The New Sudoku Players' Forum

by **rjamil** » Wed Nov 15, 2017 9:03 pm

Hi,

Please look in to basic Empty Rectangle exemplar 1 along with special case 2, where both ER-row and ER-col contain strong links:

Code: Select all: --------------+-------------+----------- --------------+-------------+----------- 01) . +Z . | . . . | . . . 02) . +Z . | . . . | . / . +Z +Z +Z | . . . | . -Z . +Z +Z +Z | . . . | . -Z . . +Z . | . . . | . . . . +Z . | . . . | . / . --------------+-------------+----------- --------------+-------------+----------- . . . | . . . | . . . . . . | . . . | . / . . . . | . . . | . . . . . . | . . . | . / . . . . | . . . | . . . . . . | . . . | . / . --------------+-------------+----------- --------------+-------------+----------- . . . | . . . | . . . . . . | . . . | . / . / +Z / | / / / | / +Z / / -Z / | / / / | / +Z / . . . | . . . | . . . . . . | . . . | . / . --------------+-------------+----------- --------------+-------------+-----------

I do understand how it works, already developed routine to catch the ERs and under testing phase.

My question relates with exemplar 2 only. Is this type of empty rectangle considered as elimination strategy or Empty Rectangle single move?

Since, imo, after eliminating from both the R2C8 and R8C2, almost always immediate next move will reveal hidden single move in R8C8.
Therefore, instead of just eliminate and detect singleton, directly solve the R8C8 cell.

Examples from Ruud50k:

Code: Select all: .816...9............4.376..6..4..5...3.....7...7..2..4..521.3............7...481. +---------+----------+--------+ | 3 8 1 | 6 4 5 | 7 9 2 | | 7 9 6 | 8 2 1 | 4 3 5 | | 5 2 4 | 9 3 7 | 6 8 1 | +---------+----------+--------+ |i6 1 89 | 4 7 -89 | 5 2 3 | | 4 3 2 | 1 5 6 | 9 7 8 | | 89 5 7 | 3 89 2 | 1 6 4 | +---------+----------+--------+ |-89 6 5 | 2 1 s89 | 3 4 7 | | 1 4 89 | 7 689 3 | 2 5 69 | | 2 7 3 | 5 69 4 | 8 1 69 | +---------+----------+--------+ 27) Found ERI Cell 27 Strong Link Cells 32 59 drop Clue 8 from Cell 54 27) Found ERI Cell 27 Strong Link Cells 54 59 drop Clue 8 from Cell 32

Code: Select all: .53.7...8........3..13......6.8....9.4..9..6.2....7.1......46..5........7...5.29. +----------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 289 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 i4 7 | 15 9 -13 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +----------------+----------------+--------------+ | 389 12389 289 | 279 1238 4 | 6 58 157 | | 5 1289 46 | 279 1268 1689 | 3 48 147 | | 7 -38 46 | 16 5 s38 | 2 9 14 | +----------------+----------------+--------------+ 37) Found ERI Cell 37 Strong Link Cells 41 77 drop Clue 3 from Cell 73 37) Found ERI Cell 37 Strong Link Cells 73 77 drop Clue 3 from Cell 41

R. Jamil

by **StrmCkr** » Tue Dec 12, 2017 11:12 pm

+---------+----------+--------+
| 3 8 1 | 6 4 5 | 7 9 2 |
| 7 9 6 | 8 2 1 | 4 3 5 |
| 5 2 4 | 9 3 7 | 6 8 1 |
+---------+----------+--------+
|i6 1 89 | 4 7 -89 | 5 2 3 |
| 4 3 2 | 1 5 6 | 9 7 8 |
| 89 5 7 | 3 89 2 | 1 6 4 |
+---------+----------+--------+
|-89 6 5 | 2 1 s89 | 3 4 7 |
| 1 4 89 | 7 689 3 | 2 5 69 |
| 2 7 3 | 5 69 4 | 8 1 69 |
+---------+----------+--------+

invalid grid at this point.

by **StrmCkr** » Tue Dec 12, 2017 11:19 pm

My question relates with exemplar 2 only. Is this type of empty rectangle considered as elimination strategy or Empty Rectangle single move?

the 2nd example is an ER elimination specifically it was classed as a dual ER {enabling the option in hudoku show cases them}

Code: Select all: .------------------.------------------.----------------. | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 289 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | :------------------+------------------+----------------: | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 13 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | :------------------+------------------+----------------: | 389 12389 289 | 279 1238 4 | 6 58 157 | | 5 1289 46 | 279 1268 1689 | 3 48 147 | | 7 38 46 | 16 5 38 | 2 9 14 | '------------------'------------------'----------------'

Dual Empty Rectangle: 3 in b4 (r67c5/r9c26) => r5c6,r7c1<>3
Dual Empty Rectangle: 3 in b8 (r6c25/r57c1) => r5c6,r9c2<>3
Dual Empty Rectangle: 3 in b8 (r57c1/r5c16) => r5c6,r7c1<>3
Dual Empty Rectangle: 3 in b8 (r6c25/r5c16) => r7c1,r9c2<>3
Dual Empty Rectangle: 3 in b7 (r67c5/r6c25) => r6c2,r7c5<>3
Dual Empty Rectangle: 3 in b4 (r67c5/r59c6) => r7c1,r9c2<>3

they can also be shown as individual empty rectangles as well, thus both views are correct.

