The New Sudoku Players' Forum

by **ronk** » Tue Apr 04, 2006 3:15 pm

Havard wrote:Now the elimination of R1C3 is known from the frankenfish/ER POV
..................
But this fish allowed another elimination... Then I realized that what is going on is really this ..............

Your theory looks perfectly legit. However, the "extra elimination" can also be made with three strong links, the empty-rectangle link in box 6 and the conjugate links in rows 2 and 4.

Do you have a practical example of your theory that doesn't have such an obvious alternative? And would you please post the starting grids for your examples?

TIA, Ron

by **Havard** » Tue Apr 04, 2006 3:27 pm

Hi.

This puzzle was posted by Ruud as one of his "require coloring" earlier in this thread.

Code: Select all: 074000000003800000100005002500307160000000000062901003900200001000003800000000670

it is enough to just look at the 4's:

Code: Select all: . . 4 | . . . | . . . . . . | . 4 4 | 4 4 4 . . . | 4 4 . | 4 4 . ------+-------+------ . 4 . | . . . | . . 4 4 . . | 4 4 4 | . . 4 4 . . | . 4 . | 4 4 . ------+-------+------ . 4 . | . 4 4 | 4 4 . 4 . . | 4 4 . | . . 4 . . . | 4 4 4 | . . 4

And then we have this one!

Code: Select all: . . 4 | . . . | . . . . . . | . 4 4 | # # 4 . . . | 4 4 . | # # . ------+-------+-|-|-- . H . | . . . | | | * 4 | . | 4 4 4 | | | 4 * | . | . 4 . | X X . --|---+-------+-|-|-- . X . | . 4 4 | X X . 4 . . | 4 4 . | . . 4 . . . | 4 4 4 | . . 4

in style with the example from my previous post, which solves the puzzle!

by the way, I noticed that my example from before is really just a varation of the patteren MJ indentified as another Frankenfish variation:

Myth Jellies wrote:
Code: Select all
Frankenfish example 1 . . . |(.) . . | # # . . . . |(.) . . | # # . . . . |(.) . . | # # . -----------+-|---------+-|---|----- . . . | | . . | | | . . . . | X . . | X X . . . . | | . . | | | . -----------+-|---------+-|---|----- . . . | | . . | # # . . . . | | . . | # # . . . . | X . . | X X * Frankenfish example 2 . . . |(.) . . | # # . . . . |(.) . . | # # . . . . |(.) . . | # # . -----------+-|---------+-|---|----- . . . | # . . | | | . . . . | X * * | X X . . . . | # . . | | | . -----------+-|---------+-|---|----- . . . | | . . | | | . . . . | | . . | | | . . . . | X . . | X X .

so that all my "fishhead" examples is really covered with the example 2!

I guess the pattern that have popped out twice now can be seen as a combination of the two like this:

Code: Select all: Frankenfish 1 and 2 combined . . . |(.) . . | # # . . . . |(.) . . | # # . . . . |(.) . . | # # . -----------+-|---------+-|---|----- . . . | X . . | | | * . . . | | * * | X X . . . . | | . . | | | . -----------+-|---------+-|---|----- . . . | | . . | | | . . . . | | . . | | | . . . . | X . . | X X .

and the conclusion... frankenfish rules!

Havard

by **Havard** » Tue Apr 04, 2006 4:04 pm

Yet another example of this pattern, with yet another Ruud puzzle:

Code: Select all: 040201000000000250000040000000004907006000130009000000005700090100908600700002000

The 8's after basic elimination, and two finned x-wing (also in 8's)

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

and then we have:

Code: Select all: . . 8 | . 8 . | # . . 8 . 8 | 8 8 . | . . . . . . | 8 . . | # # 8 ------+-------+-|-|-- 8 . . | . 8 . | | | . . . . | 8 . . | | | 8 8 X . | . 8 . | X(X). --|---+-------+-|-|-- 8 H . | . . . | | | * . . . | . . 8 | | | . . . * | . . . | X X 8

and then a simple UR1 solves the puzzle!

Havard

by **Havard** » Tue Apr 04, 2006 5:00 pm

from a previous post I found these frankenfish in Ruuds puzzle!

Code: Select all: 200104003007800100000000090100900030000271000000000500030490000000007002006002700

(note that you need to use an ALS to get rid of the 6 in R4C2 first!)

Code: Select all: . 6 . | . * . | 6 . . # X . | . X . | . . X # X . | . X # | . . X ------+-------+------ .(X). | . X . | . . X 6 6 . | . . . | . 6 6 6 . . | 6 6 6 | . . . ------+-------+------ . . . | . . 6 | 6 . . . . . | 6 6 . | . 6 . . . . | . . . | . . .

