## Introducing Benoku

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Re: Introducing Benoku

SpAce wrote:What have I missed if I don't understand what can go in r2c8?

I think he has got you there, Leren!

Mathimagics
2017 Supporter

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Location: Canberra

### Re: Introducing Benoku

Actually I got myself by not reading the Benoku rules. I've now changed my puzzle on the previous page to one that should hopefully be Benuko compliant.

Anyway, here is the original puzzle, which I'll now dub Lerenoku . The rules are the same as for Benoku, except that a blank black cell stays blank. This makes for a slightly more Str8ts compliant puzzle type. It's up to you to figure out which digit is missing in each row and column in which a blank black cell appears. Non-blank black cells are also allowed.

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`  0    0    0    0    0    0    0    0    9    0    3    0    0    0    0    0   [0]   0    0    0    0    0    0    4    0    0    0    0    0    0   [0]   7    0    0    0    0    7    0    0    0    0    8    0    0    0    0   [0]   0    0    0    0    0    5    0    0    0    0    0    0    0    5    0    0    0    0    6    0    0    2    0    0    0    0    0    3    0    0    0   [0]   0    0  `

Leren
Last edited by Leren on Tue Aug 20, 2019 11:17 am, edited 1 time in total.
Leren

Posts: 3916
Joined: 03 June 2012

### Re: Introducing Benoku

The new Benoku-compliant puzzle solves using only basics:

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`+-----------+-----------+-------------+| 7   2   5 |  9   1  6 |  3    8   4 || 6   3   4 |  8   5  2 |  7   [9]  1 || 9   1   8 |  7   3  4 |  2    5   6 |+-----------+-----------+-------------+| 3   4   2 | [1]  7  5 |  8    6   9 || 1   5   7 |  6   8  9 |  4    3   2 || 8  [9]  6 |  2   4  3 |  5    1   7 |+-----------+-----------+-------------+| 2   8   9 |  5   6  7 |  1    4   3 || 4   7   1 |  3   9  8 |  6    2   5 || 5   6   3 |  4   2  1 | [9]   7   8 |+-----------+-----------+-------------+`
SCLT

Posts: 158
Joined: 06 August 2013

### Re: Introducing Benoku

Hi SCLT,

Yes, I had to do a quick fix after reading the Benoku rules, so the puzzle was easy. You might like to try my Lerenoku puzzle above (I really should call this variant Str8tsB, but this is supposed to be a Sudoku forum). It's a bit more challenging.

For a really tough challenge try this Lerenoku puzzle.

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`  0    0    3    0    0    0    0    0    0    0    0    0    5    0    0   [0]   0    0    0    0    0    0    0   [0]   0    0    0    6    0    0    0    0    0    0    0    7    0    0    0    0    0    0    8    0    0   [0]   0    1    0    3    0    0    4    0    0    0    0    0   [0]   0    0    0    0    0   [0]   6    0    0    0    0    0    0    3    0    0    4    0    0    0    5   [0] `

Leren
Leren

Posts: 3916
Joined: 03 June 2012

### Re: Introducing Benoku

Leren wrote:Actually I got myself by not reading the Benoku rules.

Of course. It would be preposterous to even think I could "get" a genius like you. Yet your excuse is actually worse than the alternative because that's not an accident. It's just arrogance. So is the fact that you don't bother to apologize for wasting other people's time, or to thank them for bringing the mistake to your attention. Nothing new there, though. I'm done with you and your puzzles, thank you.

SpAce

Posts: 2519
Joined: 22 May 2017

### Re: Introducing Benoku

Unique solutions for both Lerenoku found by hand-solving. I'm afraid I didn't record my solving path for either puzzle, but I only used basic moves and a general observation about Lerenoku.

Lerenoku 1:

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`+---------+---------+---------+| 6  5  4 | 7  2  1 | 3  8  9 || 8  3  7 | 9  5  6 | 4  .  2 || 1  2  9 | 8  3  4 | 6  7  5 |+---------+---------+---------+| 3  4  2 | .  7  5 | 9  6  8 || 7  6  5 | 4  9  8 | 2  1  3 || 9  .  8 | 2  6  3 | 7  5  4 |+---------+---------+---------+| 2  7  1 | 3  8  9 | 5  4  6 || 4  9  6 | 5  1  2 | 8  3  7 || 5  8  3 | 6  4  7 | .  2  1 |+---------+---------+---------+`

Lerenoku 2:

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`+---------+---------+---------+| 8  7  3 | 9  4  2 | 1  6  5 || 4  1  2 | 5  6  3 | .  7  8 || 5  6  9 | 8  7  . | 4  2  3 |+---------+---------+---------+| 6  5  8 | 1  2  4 | 3  9  7 || 7  3  4 | 6  5  9 | 8  1  2 || .  2  1 | 7  3  8 | 5  4  6 |+---------+---------+---------+| 2  4  5 | 3  .  7 | 6  8  9 || 1  .  6 | 2  8  5 | 7  3  4 || 3  8  7 | 4  9  6 | 2  5  . |+---------+---------+---------+`

I actually found the second puzzle easier than the first, although that may be because by that point I was used to the Lerenoku style. Here's the Lerenoku-specific logic that proved crucial:

In any chute (stack), the numbers that are missing from each row (column) must be the same numbers that are missing from the 3x3 boxes in that chute (stack), although not necessarily in a way that corresponds to a consistent number missing from each black cell. By cross-referencing each box with a blank cell against the set of rows/columns in the same chute/stack, you can determine constraints on which numbers can or cannot be missing from a given 3x3 box, and that opens up deductions which would not a priori be available.
SCLT

Posts: 158
Joined: 06 August 2013

### Re: Introducing Benoku

SCLT wrote:Unique solutions for both Lerenoku found by hand-solving. I'm afraid I didn't record my solving path for either puzzle, but I only used basic moves and a general observation about Lerenoku.

I find that impressive. Personally I lack either the necessary IQ or the patience (or both) to solve these. The whole concept is just too unintuitive. I've had no problems with Blocku, Benoku (the fixed version), or Str8ts (now that I've looked into that too), but this melts my brain. My instincts are all off because it looks too much like a sudoku but isn't. It's also much harder to work with in Hodoku, which is a step backwards. My biggest gripe about Blocku was that it required slow p&p solving, but Benoku fixed that. This can also be entered into Hodoku, of course, but at some point the candidate maintenance gets difficult because the black cells can represent different digits for rows and columns.

That being said, let's face it that I just couldn't figure out a working process like you did, at least not in the time I was willing to put into it (now that I finally tried). Nice job! Also congrats to Leren for creating a puzzle type that beats me.

SpAce

Posts: 2519
Joined: 22 May 2017

### Re: Introducing Benoku

Brilliant observation by SCLT. If all 3 rows in a tier, or columns in a stack, must contain a digit then a box must contain that digit, even if it has a grey cell. This enables you to build a table of required digits in grey cell boxes, increasing the chances of pointing intersections, and hidden tuples in boxes, and possibly more I haven't thought of yet. Whether this variant ends up being popular is another matter.

Leren
Leren

Posts: 3916
Joined: 03 June 2012

### Re: Introducing Benoku

SpAce wrote:Personally I lack either the necessary IQ or the patience (or both) to solve these. The whole concept is just too unintuitive.

That's completely understandable.

Leaving aside the Str8ts-type compartment rule, Lerenoku is similar to a variant called "Blackout Sudoku" which occasionally appears on the competitive scene. Its rules are that each region must contain 8 distinct digits from 1-9 and a black cell, so "hidden" logic tends to fail because there are no regions that must contain every digit. It's often derided by those who take part in the competitions as being too unlike Sudoku, and too unintuitive to make a good competition puzzle.

The annoying thing about variants like this is that it's just too tempting to see hidden singles in places where they haven't been proven to exist yet!

In case you're at all interested, I re-solved the first Lerenoku puzzle and recorded my solution path. No promises that it's optimal, or that I didn't miss obvious steps or do things in the "wrong" order at any point. But if you have the time to follow through it it might help you to appreciate the logic behind the variant a bit more.