Empty Rectangle: 3 in b4 (r9c26) => r5c6<>3
Empty Rectangle: 3 in b5 (r9c26) => r6c2<>
Empty Rectangle: 3 in b4 (r67c5) => r7c1<>3
Empty Rectangle: 2 in b2 (r27c3) => r7c5<>2
Empty Rectangle: 3 in b7 (r6c25) => r7c5<>3
Empty Rectangle: 3 in b4 (r59c6) => r9c2<>3

to point something out of why i suggested keeping it basic or limited expansion, and swamping to a fish engine with a naming engine attached to it

it is also possible to extend the named strategies 2-string kites, skyscrapers etc out in the same fashion as the ERI extension: same grid above features this
Dual 2-String Kite: 3 in r5c1,r7c5/in r6c2,r9c6 (connected by r5c6,r6c5) => r7c1,r9c2<>3

so it come down to how far you wish to push the extensions and do you want to do the same for each named technique.

by **rjamil** » Wed Dec 13, 2017 1:20 pm

Hi StrmCkr,

StrmCkr wrote:invalid grid at this point.

Know, but this state reached first by wrong guess, later, program automatically rollback, change guess and solution found. Included only for showing Dual Empty Rectangle state.

StrmCkr wrote:they can also be shown as individual empty rectangles as well, thus both views are correct.

Noticed that, in dual empty rectangle, an individual empty rectangle also work the same, i.e., dual ER searching is only for entertainment purpose. There is no need to search dual ER when speed/optimization is concern.

Please also note that, after implemented W-Wings strategy searched between XY-& XYZ-Wings and WXYZ-Wings, most of the Dual ERs vanished that searched after WXYZ-Wings. (And FYI, 2-String Kite and Grouped 2-String Kite searched after ERs.)

StrmCkr wrote:to point something out of why i suggested keeping it basic or limited expansion, and swamping to a fish engine with a naming engine attached to it

it is also possible to extend the named strategies 2-string kites, skyscrapers etc out in the same fashion as the ERI extension: same grid above features this
Dual 2-String Kite: 3 in r5c1,r7c5/in r6c2,r9c6 (connected by r5c6,r6c5) => r7c1,r9c2<>3

so it come down to how far you wish to push the extensions and do you want to do the same for each named technique.

Valid point. Most of the programs available now that shows step by step to solve the puzzle, including your good self "StormDuko" too (not tested, but I guess. Requesting you to kindly share compiled program for windows executable), included these types of basic and extended naming strategies for beginners, like myself.

Comparing 01) (Grouped) 2-String Kite and 02) Sashimi Mutant X-Wings (Turbot Fish) [derived from Fig 2C inverse]:

Code: Select all: --------------+-------------+----------- --------------+-------------+----------- 01) . +Z . | . . . | . . . 02)-Z +Z -Z | . . . | . -Z . +Z / +Z | / / / | / +Z / +Z / +Z | / / / | / +Z / . +Z . | . . . | . . . -Z +Z -Z | . . . | . -Z . --------------+-------------+----------- --------------+-------------+----------- . / . | . . . | . . . . / . | . . . | . -Z . . / . | . . . | . . . . / . | . . . | . -Z . . / . | . . . | . . . . / . | . . . | . -Z . --------------+-------------+----------- --------------+-------------+----------- . / . | . . . | . . . . / . | . . . | . -Z . . +Z . | . . . | . -Z . . +Z . | . . . | . -Z . . / . | . . . | . . . . / . | . . . | . -Z . --------------+-------------+----------- --------------+-------------+-----------

The above comparision arises confusion. In second exemplar, eight eliminations may be possible from C8, but not from R8. Why?

Actually, I read Strong Links for Beginners, posted on Mon Feb 27, 2006 11:56 am, by Havard. Found interesting and easy to understand. That's why I worked on Kites and Skyscraper also included in my wish list (only one exemplar, i.e., two strong links, lying within four boxes, whose one ends lying within line but not box; and other ends lying within chute but neither line nor box). Unfortunately, all the external referenced images are gone, including practicing examples, that will help to understand and develop the routine.

Thank you and appreciate your continued support and cooperation.

R. Jamil

by **StrmCkr** » Thu Dec 14, 2017 12:40 am

The above comparison arises confusion. In second exemplar, eight eliminations may be possible from C8, but not from R8. Why?

number 2 example:

the end point at cells R2C8 and R8C2 in both directions of the chain these cells are either True or False and Both can be true at the same time!
because of this one or the other is true or both are, the elimination cell must see both of them to be eliminated

ie they only both see cell R8C8 that is the only valid elimination.
no other elimination is possible. ->> number 1 is the only correct view.

hope this visual helps:

Code: Select all: Finned X-wing {max cell count} . / . | . . . | . x . . x . | . . . |-x x-x . / . | . . . | . x . ---------------------- . / . | . . . | . / . . / . | . . . | . / . . / . | . . . | . / . ---------------------- . / . | . . . | . / . . x . | . . . | . x . . / . | . . . | . / .

Code: Select all: Finned X-wing {minimal cell count} version 1 . / . | . . . | . / . . x . | . . . |-x x-x . / . | . . . | . x . ---------------------- . / . | . . . | . / . . / . | . . . | . / . . / . | . . . | . / . ---------------------- . / . | . . . | . / . . x . | . . . | . x . . / . | . . . | . / .

Code: Select all: Finned X-wing {minimal cell count} version . / . | . . . | . x . . x . | . . . |-x x-x . / . | . . . | . / . ---------------------- . / . | . . . | . / . . / . | . . . | . / . . / . | . . . | . / . ---------------------- . / . | . . . | . / . . x . | . . . | . x . . / . | . . . | . / .

Code: Select all: Sashimi - X-wing {max cell count } . / . | . . . | . x . . x . | . . . |-x /-x . / . | . . . | . x . ---------------------- . / . | . . . | . / . . / . | . . . | . / . . / . | . . . | . / . ---------------------- . / . | . . . | . / . . x . | . . . | . x . . / . | . . . | . / .