Code: Select all: . X . | . X . | # .(X) # X . | . X . | . . X 6 6 . | . 6 6 | . . * ------+-------+------ .(X). | . X . | . . X 6 6 . | . . . | . 6 6 6 . . | 6 6 6 | . . . ------+-------+------ . . . | . . 6 | 6 . . . . . | 6 6 . | . 6 . . . . | . . . | . . .

This one has got a very interesting continuation as well.

Now you could continue with:
An x-wing for 2's in columns 3 and 5
A great example of headless swordfish for 6's in rows 1,3 and 7, opens up:
An x-wing for 6's in rows 2 and 4
and leads to these very interesting 8's:

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

first a "normal" finned swordfish columns 3,6 and 7

Code: Select all: . 8 X | . . . | X . . . . | | 8 . . | | . . 8 8 | | . . . | | . 8 ----|-+-------+-|---- . . # | . 8 . | | . 8 8 8 | | . . . | | 8 . * . X | . 8 X | | . . ------+-----|-+-|---- . . . | . . X | X . . 8 8 . | . 8 . | . 8 . 8 8 . | . 8 . | . 8 .

and then... A great example of the frankenfish type 2!

Code: Select all: . 8 X | . . . | X(X). . . | | 8 . . | | | . 8 8 | | . . . | | | 8 ----|-+-------+-|-|-- . . # | . 8 . | | | 8 * *(X)| . . . |(X)X. . . # | . 8 8 | | | . ------+-------+-|-|-- . . . | . . 8 | # | . 8 8 . | . 8 . | . # . 8 8 . | . 8 . | . # .

which solves the puzzle!

Havard
edit: some 8's went missing in the last grid. Thanks to ronk for spotting that!

by **ronk** » Tue Apr 04, 2006 5:38 pm

Havard wrote:And then we have this one!
Code: Select all
. . 4 | . . . | . . . . . . | . 4 4 | # # 4 . . . | 4 4 . | # # . ------+-------+-|-|-- . H . | . . . | | | * 4 | . | 4 4 4 | | | 4 * | . | . 4 . | X X . --|---+-------+-|-|-- . X . | . 4 4 | X X . 4 . . | 4 4 . | . . 4 . . . | 4 4 4 | . . 4

in style with the example from my previous post, which solves the puzzle!

That has an even easier two-strong-link alternative than your prior example IMO. Because of the strong links in row 4 and col 2 ... if r4c9=4 and r7c2=4, all the 4s candidates in box would be eliminated. Therefore r4c9<>4 and r7c2<>4. That pattern may also be considered as a grouped turbot fish.

Myth Jellies wrote:
Code: Select all
Frankenfish example 1 . . . |(.) . . | # # . . . . |(.) . . | # # . . . . |(.) . . | # # . -----------+-|---------+-|---|----- . . . | | . . | | | . . . . | X . . | X X . . . . | | . . | | | . -----------+-|---------+-|---|----- . . . | | . . | # # . . . . | | . . | # # . . . . | X . . | X X * Frankenfish example 2 . . . |(.) . . | # # . . . . |(.) . . | # # . . . . |(.) . . | # # . -----------+-|---------+-|---|----- . . . | # . . | | | . . . . | X * * | X X . . . . | # . . | | | . -----------+-|---------+-|---|----- . . . | | . . | | | . . . . | | . . | | | . . . . | X . . | X X .

In these Myth Jellies examples, all the cells of the 'fish' are actually involved in the deduction, as can be illustrated by "backtesting". In the first, if r9c9=X then r78c789<>X, r9c4<>X, r5c4=4 and r5c78<>X implies r123c7=X and r123c8=X which is impossible. Therefore, r9c9<>X. Many of your 'frankenfish' deductions have not been supported by this backtest method.

... and the conclusion... frankenfish rules!

I think your conclusion is based on hypothetical examples. I've been on the alert for real-life examples where the same eliminations cannot be made with simpler methods ... and I'm still looking for the first one.

Havard[/quote]

by **Havard** » Tue Apr 04, 2006 10:45 pm

ronk wrote:
Myth Jellies wrote:
Code: Select all
Frankenfish example 1 . . . |(.) . . | # # . . . . |(.) . . | # # . . . . |(.) . . | # # . -----------+-|---------+-|---|----- . . . | | . . | | | . . . . | X . . | X X . . . . | | . . | | | . -----------+-|---------+-|---|----- . . . | | . . | # # . . . . | | . . | # # . . . . | X . . | X X * Frankenfish example 2 . . . |(.) . . | # # . . . . |(.) . . | # # . . . . |(.) . . | # # . -----------+-|---------+-|---|----- . . . | # . . | | | . . . . | X * * | X X . . . . | # . . | | | . -----------+-|---------+-|---|----- . . . | | . . | | | . . . . | | . . | | | . . . . | X . . | X X .

