, so the odds of this guy surviving are small, but here he is anyway! This is actually two Franken Squirmbags (both finned) which yields the two eliminations. I await Steve's verdict.

Code: Select all: +----------------+---------------+----------------+ | *19 6 8 | *14 5 2 | 49 7 3 | | 3 124 127 | 6 -17 9 | 48 5 18 | | 159 14 57 | 147 8 3 | 2 149 6 | +----------------+---------------+----------------+ | 4 12 6 | -17 127 5 | 3 8 9 | | 8 5 9 | 3 4 6 | 1 2 7 | | 7 3 *12 | 9 *12 8 | 5 6 4 | +----------------+---------------+----------------+ | 26 7 3 | 5 9 *14 | 68 *14 28 | | *156 8 *145 | 2 3 7 | 469 *149 *15 | | *125 9 *145 | 8 6 *14 | 7 3 *125 | +----------------+---------------+----------------+

And yes, I know that two strong links will eventually result in the same eliminations, the question is, is there another fish that can perform the same trick?

After seeing one I think the name, squirmbag, is appropriate - this guy sure doesn't look like a starfish to me!

by **Havard** » Sat Jun 10, 2006 1:22 am

I guess I'll say it again then...

what about this jellyfish?

Code: Select all: Franken-Jellyfish in columns: 2 4 8 9 19 6 8 | 14X 5 2 | 49 7 3 3 124X 127 | 6 17- 9 | 48 5 18X 159 14X 57 | 147X 8 3 | 2 149X 6 ---------------+----------------+--------------- 4 12X 6 | 17X 127 5 | 3 8 9 8 5 9 | 3 4 6 | 1 2 7 7 3 12 | 9 12 8 | 5 6 4 ---------------+----------------+--------------- 26 7 3 | 5 9 14 | 68 14X 28 156 8 145 | 2 3 7 | 469 149X 15X 125 9 145 | 8 6 14 | 7 3 125X 1 . . | 1X . . | . . . . 1X 1 | . 1- . | . . 1X 1 1X . | 1X . . | . 1X . ---------+----------+--------- . 1X . | 1X 1 . | . . . . . . | . . . | 1 . . . . 1 | . 1 . | . . . ---------+----------+--------- . . . | . . 1 | . 1X . 1 . 1 | . . . | . 1X 1X 1 . 1 | . . 1 | . . 1X

Havard

by **ronk** » Sat Jun 10, 2006 3:37 am

How about these siamese franken-swordfish?

Code: Select all: 1X . .X | 1# . . | . . . . 1 1 | . 1- . | . . 1 1 1 . | 1 .- . | . 1 . ---------+----------+--------- . 1 . | 1 1 . | . . . . . . | . . . | 1 . . .X . 1X | . 1X . | . . . ---------+----------+--------- .X . .X | . . 1 | . 1 . 1X . 1X | . . . | . 1 1 1X . 1X | . . 1 | . . 1

Except for the fin in r1c4, candidates in r1, r6, and b7 are covered by c1, c3, and c5. Whether the fin is true or false, exclusions r23c5<>1 apply.

Code: Select all: 1X . .X | 1X . . | . . . . 1 1 | . 1 . | . . 1 1 1 . | 1 . . | . 1 . ---------+----------+--------- . 1 . | 1- 1 . | . . . . . . | .- . . | 1 . . .X . 1X | . 1# . | . . . ---------+----------+--------- .X . .X | . . 1 | . 1 . 1X . 1X | . . . | . 1 1 1X . 1X | . . 1 | . . 1

Except for the fin in r6c5, candidates in r1, r6, and b7 are covered by c1, c3, and c4. Whether the fin is true or false, exclusions r45c4<>1 apply.

Interestingly, at least one of the fins must be true. Actual exclusions ("1-") and potential exclusions (".-") would be valid even if candidate 1 existed at "empty" cells (".X").

by **Steve R** » Sat Jun 10, 2006 8:18 pm

Mike Barker generously wrote “I await Steve's verdict.” Unfortunately I still haven’t found the time to learn about franken fish so there is no verdict, just uncertainty.

The grid is:

Code: Select all: +----------------+---------------+----------------+ | *19 6 8 | *14 5 2 | 49 7 3 | | 3 124 127 | 6 -17 9 | 48 5 18 | | 159 14 57 | 147 8 3 | 2 149 6 | +----------------+---------------+----------------+ | 4 12 6 | -17 127 5 | 3 8 9 | | 8 5 9 | 3 4 6 | 1 2 7 | | 7 3 *12 | 9 *12 8 | 5 6 4 | +----------------+---------------+----------------+ | 26 7 3 | 5 9 *14 | 68 *14 28 | | *156 8 *145 | 2 3 7 | 469 *149 *15 | | *125 9 *145 | 8 6 *14 | 7 3 *125 | +----------------+---------------+----------------+

I, too, would think of it as containing two 5-fish rather than one. Both are based on the rows indicated by asterisks. From my viewpoint one is based on boxes 2 and 9, the other on boxes 5 and 9. Both are perfectly genuine.

I’m not so sure about the cigar, though. What is wrong with Havard’s “whale?” Admittedly (May 28) I argued for it being a 5-fish rather than a 6 but, subject to that adjustment, it seemed to make proper fishy eliminations. What’s more its dual is also a 5-fish. [I reckon the duals of Mike’s two 5-fish are 4s; that based on boxes 2 and 9 using columns 2, 5, 8 and 9; that based on boxes 5 and 9 using columns 2, 4, 8 and 9].

Do please bear in mind that different definitions of “finned fish” could well give rise to different conclusions.

Steve

by **Mike Barker** » Sat Jun 10, 2006 8:26 pm

Thanks, my solver has one fewer bug in it and now finds the jellyfish. I still can't find the swordfish. I think I'm going to leave fishing to the Norwegians.

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