Code: Select all: Sashimi - X-wing {minimum cell count } version 1 . / . | . . . | . x . . x . | . . . |-x /-x . / . | . . . | . / . ---------------------- . / . | . . . | . / . . / . | . . . | . / . . / . | . . . | . / . ---------------------- . / . | . . . | . / . . x . | . . . | . x . . / . | . . . | . / .

Code: Select all: Sashimi - X-wing {minimum cell count } version 2 -x /-x| . . . | . x . . x . | . . . | . / . . / . | . . . | . / . ---------------------- . / . | . . . | . / . . / . | . . . | . / . . / . | . . . | . / . ---------------------- . / . | . . . | . / . . x . | . . . | . x . . / . | . . . | . / .

Code: Select all: Skyscraper { 2- minimal Sashimi x-wings combined} -x /-x | . . . | . x . . x . | . . . |-x /-x . / . | . . . | . / . ---------------------- . / . | . . . | . / . . / . | . . . | . / . . / . | . . . | . / . ---------------------- . / . | . . . | . / . . x . | . . . | . x . . / . | . . . | . / .

Code: Select all: 2-String kite . x . | . . . | . . . X / / | / / / | / x / . / . | . . . | . . . ---------------------- . / . | . . . | . . . . / . | . . . | . . . . / . | . . . | . . . ---------------------- . / . | . . . | . . . . x . | . . . | .-x . . / . | . . . | . . .

Code: Select all: Grouped 2-String kite {max} . x . | . . . | . . . x / x | / / / | / x / . x . | . . . | . . . ---------------------- . / . | . . . | . . . . / . | . . . | . . . . / . | . . . | . . . ---------------------- . / . | . . . | . . . . x . | . . . | .-x . . / . | . . . | . . .

Code: Select all: Grouped 2-String kite {minimal} version 1 . x . | . . . | . . . x / x | / / / | / x / . / . | . . . | . . . ---------------------- . / . | . . . | . . . . / . | . . . | . . . . / . | . . . | . . . ---------------------- . / . | . . . | . . . . x . | . . . | .-x . . / . | . . . | . . .

Code: Select all: Grouped 2-String kite {minimal} version 2 . x . | . . . | . . . x / / | / / / | / x / . x . | . . . | . . . ---------------------- . / . | . . . | . . . . / . | . . . | . . . . / . | . . . | . . . ---------------------- . / . | . . . | . . . . x . | . . . | .-x . . / . | . . . | . . .

Code: Select all: Empty Rectangle {max count on ERI} / x / | . . . | . / . x x x | . . . | . x . / x / | . . . | . / . ---------------------- . . . | . . . | . / . . . . | . . . | . / . . . . | . . . | . / . ---------------------- . . . | . . . | . / . .-x . | . . . | . x . . . . | . . . | . / .

Code: Select all: Empty Rectangle {minimal count on ERI} / x / | . . . | . / . x / / | . . . | . x . / / / | . . . | . / . ---------------------- . . . | . . . | . / . . . . | . . . | . / . . . . | . . . | . / . ---------------------- . . . | . . . | . / . .-x . | . . . | . x . . . . | . . . | . / .

I personal use a quick search engine that was easily written to cover the above per move type within 20~ lines of code per type

why i suggest looking at building a fish algorithm,

consider that Each end point can be extended by adding another ERI, Strong link or grouped weak link
repeat !
you can add on up to 27 - {used} sectors to each of these basic named patterns whom are all Fishes in NxN class

named by what sectors they used add adding in a sub category for specific named types.{if you wish}

size 1 - 7 { cyclops, x-wing,sword fish, jelly fish, Squimbag/ starfish , leviathan, whale. }
basic {rows or cols},
Fraken {Box + Row or Col }
mutant {Box + Row + Col }

plus some other rules outlined Here For NxN fish
personally i opt for my own variation of NxN+K fish with less worrying over endo/exo cells and which cell is finned
seen here

so if you choose to write 1 code for each type and consider expansion your basically writing an overly specific fish code for case - case
i Tried this my self for ERI's and ended up with a 1k+ line code that still missed cases, when i was discovering new ones as i increased the N size

at most a depth of 2 links per case is feasible quickly with marginal increases in run time: however as you mentioned this has minimal effects on outcome.

if you do want to delve into a fish finder i can help with that to a point i do not use speed up techniques though : a lot of programmers use templating to aid in narrowing down search time.
my current build for ALL valid cases {with/with out eliminations takes ~4 hours to execute size 1- 7 + 2 finned sectors on digits 1-9
via worst case scenario dry runs. { looking for a 7x7+2 }

size 1-4 runs very quickly, plus a large % of bigger fish "usually" have a complementary lower size fish to match

Code: Select all: * X * | . . . | . * . / X / | . . . | . / . X / X | / / / | / X / X *X X | * * * | * X * * X * | . . . | . * . / X / | . . . | . / . ---------+----------+---------- ---------+----------+---------- . / . | . . . | . * . . * . | . . . | . / . . / . | . . . | . * . . * . | . . . | . / . . / . | . . . | . * . . * . | . . . | . / . ---------+----------+---------- ---------+----------+---------- . / . | . . . | . * . . * . | . . . | . / . . # . | . . . | . ** . . ** . | . . . | . # . . / . | . . . | . * . . * . | . . . | . / . Fig 2C: rc\cb Fig 2C inverse: cb\rc rc\rb transpose rb\rc transpose turbot fish (2-stringed kite) turbot fish (ER + conjugate link) sashimi mutant x-wing

the confusion arises as you are assuming all the * cells are eliminated:
when the # cell is missing all the * and ** cells are eliminated.
if the # cell is present only the ** cell is eliminated.

