The New Sudoku Players' Forum

by **Ruud** » Wed Aug 16, 2006 12:07 pm

I'm not sure if this technique has been introduced earlier with a different name. If that is true, please ignore this post and my ignorance. A pointer to the message would be wonderful. I did search the forum. For now, I will proceed as if this is something new.

I would like to start with a few definitions:

Primary Lines: a number of lines which are either all rows or all columns.
Secundary Lines: a number of lines perpendicular to the primary lines.

Row-Column Subset: a pattern where N primary lines have all candidates for a selected digit D confined to N secundary lines, allowing us to eliminate the candidates for digit D in these N secundary lines which are not located in any of the primary lines. There are many different names (Hidden single, X-Wing, Swordfish, Jellyfish), depending in the value of N.

Almost Row-Column Subset (ARCS): a pattern where N primary lines have all candidates for a selected digit confined to N+1 secundary lines. The smallest ARCS, where N = 1, is a bilocation unit, a.k.a. conjugate pair.

There is an overlap between ARCS and Finned Fish, in case all fin cells are located in a single secundary line. Theoretically, the ARCS can have any number of surplus candidates, but for practical purposes, confinement to a single box is the only way to see all of them at the same time. The main difference is in how we use them. In finned fish, we focus on the box containing the fin. In ARCS, this box is only a possible "point of entry" for using it in an AIC.

Following ALS rules, we can now define ARCS rules.

Any move that eliminates all ARCS candidates in one of the secundary lines, automatically forces the remainder of the pattern into a row-column subset.
Following such a move, there is weak inference between the pattern as a whole, and any surplus candidate on any of the remaining secundary lines.

I'm not an expert in formal definitions nor in any of the notation systems, so I'll leave the task of embedding ARCS into nice loops and AIC's to the qualified people.

For now, this is only an introduction (see title) and pure theory. We need practical examples, maybe from our finned fish collections.

Ruud

by **Mike Barker** » Thu Aug 17, 2006 4:08 am

Its probably not too surprising that I've also been thinking about how to extend constraint subsets (the Franken Squirmbags were the first step in this process). I've been looking at almost fish with the basis set consisting of "n" columns (rows) and the covering set consisting of "n+m" rows (columns) and boxes. As in the case of the standard fish (basic, finned, big fin, and franken) the idea is to locate a candidate in a cell not part of the fish which reduces the size of the covering set to "n-1". For standard fish this extra candidate is the same as the candidate which makes up the fish and the candidate cell is directly visible by enough cells of the fish to achieve the requisite dimension reduction. Although the almost fish can be linked into a nice loop (see here), I think I prefer the approach Anne Morelot used to solve #5000 of Ruuds Top50000 (see here). The key in her solution is to connect a non-fish cell to enough fish cells via ALSs to reduce the size of the covering set to "n-1". Although she used ALSs here, basic and grouped strong links as well as directly visible cells can be used as well. As seen in the example, the elimnated candidate need not be the same as the candidate making up the fish.

The puzzle and her Almost Swordfish elimination are:

Code: Select all: .....14..7...6..1..2.8..7...3...2...6...8...9...4...2...1..9.5..4..1...6..73..... +--------------------+-------------------+---------------------+ | 3589 *589 35689 | *2579 2379 1 | 4 3689 2358 | | 7 *589 34589 | *259 6 345 | *23589 1 2358 | | 1 2 34569 | 8 359 345 | 7 369 35 | +--------------------+-------------------+---------------------+ | 4 3 @589 | *1569 @59 2 | 1568 -78 1578 | | 6 17 2 | 157 8 357 | 135 4 9 | | 589 17 589 | 4 3579 3567 | 13568 2 1358 | +--------------------+-------------------+---------------------+ | 238 68 1 | 267 247 9 | 238 5 23478 | | 2359 4 359 | 257 1 578 | *2389 3789 6 | | 2589 *5689 7 | 3 245 4568 | 128 #89 1248 | +--------------------+-------------------+---------------------+

The almost fish consists of 3 columns (c247) which contain "9". r12 are the "n-1" constraint subsets which will remain. An ALS (r4c35) and a bivalue (r9c8) are used to link r4c8 to the remaining cells of the almost fish (r4c4, r9c2, and r8c7), so that if r4c8=8 then these cells cannot contain "9" and thus r4c8<>8.