In these Myth Jellies examples, all the cells of the 'fish' are actually involved in the deduction, as can be illustrated by "backtesting". In the first, if r9c9=X then r78c789<>X, r9c4<>X, r5c4=4 and r5c78<>X implies r123c7=X and r123c8=X which is impossible. Therefore, r9c9<>X. Many of your 'frankenfish' deductions have not been supported by this backtest method.

It is always nice to hear your critical voice ronk, but this time you are a bit out of line. When you say that these Myth Jellies examples somehow are more valid than my examples, you are contradicting yourself, because ALL my examples uses exactly the patterns described. Most of them the type 1 pattern, but the few latest ones the type 2 as well. Then you talk about "backtesting"... When you say that they have not been supported by backtesting as assume you mean that you have not been able to perform this technique on my examples. (since this is the first time in this thread that anyone bring this up) Then all I can tell you is to look again and maybe a bit closer, because if you can do a "backtest" on the two example patterns (where I might remind you that I was the one to first describe the type 1, so it is in fact my example) then you can do it on ALL my frankenfish examples. You will be pleased to find that your "backtesting" will work for every example I have given.

ronk wrote:
... and the conclusion... frankenfish rules!

I think your conclusion is based on hypothetical examples. I've been on the alert for real-life examples where the same eliminations cannot be made with simpler methods ... and I'm still looking for the first one.

again you should maybe look a bit closer before making statements like that. You and I and everyone know that the Frankenfish type 1 is the same as an Empty Rectangle/hinge/two strong links, but in the last few posts I have shown eliminations that the ER does NOT pick up on. I would be very interested to see what simpler method would be applied to do the same elimination as I did show in this previous post:

Code: Select all: . 8 X | . . . | X(X). . . | | 8 . . | | | . 8 8 | | . . . | | | 8 ----|-+-------+-|-|-- . . # | . 8 . | | | 8 * *(X)| . . . |(X)X. . . # | . 8 8 | | | . ------+-------+-|-|-- . . . | . . 8 | # | . 8 8 . | . 8 . | . # . 8 8 . | . 8 . | . # .

which is a great example of the frankenfish type 2, and follows the exact same rules as the generalized pattern posted first by MJ. With this one I can't see any simpler counterpart, and yet your post claims that there is. It would then be nice if you could point out this, and not just throwing statements like that out in the air.

...or maybe I simply can't wait until the day I will get a postive response from you...

Havard
edit: added those missing 8's.

by **ronk** » Wed Apr 05, 2006 4:38 pm

Havard wrote:When you say that these Myth Jellies examples somehow are more valid than my examples, you are contradicting yourself, because ALL my examples uses exactly the patterns described. Most of them the type 1 pattern, but the few latest ones the type 2 as well.

AFAICS, those examplars (models) do not intrinsically contain two strong links -- in the form of an empty rectangle and a conjugate link -- as your "type 1" examples have had. If either does, please show me where.

Havard wrote:You and I and everyone know that the Frankenfish type 1 is the same as an Empty Rectangle/hinge/two strong links ...

I believe you're helping me make my point.

Havard wrote:... but in the last few posts I have shown eliminations that the ER does NOT pick up on. I would be very interested to see what simpler method would be applied to do the same elimination as I did show in this previous post:
Code: Select all
. 8 X | . . . | X(X). . . | | 8 . . | | | . 8 8 | | . . . | | | 8 ----|-+-------+-|-|-- . . # | . 8 . | | | 8 * *(X)| . . . |(X)X. . . # | . 8 . | | | . ------+-------+-|-|-- . . . | . . . | # | . 8 8 . | . 8 . | . # . 8 8 . | . 8 . | . # .

which is a great example of the frankenfish type 2, and follows the exact same rules as the generalized pattern posted first by MJ. With this one I can't see any simpler counterpart, and yet your post claims that there is.

At the time I posted, I hadn't seen that 'type 2' example from your third post. For that example, as you say, there doesn't seem to be a smaller set of cells to produce those same two exclusions ... and it definitely fits the 'frankenfish type 2' definition. Congratulations on the find.

However, grouped coloring of those same cells yields six exclusions instead of two ...