KEY:

ps if you need an example puzzle that show cases the outlined stuff above

this grid has multiple examples for each type.

.------------------.------------------.----------------.
| 469 5 3 | 49 7 169 | 149 2 8 |
| 4689 2789 289 | 249 1268 15689 | 1459 457 3 |
| 489 2789 1 | 3 28 589 | 459 457 6 |
:------------------+------------------+----------------:
| 1 6 5 | 8 4 2 | 7 3 9 |
| 38 4 7 | 15 9 13 | 58 6 2 |
| 2 389 89 | 56 36 7 | 458 1 45 |
:------------------+------------------+----------------:
| 389 12389 289 | 279 1238 4 | 6 58 157 |
| 5 1289 46 | 279 1268 1689 | 3 48 147 |
| 7 38 46 | 16 5 38 | 2 9 14 |
'------------------'------------------'----------------'

by **rjamil** » Thu Dec 14, 2017 2:11 pm

Hi StrmCkr,

Thank you for such a detail response that clear my confusion (regarding # cell) but raise couple of questions in my mind (i.e., confusion converted to questions).

In example Fig 2c inverse, if R8C8 does not contain X clue, then there is only one occurrence of X in C8, correct? The question is, why someone search either manually or by optimized for speed program to search advance/complex strategies when there is simple hidden single column present in R2C8?

As far as one module for Wings and one module for Fishes is concern, am not that much too smart right now. Also, not familiar with Pascal language to decipher your set wise 20~ lines of code per type. Since, I use bit based pattern searching program with only pencil mark grid maintain and rest views calculated on demand basis, therefore, need more time to understand all aspect of wings/fishes first in order to write one module to cater entire family respectively.

FYI, I revised my program thoroughly and updated as and when needed. Recently, I merged XY-Wings chute wise and XYZ-Wings in to one module as they both look like same except apex contain wings common clue or not. Similarly, all XY-Wings Row-Column wise are coded in separate module. Also, WXYZ-Wings coded in two separate modules, i.e., one for wings within Box-Row-Column, and other for wings within Box-Line.

Just for your information and record, with XY-Wings Type 1 Transport first 32 examples included, my program searched following puzzles without guess and debugging as follows:

Code: Select all: File Description Puzzles Without Guess Approximate Seconds Max Guess Max Depth Ruud Specialty 50000 7011 120 36 9 Tarek Pearl 6000 0 80 244 14 Dobrichev PG 28655 9893 100 129 13 17 Clue 49157 47514 26 82 11 Top 1465 494 5 54 8

And, number of lines per module out of 2836 total lines, including comments, debugging and one command per line as follows:

Code: Select all: Lines Strategy 004 Naked single 016 Hidden single 249 / Naked pair | Hidden pair | Naked triplet | Hidden triplet | Naked quad \ Hidden quad 067 / Box-line reduction intersection \ Pointing and claiming intersection 101 / X-Wings | Sword Fish \ Jelly Fish 780 / XY-Wings Type 1 & 2* \ XY-Wings Type 1 Transport (32 out of 36 exemplars) 134 / XYZ-Wings* \ XYZ-Wings Hybrid Type 1 & 2 253 W-Wings Type 1 & 2 061+167 WXYZ-Wings Type 1, 2a, 2b, 3, 4a & 4b 126 Empty Rectangle & Dual Linked Empty Rectangle 054 2-String Kite & Grouped 2-String Kite 0 Trial & Error

R. Jamil

by **StrmCkr** » Thu Dec 14, 2017 5:46 pm

Re: Empty Rectangle Row&Col Strong Link

Postby rjamil » Thu Dec 14, 2017 2:11 pm
Hi StrmCkr,

Thank you for such a detail response that clear my confusion (regarding # cell) but raise couple of questions in my mind (i.e., confusion converted to questions).

In example Fig 2c inverse, if R8C8 does not contain X clue, then there is only one occurrence of X in C8, correct? The question is, why someone search either manually or by optimized for speed program to search advance/complex strategies when there is simple hidden single column present in R2C8?

In an all incumbent solver, degenerative cases are still valid under any technique set.

Based on order of operations using basic first these types won't be found.
But that dosen't mean they aren't there.
Technically a speed solution can be shorter by skipping basic steps and including them via the more complex step.
Run time however increases the more complex the search depth lenght and size is.