I'm still in the process of checking out the algorithm in my solver, but I think it is pretty much working. Here is a complete non-minimal solution to #5000 including 5 almost X-wings and 2 almost Swordfish. I'm not sure if it is a bug or real, but most of the eliminations occur using ALS. There is one grouped strong link as well. Prior to implementing this technique, my solver couldn't solve #5000, but with it it can. I don't know how generally useful the technique is yet.

Code: Select all: Hidden Single: r3c1 => r3c1=1,r46c1<>1 Hidden Single: r4c1 => r4c1=4,r45c3<>4,r4c89<>4 Hidden Single: r5c3 => r5c3=2,r8c3<>2 Hidden Single: r5c8 => r5c8=4,r9c8<>4 Locked Column Line/Box: r79c2 => r1c2<>6 Locked Column Line/Box: r13c8 => r4c8<>6 Naked Block Triple: r6c13|r4c3 => r5c2<>5,r6c2<>589 Column X-Wing Fillet-o-Fish: r568c6|r48c8 => r4c45<>7 Locked Row Box/Box: r5c246|r6c256 => r6c9<>7 WXYZ-wing: r3c56|r2c6, r4c5 => r1c5<>5 UVWXYZ-wing: r78c4|r79c5|r8c6, r7c2 => r8c13<>8 5-element Nice Loop: r9c7=1=r9c9=4=r7c9=7=r4c9-7-r4c8-8-r9c8~9~r9c7 => r9c7<>9 Almost X-Wing (r56c7-3-r78c7|r9c8-2-, r6c9-3-r123c9-2-): r5c67|r6c5679 => r7c9<>2,r2c7<>2,r9c9<>2 Almost X-Wing (r56c7-3-r78c7|r9c8-8-, r6c9-3-r123c9-8-): r5c67|r6c5679 => r79c9<>8 +--------------------+-------------------+---------------------+ | 3589 589 35689 | 2579 2379 1 | 4 3689 @2358 | | 7 589 34589 | 259 6 345 | -23589 1 @2358 | | 1 2 34569 | 8 3459 345 | 7 369 @35 | +--------------------+-------------------+---------------------+ | 4 3 589 | 1569 59 2 | 1568 78 1578 | | 6 17 2 | 157 8 *357 | *135 4 9 | | 589 17 589 | 4 *3579 *3567 | *13568 2 *1358 | +--------------------+-------------------+---------------------+ | 238 68 1 | 267 247 9 | @238 5 -23478 | | 2359 4 359 | 257 1 578 | @2389 3789 6 | | 2589 5689 7 | 3 245 4568 | 128 @89 -1248 | +--------------------+-------------------+---------------------+ Almost X-Wing (r78c1|r8c8-3-r468c3-5-, r3c8-3-r3c9-5-): r178c1|r138c8 => r3c3<>5 +--------------------+-------------------+--------------------+ | *3589 589 35689 | 2579 2379 1 | 4 *3689 2358 | | 7 589 34589 | 259 6 345 | 3589 1 2358 | | 1 2 -34569 | 8 3459 345 | 7 *369 @35 | +--------------------+-------------------+--------------------+ | 4 3 @589 | 1569 59 2 | 1568 78 1578 | | 6 17 2 | 157 8 357 | 135 4 9 | | 589 17 @589 | 4 3579 3567 | 13568 2 1358 | +--------------------+-------------------+--------------------+ | *238 68 1 | 267 247 9 | 238 5 347 | | *2359 4 @359 | 257 1 578 | 2389 *3789 6 | | 2589 5689 7 | 3 245 4568 | 128 89 14 | +--------------------+-------------------+--------------------+ Almost X-Wing (r78c1|r8c8-3-r468c3-5-, r3c8-3-r2c2479-5-): r178c1|r138c8 => r2c3<>5 Almost X-Wing (r78c1|r8c8-3-r468c3-8-, r3c8-3-r2c2479-8-): r178c1|r138c8 => r2c3<>8 +--------------------+-------------------+--------------------+ | *3589 589 35689 | 2579 2379 1 | 4 *3689 2358 | | 7 @589 -34589 | @259 6 345 | @3589 1 @2358 | | 1 2 3469 | 8 3459 345 | 7 *369 35 | +--------------------+-------------------+--------------------+ | 4 3 @589 | 1569 59 2 | 1568 78 1578 | | 6 17 2 | 157 8 357 | 135 4 9 | | 589 17 @589 | 4 3579 3567 | 13568 2 1358 | +--------------------+-------------------+--------------------+ | *238 68 1 | 267 247 9 | 238 5 347 | | *2359 4 @359 | 257 1 578 | 2389 *3789 6 | | 2589 5689 7 | 3 245 4568 | 128 89 14 | +--------------------+-------------------+--------------------+ Almost X-Wing (r78c1|r8c8-3-r468c3-9-, r3c8-3-r3c569-9-): r178c1|r138c8 => r3c3<>9 +--------------------+-------------------+--------------------+ | *3589 589 35689 | 2579 2379 1 | 4 *3689 2358 | | 7 589 349 | 259 6 345 | 3589 1 2358 | | 1 2 -3469 | 8 @3459 @345 | 7 *369 @35 | +--------------------+-------------------+--------------------+ | 4 3 @589 | 1569 59 2 | 1568 78 1578 | | 6 17 2 | 157 8 357 | 135 4 9 | | 589 17 @589 | 4 3579 3567 | 13568 2 1358 | +--------------------+-------------------+--------------------+ | *238 68 1 | 267 247 9 | 238 5 347 | | *2359 4 @359 | 257 1 578 | 2389 *3789 6 | | 2589 5689 7 | 3 245 4568 | 128 89 14 | +--------------------+-------------------+--------------------+ 5-element Nice Loop: r7c5=4=r7c9=7=r4c9-7-r4c8-8-r9c8-9-r3c8=9=r3c5~4~r7c5 => r3c5<>4 Locked Column Line/Box: r23c6 => r9c6<>4 Almost X-Wing (r78c1|r8c8-3-r468c3-9-, r3c8-3-r2c2479-9-): r178c1|r138c8 => r2c3<>9 +--------------------+-------------------+--------------------+ | *3589 589 35689 | 2579 2379 1 | 4 *3689 2358 | | 7 @589 -349 | @259 6 345 | @3589 1 @2358 | | 1 2 346 | 8 359 345 | 7 *369 35 | +--------------------+-------------------+--------------------+ | 4 3 @589 | 1569 59 2 | 1568 78 1578 | | 6 17 2 | 157 8 357 | 135 4 9 | | 589 17 @589 | 4 3579 3567 | 13568 2 1358 | +--------------------+-------------------+--------------------+ | *238 68 1 | 267 247 9 | 238 5 347 | | *2359 4 @359 | 257 1 578 | 2389 *3789 6 | | 2589 5689 7 | 3 245 568 | 128 89 14 | +--------------------+-------------------+--------------------+ UR+3C/2SL (34): r23c36 => r3c6<>3 UR+4X/2SL (63): r13c38, r3c69 => r1c8<>3 Almost Swordfish (r13c5|r5c6-3-r23c6-5-, r5c7-3-r5c7=5=r5c46): r1c1359|r3c3589|r5c67 => r6c6<>5 +--------------------+-------------------+--------------------+ | *3589 589 *35689 | 2579 *2379 1 | 4 689 *2358 | | 7 589 34 | 259 6 @345 | 3589 1 2358 | | 1 2 *346 | 8 *359 @45 | 7 *369 *35 | +--------------------+-------------------+--------------------+ | 4 3 589 | 1569 59 2 | 1568 78 1578 | | 6 17 2 | @157 8 @357 | @135 4 9 | | 589 17 589 | 4 3579 -3567 | 13568 2 1358 | +--------------------+-------------------+--------------------+ | 238 68 1 | 267 247 9 | 238 5 347 | | 2359 4 359 | 257 1 578 | 2389 3789 6 | | 2589 5689 7 | 3 245 568 | 128 89 14 | +--------------------+-------------------+--------------------+ Almost Swordfish (r9c2|r8c7-9-r49c8-7-, r4c4-9-r4c358-7-): r129c2|r124c4|r28c7 => r8c8<>7 +--------------------+------------------+--------------------+ | 3589 *589 35689 | *2579 2379 1 | 4 689 2358 | | 7 *589 34 | *259 6 345 | *3589 1 2358 | | 1 2 346 | 8 359 45 | 7 369 35 | +--------------------+------------------+--------------------+ | 4 3 @589 | *1569 @59 2 | 1568 @78 1578 | | 6 17 2 | 157 8 357 | 135 4 9 | | 589 17 589 | 4 3579 367 | 13568 2 1358 | +--------------------+------------------+--------------------+ | 238 68 1 | 267 247 9 | 238 5 347 | | 2359 4 359 | 257 1 578 | *2389 -3789 6 | | 2589 *5689 7 | 3 245 568 | 128 @89 14 | +--------------------+------------------+--------------------+ Hidden Single: r4c8 => r4c8=7,r4c9<>7 Hidden Single: r7c9 => r7c9=7,r7c45<>7 Hidden Single: r7c5 => r7c5=4,r9c5<>4 Hidden Single: r9c9 => r9c9=4 Hidden Single: r9c7 => r9c7=1,r456c7<>1 Three Grouped Strong Links: r2c2=8=r1c123-8-r1c8=8=r89c8-8-r7c7=8=r7c12~8~ => r9c2<>8 3-element Nice Loop: r9c1=2=r9c5-2-r7c4-6-r7c2~8~r9c1 => r9c1<>8 Locked Row Line/Box: r7c12 => r7c7<>8 3-element Nice Loop: r9c2=6=r7c2-6-r7c4-2-r9c5~5~r9c2 => r9c2<>5 Locked Column Box/Box: r689c1|r468c3 => r1c13<>5 4-element Nice Loop: r5c6=3=r5c7-3-r7c7-2-r7c4-6-r4c4=6=r6c6~3~r5c6 => r6c6<>3 4-element Nice Loop: r9c5=2=r1c5=7=r6c5-7-r6c6-6-r4c4=6=r7c4~2~r9c5 => r7c4<>2 Naked Single: r7c4 => r7c4=6,r7c2<>6,r4c4<>6,r9c6<>6 Naked Single: r7c2 => r7c2=8,r7c1<>8,r12c2<>8 Hidden Single: r9c2 => r9c2=6 Hidden Single: r6c6 => r6c6=6,r6c7<>6 Hidden Single: r4c7 => r4c7=6 Locked Row Line/Box: r1c13 => r1c89<>8 Locked Column Box/Box: r689c1|r468c3 => r1c13<>9 Locked Column Box/Box: r26c7|r246c9 => r8c7<>8 UR+2B/1SL (23): r78c17 => r8c1<>3 ALS xz-rule with A=1 cells: r5c7-3-r6c137 => r6c9<>5 UR+4X/2SL (75): r58c46, r5c2 => r8c6<>5 4-link Advanced Colouring: r5c7=3=r5c6=7=r8c6=8=r8c8=3=r3c8~3~ => r2c7<>3 4-link Advanced Colouring: r6c5=3=r5c6=7=r8c6=8=r8c8=3=r3c8~3~ => r3c5<>3 Naked Column Pair: r34c5 => r1c5<>9,r6c5<>59,r9c5<>5 Naked Single: r9c5 => r9c5=2,r9c1<>2,r1c5<>2,r8c4<>2 Locked Row Line/Box: r4c45 => r4c3<>9 Three Strong Links: r5c7=3=r5c6-3-r6c5=3=r1c5-3-r1c1=3=r7c1~3~ => r7c7<>3 Naked Single: r7c7 => r7c7=2,r7c1<>2,r8c7<>2 Naked Single: r7c1 => r7c1=3,r1c1<>3,r8c3<>3 Naked Single: r1c1 => r1c1=8,r1c3<>8,r6c1<>8 Hidden Single: r8c1 => r8c1=2 5-node XY-chain: r58c7|r9c68 => r5c6<>5 Naked Block Pair: r6c5|r5c6 => r5c4<>7 Empty Column Rectangle : r48c4|r8c3 => r4c3<>5 Naked Single: r4c3 => r4c3=8,r4c9<>8,r6c3<>8 Locked Row Line/Box: r6c13 => r6c7<>5 2-line BUG Lite Elimination (SL:r1c9=2=r1c4, SL:r1c9=2=r2c9, XY:r2c2|r1c4-9-r2c4): r12c249 => r1c9<>5,r2c4<>9 7-node XY-chain: r1c358|r9c68|r8c4 => r1c4<>7 The Solution is completed with singles