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

P.S. Your 8s candidate grid is used as given. However, I couldn't figure out how to get the r8c3<>8 and r4c6<>8 exclusions.

... because two 8s in row 1 is impossible. Note also the use of two hinges/empty rectangles in the single chain of six strong links.

by **Havard** » Wed Apr 05, 2006 5:42 pm

ronk wrote:P.S. Your 8s candidate grid is used as given. However, something's amiss in col 6 and row 7. I couldn't figure out the r8c3<>8 exclusion either.

Looks like I left the two 8's in column 6 out after I drew up the first finned swordfish. I will edit them back in! Thanks for spotting that!

About the r8c3<>8. That follows after the elimination first pointed out by the frankenfish/er where the 6 in r1c5 is eliminated. Did you notice the nice headless swordfish for 6's in rows 1,3 and 7 a bit after:

Code: Select all: . #--------(X)--X . . * * . | . 6 . | . . 6 #-#---------X--(X). . ------+-------+------ . . . | . 6 . | . . 6 6 6 . | . . . | . 6 . 6 . . | 6 6 * | . . . ------+-------+------ . . . | . . X---X . . . . . | 6 . . | . 6 . . . 6 | . . . | . . .

very nice if you ask me!

Havard

by **ronk** » Wed Apr 05, 2006 6:12 pm

Hi Havard, can you identify a 'frankenfish' that might explain all, or most, of the digit 4 exclusions shown below?

From #422 of the top1465:
000000000072060100005100082080001300400000000037090010000023800504009000000000790

Code: Select all: . 4 . |-4 4 -4 | 4 -4 -4 . . . | 4 . 4 | . 4 4 . 4 . | . 4 -4 | 4 . . ----------+----------+---------- . . . | 4 4 . | . -4 -4 4 . . | . . . | . . . . . . |-4 . -4 | 4 . 4 ----------+----------+---------- . . . | 4 . . | . 4 4 . . 4 | . . . | . . . . . . | 4 . 4 | . . 4

TIA, Ron

by **Havard** » Wed Apr 05, 2006 7:01 pm

ronk wrote:Hi Havard, can you identify a 'frankenfish' that might explain all, or most, of the digit 4 exclusions shown below?

From #422 of the top1465:
000000000072060100005100082080001300400000000037090010000023800504009000000000790

Code: Select all
. 4 . |-4 4 -4 | 4 -4 -4 . . . | 4 . 4 | . 4 4 . 4 . | . 4 -4 | 4 . . ----------+----------+---------- . . . | 4 4 . | . -4 -4 4 . . | . . . | . . . . . . |-4 . -4 | 4 . 4 ----------+----------+---------- . . . | 4 . . | . 4 4 . . 4 | . . . | . . . . . . | 4 . 4 | . . 4

TIA, Ron

As far as I can tell four of those is pretty straight forward with a case of finned swordfish where you can swap between what is fish and what is fin to do the elimination in the traditional way:

Code: Select all: . X . | 4 X 4 | X 4 4 . | . | 4 | 4 | | 4 4 . X . | . X 4 | X . . ----------+----|-----+-|-------- . . . | 4 #X . | | * * 4 . . | . . . | | . . . . . | * . * | X# . 4 ----------+----------+---------- . . . | 4 . . | . 4 4 . . 4 | . . . | . . . . . . | 4 . 4 | . . 4

As for r1c4689 and r3c6 I would be very interested to see how you arrive at those eliminations, using just those 4's. I tried to set them to be true to see if I could reach any contradiction, but the 4's look perfectly good to me. Are you sure those eliminations are valid? My Nishio did not find them either...

So unless you can prove me wrong, I would claim that the swordfish picks up on all possible eliminations of these 4's.

Havard

by **ronk** » Wed Apr 05, 2006 7:43 pm

Havard wrote:
Code: Select all
. X . | 4 X 4 | X 4 4 . | . | 4 | 4 | | 4 4 . X . | . X 4 | X . . ----------+----|-----+-|-------- . . . | 4 #X . | | * * 4 . . | . . . | | . . . . . | * . * | X# . 4 ----------+----------+---------- . . . | 4 . . | . 4 4 . . 4 | . . . | . . . . . . | 4 . 4 | . . 4

As for r1c4689 and r3c6 I would be very interested to see how you arrive at those eliminations, using just those 4's. I tried to set them to be true to see if I could reach any contradiction, but the 4's look perfectly good to me. Are you sure those eliminations are valid?

I don't actually think those elims are valid ... and was just hoping maybe you could tell me they were.