by **rjamil** » Tue Dec 19, 2017 1:33 pm

Hi StrmCkr,

Hidden Text: Show

Code: Select all: .53.7..28........3..13....6165842739.47.9..622....7.1......46..5.....3..7...5.29. +------------------+-------------------+--------------+ | 469 5 3 | 1469 7 169 | 149 2 8 | | 4689 2789 24689 | 124569 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +------------------+-------------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 135 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +------------------+-------------------+--------------+ | 389 12389 289 | 1279 1238 4 | 6 578 157 | | 5 1289 24689 | 12679 1268 1689 | 3 478 147 | | 7 138 468 | 16 5 1368 | 2 9 14 | +------------------+-------------------+--------------+ 0) Found Naked triplet Cells 39 48 75 Unit 12 drop Clues 156 from Cells 3 12 21 30 57 66 +------------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 24689 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +------------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 135 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +------------------+----------------+--------------+ | 389 12389 289 | 279 1238 4 | 6 578 157 | | 5 1289 24689 | 279 1268 1689 | 3 478 147 | | 7 138 468 | 16 5 1368 | 2 9 14 | +------------------+----------------+--------------+ 0) Found Hidden pair Cells 65 74 Unit 24 Clues 46 +------------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 24689 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +------------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 135 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +------------------+----------------+--------------+ | 389 12389 289 | 279 1238 4 | 6 578 157 | | 5 1289 46 | 279 1268 1689 | 3 478 147 | | 7 138 46 | 16 5 1368 | 2 9 14 | +------------------+----------------+--------------+ 0) Found Hidden pair Cells 73 77 Unit 8 Clues 38 +------------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 24689 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +------------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 135 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +------------------+----------------+--------------+ | 389 12389 289 | 279 1238 4 | 6 578 157 | | 5 1289 46 | 279 1268 1689 | 3 478 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +------------------+----------------+--------------+ 0) Found Naked pair Cells 65 74 Unit 11 drop Clues 46 from Cells 2 11 20 29 38 47 56 +----------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 289 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 135 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +----------------+----------------+--------------+ | 389 12389 289 | 279 1238 4 | 6 578 157 | | 5 1289 46 | 279 1268 1689 | 3 478 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +----------------+----------------+--------------+ 0) Found Box-Line Reduction Intersection Removal Cells 30 39 48 drop Clue 5 from Cells 31 32 40 41 49 50 +----------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 289 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 13 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +----------------+----------------+--------------+ | 389 12389 289 | 279 1238 4 | 6 578 157 | | 5 1289 46 | 279 1268 1689 | 3 478 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +----------------+----------------+--------------+ 0) Found Pointing and Claiming Intersection Removal Cells 7 16 25 drop Clue 7 from Cells 34 43 52 61 70 79 +----------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 289 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 13 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +----------------+----------------+--------------+ | 389 12389 289 | 279 1238 4 | 6 58 157 | | 5 1289 46 | 279 1268 1689 | 3 48 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +----------------+----------------+--------------+ 0) Found 2-String Kite Strong Link Row between Cell 19 and Cell 22 Column between Cell 11 and Cell 56 drop Clue 2 from Cell 58 +----------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 289 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 13 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +----------------+----------------+--------------+ | 389 12389 289 | 279 138 4 | 6 58 157 | | 5 1289 46 | 279 1268 1689 | 3 48 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +----------------+----------------+--------------+ 0) Found 2-String Kite Strong Link Row between Cell 41 and Cell 36 Column between Cell 49 and Cell 58 drop Clue 3 from Cell 54 +----------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 289 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 38 4 7 | 15 9 13 | 58 6 2 | | 2 389 89 | 56 36 7 | 458 1 45 | +----------------+----------------+--------------+ | 89 12389 289 | 279 138 4 | 6 58 157 | | 5 1289 46 | 279 1268 1689 | 3 48 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +----------------+----------------+--------------+ 0) Apply Hidden single Cell 36 Clue 3 from Clues 38 +----------------+----------------+--------------+ | 469 5 3 | 49 7 169 | 149 2 8 | | 4689 2789 289 | 249 1268 15689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+----------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 15 9 1 | 58 6 2 | | 2 89 89 | 56 36 7 | 458 1 45 | +----------------+----------------+--------------+ | 89 12389 289 | 279 138 4 | 6 58 157 | | 5 1289 46 | 279 1268 1689 | 3 48 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +----------------+----------------+--------------+ 1) Apply Naked single Cell 41 Clue 1 from Clues 1 +----------------+---------------+--------------+ | 469 5 3 | 49 7 69 | 149 2 8 | | 4689 2789 289 | 249 1268 5689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+---------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 58 6 2 | | 2 89 89 | 56 36 7 | 458 1 45 | +----------------+---------------+--------------+ | 89 12389 289 | 279 138 4 | 6 58 157 | | 5 1289 46 | 279 1268 689 | 3 48 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +----------------+---------------+--------------+ 2) Apply Naked single Cell 39 Clue 5 from Clues 5 +----------------+---------------+--------------+ | 469 5 3 | 49 7 69 | 149 2 8 | | 4689 2789 289 | 249 1268 5689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+---------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 36 7 | 458 1 45 | +----------------+---------------+--------------+ | 89 12389 289 | 279 138 4 | 6 58 157 | | 5 1289 46 | 279 1268 689 | 3 48 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +----------------+---------------+--------------+ 3) Apply Naked single Cell 42 Clue 8 from Clues 8 +----------------+---------------+--------------+ | 469 5 3 | 49 7 69 | 149 2 8 | | 4689 2789 289 | 249 1268 5689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+---------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 36 7 | 45 1 45 | +----------------+---------------+--------------+ | 89 12389 289 | 279 138 4 | 6 58 157 | | 5 1289 46 | 279 1268 689 | 3 48 147 | | 7 38 46 | 16 5 38 | 2 9 14 | +----------------+---------------+--------------+ 4) Apply Naked single Cell 48 Clue 6 from Clues 6 +----------------+---------------+--------------+ | 469 5 3 | 49 7 69 | 149 2 8 | | 4689 2789 289 | 249 1268 5689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+---------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +----------------+---------------+--------------+ | 89 12389 289 | 279 138 4 | 6 58 157 | | 5 1289 