by **Ruud** » Thu Aug 17, 2006 1:28 pm

Hi Mike,

Thanks for your response.

Anne already pointed me to her solution of my #5000. Never thought it had already been used in one of my own puzzles.

I agree with you that this is a logical extension of constraint subsets. This idea struck me when I was trying to code a generic constraint subset method in SudoCue. (it's not working yet

)

Using an ALS as the fuse for ARCS makes sense. At first, I was only looking at bivalue cells to link to the entry point, but a bivalue cell is also an ALS. With multiple fin cells, a larger ALS may be able to hit them all.

About the method to find these:

Finned fish are abundant, and so are ALS. We need a search program that can scan collections for combinations of the two.

When we find such a combination, the only thing we need is an AIC from one of the 'victims' of the RCS to the ALS that forces it into play.

Ruud

by **ronk** » Thu Aug 17, 2006 2:23 pm

Backtest [edit: comment was based on an incorrect candidate grid.]

by **Mike Barker** » Thu Aug 17, 2006 8:48 pm

Edit: Comment no longer pertinent.

by **ronk** » Thu Aug 17, 2006 9:05 pm

Mike Barker wrote:if r4c8=8 then r4c3={59} and r4c5={59} forming a naked pair so r4c4<>9. The backtest solutions include r4c4=9. Does it handle ALS?

Sorry, my candidate grid had r4c5={579} when backtest applied.

by **Ruud** » Wed Aug 23, 2006 5:48 pm

The responses to this thead are a bit below expectations, so maybe I can attract your attention by posting an example recently found:

This is a One-Trick Puzzle. I originally posted it as a Nishio example, but somebody already spotted a Broken Wing in this pattern.

Code: Select all: 000050200507094030000700009070100040600000051002007600008300090090006000700020010 .------------------.------------------.------------------. | 1389 1368 1369 | 68 5 138 | 2 7 4 | | 5 26 7 | 26 9 4 | 1 3 8 | | 12348 12348 134 | 7 138 1238 | 5 6 9 | :------------------+------------------+------------------: | 38 7 35 | 1 6 358 | 9 4 2 | | 6 348 349 | 289 38 2389 | 7 5 1 | | 19 15 2 | 59 4 7 | 6 8 3 | :------------------+------------------+------------------: | 12 1256 8 | 3 17 15 | 4 9 567 | | 134 9 1345 | 458 178 6 | 38 2 57 | | 7 3456 3456 | 4589 2 589 | 38 1 56 | '------------------'------------------'------------------'

And an image for the ASCII-impaired:

The blue candidates occupy 4 rows and 2 columns. Any move that would eliminate all candidates from 2 of those rows would force the remaining pattern into an X-Wing (Row-Column set of size 2)

The green candidates fit the same description.

The 2 patterns are connected by the candidates in rows 4 and 6. These 4 candidates can be grouped, because each of them can see all members of the other group: (r4c6]+[r6c2]) is strongly linked with ([r4c3]+[r6c4]).

It is now possible to define a (not so nice) loop that includes these twin ARCS:

[r9c9]-5-[ARCS:r79c26]=5=([r4c6]+[r6c2])-5-([r4c3]+[r6c4])=5=[ARCS:r89c34]-5-[r9c9] => [r9c9]<>5

I'm not sure if Jeff will approve of this addition, but on an abstract level, there is no difference between an ALS and an ARCS. Both are constraint subsets with a surplus.