Because the swordfish is headless, there exists a strong inference between r4c5 and r6c7. For me, that's a new way of looking at this scenario ... and it's then easy to understand the exclusions at r4c89 and r6c46 because each cell "sees" both r4c5 and r6c7.

However, both r4c5 and r6c7 might be true, so it stops there AFAICT.

[edit: added the following]

Indeed, here is a puzzle where both such fin cells ultimately turn out to be true. It is #716 of the top1465:
004000003000701000076000001002000000000900008080200050250800090060007040000003000

Code: Select all: . . . | . . . | . . 3 3 #3 . | . -3 . | . . . -3 . . | #3 3 . | . . . ----------+-----------+---------- . *3 . | *3 3 . | . *3 . . *3 3 | . 3 . | . *3 . . . 3 | . 3 . | . . . ----------+-----------+---------- . . 3 | . . . | 3 . . 3 . 3 | . . . | 3 . . . . . | . . . | . . .

by **Havard** » Thu Apr 06, 2006 3:30 pm

Ever since I found this set of rules for finding fish, I have been running a lot of puzzles in search of new weird fish, and today it came up with a strange looking one from a puzzle we all know quite well: Vidar's Monster #4:

Code: Select all: +-------+-------+-------+ | 1 . . | . . 6 | . . . | | . 6 . | 9 . . | . . 8 | | 8 . . | . . 4 | 3 6 . | +-------+-------+-------+ | . . 8 | . . . | 4 . . | | . . 6 | . 4 3 | 9 . 5 | | . 4 . | 5 . . | . . . | +-------+-------+-------+ | . 2 . | . . . | . 7 . | | 4 . 1 | . 7 . | . . . | | . . 3 | . 1 . | 2 . . | +-------+-------+-------+

Now lets look at the 7's from go:

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 ------+-------+------ . . . | . . . | . 7 . . . . | . 7 . | . . . 7 7 . | . . . | . . .

Now just looking at columns 3,6,7 and 9 you can make the following eliminations:

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 . | . . . 7 7 . | . . . | . . .

lets look at why:

Code: Select all: . * a | 7 . . | f . i * . a | . . c | g . | . 7 a | 7 . | | | . i ----|-+-----|-+-|---| 7 7 | | 7 . d | | . j 7 7 | | 7 . | | | . | 7 . b | . . e | h . k ------+-------+------ . . . | . . . | . 7 . . . . | . 7 . | . . . 7 7 . | . . . | . . .

Now if aaa is true, the * elimination is obvious, so let's start with a as false.
Then b=true and e, h,k=false. Now f or g are true, which means ii is false, and then j=true and d=false and then c=true and then g=false and then finally f=true...
so with f and c true, the two * gets killed off anyway...

now I know this is a bit of a freak, but I would purpose this general pattern:

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

easiest to prove by saying: the only way r2c12 is NOT going to be eliminated, is to set the "head" (r2c6) as false, and the fins in box 1 as false. But this creates an x-wing in columns 3 and 6, and that makes it impossible to place a number in both column 7 and 9.

so what do you think... is it fish?

Havard

by **Myth Jellies** » Thu Apr 06, 2006 4:29 pm

Havard wrote:so what do you think... is it fish?

...Perhaps more like something that swallowed a fish (those dang little x-bones are hard to digest, and can get stuck in your throat if you don't chew thoroughly

)

by **Havard** » Thu Apr 06, 2006 4:34 pm

Myth Jellies wrote:
Havard wrote:so what do you think... is it fish?

...Perhaps more like something that swallowed a fish (those dang little x-bones are hard to digest, and can get stuck in your throat if you don't chew thoroughly )

hehe! Did you see that I caught an specimen of your frankenfish-variation?

Havard

by **ronk** » Thu Apr 06, 2006 5:32 pm

Havard wrote:Now just looking at columns 3,6,7 and 9 you can make the following eliminations:
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 . | . . . 7 7 . | . . . | . . .

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 ----------+-----------+----------- . . . | . . . | . 7 . . . . | . 7 . | . . . 7 7 . | . . . | . . .

Ultimately, either a) r1246c3679 is a jellyfish in 7s or it's not. If not a jellyfish, it's because either b) fin r3c3=7 or c) fin r3c9=7.

if a jellyfish, r12c12<>7
if fin r3c3=7, then r12c12<>7
if fin r3c9=7, then r46c9<>7, r6c7=7, r6c3<>7, r123c3=7, and r12c12<>7

Therefore r12c12<>7. Of course, the argument works just as well with the swordfish in r359c124.

The New Sudoku Players' Forum

Big fish