46 | 279 1268 689 | 3 48 147 | | 7 38 46 | 1 5 38 | 2 9 14 | +----------------+---------------+--------------+ 5) Apply Naked single Cell 49 Clue 3 from Clues 3 +----------------+---------------+--------------+ | 469 5 3 | 49 7 69 | 149 2 8 | | 4689 2789 289 | 249 1268 5689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+---------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +----------------+---------------+--------------+ | 89 12389 289 | 279 18 4 | 6 58 157 | | 5 1289 46 | 279 1268 689 | 3 48 147 | | 7 38 46 | 1 5 38 | 2 9 14 | +----------------+---------------+--------------+ 6) Apply Naked single Cell 75 Clue 1 from Clues 1 +----------------+---------------+--------------+ | 469 5 3 | 49 7 69 | 149 2 8 | | 4689 2789 289 | 249 1268 5689 | 1459 457 3 | | 489 2789 1 | 3 28 589 | 459 457 6 | +----------------+---------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +----------------+---------------+--------------+ | 89 12389 289 | 279 8 4 | 6 58 157 | | 5 1289 46 | 279 268 689 | 3 48 147 | | 7 38 46 | 1 5 38 | 2 9 4 | +----------------+---------------+--------------+ 7) Apply Naked single Cell 58 Clue 8 from Clues 8 +---------------+--------------+--------------+ | 469 5 3 | 49 7 69 | 149 2 8 | | 4689 2789 289 | 249 126 5689 | 1459 457 3 | | 489 2789 1 | 3 2 589 | 459 457 6 | +---------------+--------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +---------------+--------------+--------------+ | 9 1239 29 | 279 8 4 | 6 5 157 | | 5 1289 46 | 279 26 69 | 3 48 147 | | 7 38 46 | 1 5 3 | 2 9 4 | +---------------+--------------+--------------+ 8) Apply Naked single Cell 22 Clue 2 from Clues 2 +---------------+-------------+--------------+ | 469 5 3 | 49 7 69 | 149 2 8 | | 4689 2789 289 | 49 16 5689 | 1459 457 3 | | 489 789 1 | 3 2 589 | 459 457 6 | +---------------+-------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +---------------+-------------+--------------+ | 9 1239 29 | 279 8 4 | 6 5 157 | | 5 1289 46 | 279 6 69 | 3 48 147 | | 7 38 46 | 1 5 3 | 2 9 4 | +---------------+-------------+--------------+ 9) Apply Naked single Cell 54 Clue 9 from Clues 9 +--------------+-------------+--------------+ | 46 5 3 | 49 7 69 | 149 2 8 | | 468 2789 289 | 49 16 5689 | 1459 457 3 | | 48 789 1 | 3 2 589 | 459 457 6 | +--------------+-------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +--------------+-------------+--------------+ | 9 123 2 | 27 8 4 | 6 5 157 | | 5 128 46 | 279 6 69 | 3 48 147 | | 7 38 46 | 1 5 3 | 2 9 4 | +--------------+-------------+--------------+ 10) Apply Naked single Cell 56 Clue 2 from Clues 2 +-------------+-------------+--------------+ | 46 5 3 | 49 7 69 | 149 2 8 | | 468 2789 89 | 49 16 5689 | 1459 457 3 | | 48 789 1 | 3 2 589 | 459 457 6 | +-------------+-------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+-------------+--------------+ | 9 13 2 | 7 8 4 | 6 5 157 | | 5 18 46 | 279 6 69 | 3 48 147 | | 7 38 46 | 1 5 3 | 2 9 4 | +-------------+-------------+--------------+ 11) Apply Naked single Cell 57 Clue 7 from Clues 7 +-------------+------------+--------------+ | 46 5 3 | 49 7 69 | 149 2 8 | | 468 2789 89 | 49 16 5689 | 1459 457 3 | | 48 789 1 | 3 2 589 | 459 457 6 | +-------------+------------+--------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+------------+--------------+ | 9 13 2 | 7 8 4 | 6 5 15 | | 5 18 46 | 29 6 69 | 3 48 147 | | 7 38 46 | 1 5 3 | 2 9 4 | +-------------+------------+--------------+ 12) Apply Naked single Cell 61 Clue 5 from Clues 5 +-------------+------------+-------------+ | 46 5 3 | 49 7 69 | 149 2 8 | | 468 2789 89 | 49 16 5689 | 1459 47 3 | | 48 789 1 | 3 2 589 | 459 47 6 | +-------------+------------+-------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+------------+-------------+ | 9 13 2 | 7 8 4 | 6 5 1 | | 5 18 46 | 29 6 69 | 3 48 147 | | 7 38 46 | 1 5 3 | 2 9 4 | +-------------+------------+-------------+ 13) Apply Naked single Cell 62 Clue 1 from Clues 1 +-------------+------------+------------+ | 46 5 3 | 49 7 69 | 149 2 8 | | 468 2789 89 | 49 16 5689 | 1459 47 3 | | 48 789 1 | 3 2 589 | 459 47 6 | +-------------+------------+------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+------------+------------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 46 | 29 6 69 | 3 48 47 | | 7 38 46 | 1 5 3 | 2 9 4 | +-------------+------------+------------+ 14) Apply Naked single Cell 55 Clue 3 from Clues 3 +-------------+------------+------------+ | 46 5 3 | 49 7 69 | 149 2 8 | | 468 2789 89 | 49 16 5689 | 1459 47 3 | | 48 789 1 | 3 2 589 | 459 47 6 | +-------------+------------+------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+------------+------------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 46 | 29 6 69 | 3 48 47 | | 7 8 46 | 1 5 3 | 2 9 4 | +-------------+------------+------------+ 15) Apply Naked single Cell 67 Clue 6 from Clues 6 +-------------+-----------+------------+ | 46 5 3 | 49 7 69 | 149 2 8 | | 468 2789 89 | 49 1 5689 | 1459 47 3 | | 48 789 1 | 3 2 589 | 459 47 6 | +-------------+-----------+------------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+-----------+------------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 29 6 9 | 3 48 47 | | 7 8 46 | 1 5 3 | 2 9 4 | +-------------+-----------+------------+ 16) Apply Naked single Cell 13 Clue 1 from Clues 1 +-------------+-----------+-----------+ | 46 5 3 | 49 7 69 | 149 2 8 | | 468 2789 89 | 49 1 5689 | 459 47 3 | | 48 789 1 | 3 2 589 | 459 47 6 | +-------------+-----------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+-----------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 29 6 9 | 3 48 47 | | 7 8 46 | 1 5 3 | 2 9 4 | +-------------+-----------+-----------+ 17) Apply Naked single Cell 65 Clue 4 from Clues 4 +-------------+-----------+-----------+ | 46 5 3 | 49 7 69 | 149 2 8 | | 468 2789 89 | 49 1 5689 | 459 47 3 | | 48 789 1 | 3 2 589 | 459 47 6 | +-------------+-----------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+-----------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 29 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------------+-----------+-----------+ 18) Apply Naked single Cell 68 Clue 9 from Clues 9 +-------------+----------+-----------+ | 46 5 3 | 49 7 6 | 149 2 8 | | 468 2789 89 | 49 1 568 | 459 47 3 | | 48 789 1 | 3 2 58 | 459 47 6 | +-------------+----------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+----------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------------+----------+-----------+ 19) Apply Naked single Cell 5 Clue 6 from Clues 6 +-------------+---------+-----------+ | 4 5 3 | 49 7 6 | 149 2 8 | | 468 2789 89 | 49 1 58 | 459 47 3 | | 48 789 1 | 3 2 58 | 459 47 6 | +-------------+---------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+---------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------------+---------+-----------+ 20) Apply Naked single Cell 66 Clue 2 from Clues 2 +-------------+---------+-----------+ | 4 5 3 | 49 7 6 | 149 2 8 | | 468 2789 89 | 49 1 58 | 459 47 3 | | 48 789 1 | 3 2 58 | 459 47 6 | +-------------+---------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +-------------+---------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------------+---------+-----------+ 21) Apply Naked single Cell 0 Clue 4 from Clues 4 +------------+---------+-----------+ | 4 5 3 | 9 7 6 | 19 2 8 | | 68 2789 89 | 49 1 58 | 459 47 3 | | 8 789 1 | 3 2 58 | 459 47 6 | +------------+---------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +------------+---------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +------------+---------+-----------+ 22) Apply Naked single Cell 18 Clue 8 from Clues 8 +----------+---------+-----------+ | 4 5 3 | 9 7 6 | 19 2 8 | | 6 279 9 | 49 1 58 | 459 47 3 | | 8 79 1 | 3 2 5 | 459 47 6 | +----------+---------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +----------+---------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +----------+---------+-----------+ 23) Apply Naked single Cell 9 Clue 6 from Clues 6 +----------+---------+-----------+ | 4 5 3 | 9 7 6 | 19 2 8 | | 6 279 9 | 49 1 58 | 459 47 3 | | 8 79 1 | 3 2 5 | 459 47 6 | +----------+---------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +----------+---------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +----------+---------+-----------+ 24) Apply Naked single Cell 3 Clue 9 from Clues 9 +----------+--------+-----------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 279 9 | 4 1 58 | 459 47 3 | | 8 79 1 | 3 2 5 | 459 47 6 | +----------+--------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +----------+--------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +----------+--------+-----------+ 25) Apply Naked single Cell 12 Clue 4 from Clues 4 +----------+--------+-----------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 279 9 | 4 1 58 | 59 7 3 | | 8 79 1 | 3 2 5 | 459 47 6 | +----------+--------+-----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +----------+--------+-----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +----------+--------+-----------+ 26) Apply Naked single Cell 16 Clue 7 from Clues 7 +---------+--------+----------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 58 | 59 7 3 | | 8 79 1 | 3 2 5 | 459 4 6 | +---------+--------+----------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +---------+--------+----------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+--------+----------+ 27) Apply Naked single Cell 25 Clue 4 from Clues 4 +---------+--------+---------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 58 | 59 7 3 | | 8 79 1 | 3 2 5 | 59 4 6 | +---------+--------+---------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +---------+--------+---------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+--------+---------+ 28) Apply Naked single Cell 23 Clue 5 from Clues 5 +---------+-------+---------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 59 7 3 | | 8 79 1 | 3 2 5 | 9 4 6 | +---------+-------+---------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +---------+-------+---------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+---------+ 29) Apply Naked single Cell 24 Clue 9 from Clues 9 +---------+-------+---------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+---------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 45 1 45 | +---------+-------+---------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+---------+ 30) Apply Naked single Cell 15 Clue 5 from Clues 5 +---------+-------+--------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+--------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 4 1 45 | +---------+-------+--------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+--------+ 31) Apply Naked single Cell 51 Clue 4 from Clues 4 +---------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 4 1 5 | +---------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+-------+ 32) Apply Naked single Cell 19 Clue 7 from Clues 7 +---------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 4 1 5 | +---------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+-------+ 33) Apply Naked single Cell 53 Clue 5 from Clues 5 +---------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 4 1 5 | +---------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+-------+ 34) Apply Naked single Cell 14 Clue 8 from Clues 8 +---------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 4 1 5 | +---------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+-------+ 35) Apply Naked single Cell 6 Clue 1 from Clues 1 +---------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 4 1 5 | +---------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 18 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+-------+ 36) Apply Naked single Cell 70 Clue 8 from Clues 8 +---------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 4 1 5 | +---------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+-------+ 37) Apply Naked single Cell 64 Clue 1 from Clues 1 +---------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 4 1 5 | +---------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+-------+ 38) Apply Naked single Cell 71 Clue 7 from Clues 7 +---------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 89 89 | 6 3 7 | 4 1 5 | +---------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+-------+ 39) Apply Naked single Cell 73 Clue 8 from Clues 8 +---------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 29 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +---------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 9 89 | 6 3 7 | 4 1 5 | +---------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +---------+-------+-------+ 40) Apply Naked single Cell 46 Clue 9 from Clues 9 +-------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 2 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +-------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 9 8 | 6 3 7 | 4 1 5 | +-------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------+-------+-------+ 41) Apply Naked single Cell 10 Clue 2 from Clues 2 +-------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 2 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +-------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 9 8 | 6 3 7 | 4 1 5 | +-------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------+-------+-------+ 42) Apply Naked single Cell 47 Clue 8 from Clues 8 +-------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 2 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +-------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 9 8 | 6 3 7 | 4 1 5 | +-------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------+-------+-------+ 43) Apply Naked single Cell 74 Clue 6 from Clues 6 +-------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 2 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +-------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 9 8 | 6 3 7 | 4 1 5 | +-------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------+-------+-------+ 44) Apply Naked single Cell 11 Clue 9 from Clues 9 +-------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 2 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +-------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 9 8 | 6 3 7 | 4 1 5 | +-------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------+-------+-------+ 45) Apply Naked single Cell 77 Clue 3 from Clues 3 +-------+-------+-------+ | 4 5 3 | 9 7 6 | 1 2 8 | | 6 2 9 | 4 1 8 | 5 7 3 | | 8 7 1 | 3 2 5 | 9 4 6 | +-------+-------+-------+ | 1 6 5 | 8 4 2 | 7 3 9 | | 3 4 7 | 5 9 1 | 8 6 2 | | 2 9 8 | 6 3 7 | 4 1 5 | +-------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 5 1 4 | 2 6 9 | 3 8 7 | | 7 8 6 | 1 5 3 | 2 9 4 | +-------+-------+-------+ 46) Apply Naked single Cell 80 Clue 4 from Clues 4 1) 453976128629418573871325946165842739347591862298637415932784651514269387786153294 # S1# C34 # N46 # H1 # G0 # D0 # 0.000000