Ruud

by **tarek** » Wed Aug 23, 2006 6:29 pm

Wasn't this mentioned before & described (In the general context) as Counting, This helped us when we discussed Big fish, especially when we realised that there shouldn't be a finned Squirmbag, & that there existed finned fish with 0 vetices lines........

It also swallows ALS completely & even provides simpler alternatives to ALS that go beyond ALS xy rules.....

It is powerful but how can we capture its power..........

tarek

by **daj95376** » Wed Aug 23, 2006 7:01 pm

This topic is way beyond my understanding of Sudoku techniques, but the example above is ripe for pseudo-Double Implication Chain logic.

Code: Select all: [r6c2]=5 => (X-Wing) => [r89c9]<>5. [r6c4]=5 => (X-Wing) => [r79c9]<>5.

Therefore, [r9c9]<>5.

by **Myth Jellies** » Sat Aug 26, 2006 12:16 am

Perhaps a little late (I've been on holiday) and off topic, but I see a grouped coloring method to make the following deduction, similar to what daj has seen...

Code: Select all: .------------------.------------------.------------------. | 1389 1368 1369 | 68 5 138 | 2 7 4 | | 5 26 7 | 26 9 4 | 1 3 8 | | 12348 12348 134 | 7 138 1238 | 5 6 9 | :------------------+------------------+------------------: | 38 7 X35 | 1 6 x358 | 9 4 2 | | 6 348 349 | 289 38 2389 | 7 5 1 | | 19 x15 2 |X59 4 7 | 6 8 3 | :------------------+------------------+------------------: | 12 X1256 8 | 3 17 X15 | 4 9 567 | | 134 9 x1345 |x458 178 6 | 38 2 57 | | 7 X3456 x3456 |x4589 2 X589 | 38 1 -56 | '------------------'------------------'------------------'

x=X => r9c9 <> 5

Either the x's or the X's are true, and as daj pointed out, the x's occupy an r89c34 x-wing, while the X's occupy an r79c26 x-wing, either of which will wipe out any other fives in r9.

I've used this type of logic to color x-wings and box-box interactions before, but I've never noticed a case like this which has two colored x-wings. Very interesting.

by **daj95376** » Sat Aug 26, 2006 1:08 am

Look here for another example with two X-Wings.

by **Myth Jellies** » Sat Aug 26, 2006 2:44 am

I hadn't seen that one yet, daj....

Code: Select all: *-----------------------------------------------------------------------------* | 1 279 3 | 4 2689 268 | 5 27 268 | | 67 8 24 |A136 5 A1236 | 1467 9 1267 | | 269 249 5 | 1689 12689 7 | 1468 124 3 | |-------------------------+-------------------------+-------------------------| | 4 B359 8 | 135679- 1369 1356- | 2 B137 167 | | 239 6 1 | 379 234 234 | 47 8 5 | | 235 B235 7 | 13568- 12348 12358- | 146 B134 9 | |-------------------------+-------------------------+-------------------------| | 8 457 9 | 2 146 1456 | 3 157 17 | |a235 1 24 |A358 7 A3458 | 9 6 28 | | 2357 b2357 6 | 1358 138 9 | 178 125 4 | *-----------------------------------------------------------------------------*

A = a - b = B => color A or color B or both is true. Either the colored box-box interaction A or the colored box-box interaction B will eliminate the threes in r46c46. These also work as colored x-wings.

Ruud, can you tell us if your ARCS basically say the same thing, only in a more general & programming friendly fashion?

by **Ruud** » Sat Aug 26, 2006 2:35 pm

The symmetry of this pattern is very nice

Any of the 4 red candidates can be eliminated with a nice loop:

[r4c4]-3-[ARCS:r28c46=r8c1]-3-[ARCS:r9c2=r46c28]-3-[r4c4] => [r4c4]<>3

The coloring version is obviously shorter.

Ruud

by **Havard** » Sun Aug 27, 2006 5:55 pm

Hi.