I am happy as long as my program solve the puzzle without guess. I am only concentrating to solve bulk puzzle without guess. Presently, I have no plan to convert my program in to interactive, GUI based, one puzzle at a time, etc., scenario basis. But, believe me, I will develop GUI based, when at least my solver will have as much strategies that will solve sufficient Ruud50k puzzles without guessing.

I have an impression that almost all Tarek Pearl 6000 puzzles are unsolvable by up to date known strategies (But may be I am wrong).

FYI, I normally coded single module that cover single strategy all minimum and maximum possible scenarios.

Wish to share that I have expanded my current Empty Rectangle module searching and able to solve 7021 Ruud50k puzzles without guess. Now it will cover all possible patterns (including min and max formations) as follows:

Hidden Text: Show

Code: Select all: Empty Rectangle (1 exemplar, 1 exclusion) Dual Empty Rectangle (1 exemplar, up to 2 exclusions) --------------+-------------+----------- --------------+-------------+----------- 01) / +Z / | . . . | . . . 02) / +Z / | . . . | . / . +Z +Z +Z | . . . | . -Z . +Z +Z +Z | . . . | . -Z . / +Z / | . . . | . . . / +Z / | . . . | . / . 12--------------+-------------+-----------12--------------+-------------+----------- . . . | . . . | . . . . . . | . . . | . / . . . . | . . . | . . . . . . | . . . | . / . . . . | . . . | . . . . . . | . . . | . / . --------------+-------------+----------- --------------+-------------+----------- . . . | . . . | . . . . . . | . . . | . / . / +Z / | / / / | / +Z / / -Z / | / / / | / +Z / . . . | . . . | . . . . . . | . . . | . / . --------------+-------------+----------- --------------+-------------+----------- Dual Linked Empty Rectangle (1 exemplar, up to 15 exclusions) --------------+-------------+----------- 01) / +Z / | . . . | / +Z / +Z -Z +Z | -Z -Z -Z | +Z -Z +Z / +Z / | . . . | / +Z / 12--------------+-------------+----------- . -Z . | . . . | . -Z . . -Z . | . . . | . -Z . . -Z . | . . . | . -Z . --------------+-------------+----------- . -Z . | . . . | . -Z . / +Z / | / / / | / +Z / . -Z . | . . . | . -Z . --------------+-------------+----------- Grouped Empty Rectangle (1 exemplar, 1 exclusion) --------------+-------------+----------- 01) / +Z / | . . . | . . . +Z +Z +Z | . . . | . -Z . / +Z / | . . . | . . . 72--------------+-------------+----------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . --------------+-------------+----------- / +Z / | . . . | / +Z / +Z +Z +Z | . . . | +Z +Z +Z / +Z / | . . . | / +Z / --------------+-------------+----------- Dual Linked Grouped Empty Rectangle (1 exemplar, up to 16 exclusions) --------------+-------------+----------- 01) / +Z / | . . . | / +Z / +Z -Z +Z | -Z -Z -Z | +Z -Z +Z / +Z / | . . . | / +Z / 72--------------+-------------+----------- . -Z . | . . . | . -Z . . -Z . | . . . | . -Z . . -Z . | . . . | . -Z . --------------+-------------+----------- / +Z / | . . . | / +Z / +Z -Z +Z | -Z -Z -Z | +Z -Z +Z / +Z / | . . . | / +Z / --------------+-------------+-----------

Note: number shown between first and second band's separator line is total number of combinations searched for each first ERI formation (that are 81).
Also, for each ERI combination searched, ERI digit present in at least one within mini-row and one within mini-column.

Edit on 20171230 12:25am (GMT+05:00): Hiding long code only.

R. Jamil