Great to see some innovation again!

I have to try and sum this up for myself. (read: to see if even I can understand this...)

so, basically we are dealing with fish that has extra bits. let's take a very basic example:

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

now, if it were not for that pesky little candidate in r8c2, we could eliminate all the little "*" in good old X-wing style.

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

now, what Ruud is saying(i think...) is that that extra candidate that prevents us from formin the fish, is connected to all the potential eliminations(*), by that if one of the * were to be TRUE, and that somehow made the r8c2 X false, then the * can't be true, and we can eliminate it anyway...

not being to imaginative, I can think of this little xy-chain connecting those:

Code: Select all: . . . | . . . | . . . . 3 . | . 3 . | . * . . | . | . | . | . . . --|---+---|---+------ . | . | . | . | . . . . 3 . | . 3 . | . * . . | . | . . . | . . . --|---+-------+------ . | . | . .45 | .53 . . 3 . |34 . . | . . . . . . | . . . | . . .

now if any of the * were to be TRUE, then (53) would become 5,
(45) would become 4,
(34) would become 3,
and our 3 in r8c2 could then not be a 3,
and we would get an X-wing formed with the 4 other 3's, so that both our * would be eliminated = CONTRADICTION!
hence we can safely eliminate the two "*".

Another way of looking at it is that the "extra 3" get's an "extended arm" using the little xy-chain to be able to "see" the X-wing-casualties(*).

Ruud demonstrated this with two X-wing's that can see eachothers "extrabits", and hence any eliminations the two x-wings have in common must be true.

Now Mike Barker presented (in my opinion) a bit of a twist to this. Instead of concidering the link between a fishes extra bits and its potential elimination, he finds ALS that will potentially reduce the fish to an "illegal" state. The proper definition is that if a fish concists of n columns, but only n-1 rows, it is in fact not possible... It is rotten fish, unfit for consumption!

now to again use an example as simple as possible, we can take the same fish from before:

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

We all agree that this is a column-fish, that takes up columns 2 and 5. It also occupies three rows, 2, 5 and 8. Now if the placement of any candidate would destroy the fish so much that it actually took away two of the three rows, we would have a rotten fish, 2 columns and only one row, and that would be a contradiction!

again not to imaginative, we could do this with two ALS placed like so:

Code: Select all: . . . | . . . | . . . . 3 . | . 3 . | . . . . | . | . | . | . . . --|---+---|---+------ . | . | . | . | . . . . 3 . | . 3 . | .34 . . | . | . . . | . . . --|---+-------+------ . | . | . . . | . . . . 3 . | . . . | . . . . .34 | . . . | . . .

now, one of our (34) sees "all" the candidates in the fishes third row. (row 8, yes there are only one...), and our other (34) sees all the candidates in the fish'es second row (row 5).

Now as some of you will see, these ALS makes it impossible for a 4 to vacate r9c8, so any 4 there can be eliminated(*):

Code: Select all: . . . | . . . | . . . . 3 . | . 3 . | . . . . | . | . | . | . . . --|---+---|---+------ . | . | . | . | . . . . 3 . | . 3 . | .34 . . | . | . . . | . . . --|---+-------+------ . | . | . . . | . . . . 3 . | . . . | . . . . .34 | . . . | . * .

why? because a 4 in r9c8 would make both (34)'s into 3's, and that would again destroy all the 3's in r8c2 and r5c25. That again would make it a rotten fish, hence no 4 in r9c8.

Now are these two (the Ruud and Barker ways) the same thing? I don't quite think so, but they are both great contributions to the solvingtechnique-librarys in my opinion. (but it should be added that I have found almost-x-wings in Carculs and others loops a long, long time ago, so there are probably a lot of credit to be handed out for this)

Havard

by **Myth Jellies** » Sun Aug 27, 2006 8:43 pm

A very nice job, Havard, of showing the difference in approach for the two multi-digit cases. Thumbs up all around.

The New Sudoku Players' Forum

Introducing ARCS: Almost Row-Column Subsets

Introducing ARCS: Almost Row-Column Subsets