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`Basic compartments: -3467r1c8, -4567r2c9, -5r123c4, -56r4c123, +23r4c123, -12r4c6789, +5r4c6, +6r4c789, -346r6c1, +789r789c2, -46r9c89Row 4 and Row 6 must both contain a 6 --> Block 4 must contain a 6 --> +6r5c2Compartments: -1c2Column 2 missing a 1 --> Block 4 must also be missing a 1Naked triples 234 r4c123, 689r4c789Singles: +5r5c3, +3r4c1Pointing 5b7: -5r123c1Row 2 must contain 5 --> Block 3 contains 5 --> +5r3c9 --> +5r1c2 --> +5r2c5Singles +2r3c2, +4r4c2, +2r4c3Compartments and pair 89r6: +9r6c1, +8r6c3Compartments: -1r6Row 4 and Row 6 both missing 1 --> Block 5 must also be missing 1Block 5 missing 1 --> Column 4 must be missing 1Compartments: -23r123c4, +789r123c4, +23456r56789c4Claiming 2b5: -2r56c5Singles: +2r1c5, +9r5c5Row 2 must contain 4 --> Block 3 must also contain 4Pointing 4b3: -4r568c7Block 3 contains a 9 --> Row 2 must contain a 9 --> -12r2c1234567Singles (in order): +6r2c6, +3r6c6, +1r1c6, +3r3c5, +8r1c8, +7r1c4, +4r1c3, +6r1c1, +8r2c1, +1r3c1, +1r7c3, +9r2c4, +7r2c3, +9r3c3, +8r3c4, +4r2c7, +3r1c7Compartments: -9c8Column 8 missing a 9 and Block 3 contains a 9 --> Block 9 must be missing 9Singles: +9r4c7, +6r4c8, +8r4c9, +9r8c2, +6r3c7, +7r3c8Compartments: -1c7Singles: +2r5c7, +7r6c7, +8r8c7, +4r6c9, +6r6c5, +2r6c4, +4r5c4Row 9 must contain a 6 --> Block 9 must contain a 6 --> +6r7c9Singles to the end`
SCLT

Posts: 158
Joined: 06 August 2013

### Re: Introducing Benoku

Hi SCLT. There is a wealth of good information in your solution path. I'll try to improve the speed of my solver so that it more closely matches what a human solver would do.

Leren
Leren

Posts: 3916
Joined: 03 June 2012

### Re: Introducing Benoku

Many thanks to all of you for posting comments, specially SCLT and Leren,

I'm working on a way that it shows the optimal way to solve the puzzle, any software programmer here who is willing to help?

Posts: 75
Joined: 31 July 2019

### Re: Introducing Benoku

SCLT wrote:Leaving aside the Str8ts-type compartment rule, Lerenoku is similar to a variant called "Blackout Sudoku" which occasionally appears on the competitive scene. Its rules are that each region must contain 8 distinct digits from 1-9 and a black cell, so "hidden" logic tends to fail because there are no regions that must contain every digit. It's often derided by those who take part in the competitions as being too unlike Sudoku, and too unintuitive to make a good competition puzzle.

Thanks for that piece of information! Never heard of Blackout Sudoku before, but I'm not familiar with many variants anyway.

The annoying thing about variants like this is that it's just too tempting to see hidden singles in places where they haven't been proven to exist yet!

Exactly. Or, unless you're really certain about the rules and their implications, it might feel too scary to take even valid ones. That's kind of what happened to me.

In case you're at all interested, I re-solved the first Lerenoku puzzle and recorded my solution path. No promises that it's optimal, or that I didn't miss obvious steps or do things in the "wrong" order at any point. But if you have the time to follow through it it might help you to appreciate the logic behind the variant a bit more.

Thanks for that! I really appreciate it! Before I dive deeper into it, let me just make sure I understand the crucial bit:

Row 4 and Row 6 must both contain a 6 --> Block 4 must contain a 6 --> +6r5c2

I think I see that now. I didn't understand that part in your original post, but now it makes sense. You could make it even clearer by saying that explicit 6s are needed in both rows (as opposed to implicit ones within the black cells). I think the logic is also easier to understand if I focus on row 5, instead of box 4 (btw, "block" can be confused with the black cells, as in Blocku). Unlike box 4, row 5 doesn't have any black cells so it must contain all digits explicitly, which makes it easier to deal with. If we recognize the fact that neither black cell in r4 or r6 can represent a 6 row-wise (for different reasons), it leaves just r4c789 and r6c456 for 6s in those rows, and then claiming gives us a row-wise hidden single 6r5c2. Would you agree with that logic, and that it yields the same result? I'm sure yours works nicely and more quickly once internalized, but until then this POV might feel safer and more intuitive for me.

Perhaps it's conceptually wrong, but I still see the black cells as digits (or sets of candidates), though possibly different ones depending on the context (row, col, box). That's why I tried to maintain candidates within them, but color-coded to specify whether one applied to the row or col or both. It was a bit messy, but now that I might understand the logic a bit better, I think it might actually work that way too. I'll have to try again.

SpAce

Posts: 2519
Joined: 22 May 2017

### Re: Introducing Benoku

SpAce wrote:Would you agree with that logic, and that it yields the same result?

Upon re-reading a couple of times, yes, I believe so. But I find it a bit uncomfortable, presumably in the same way my description is unintuitive to you. That's probably a result of you still seeing the black cells containing candidates, and me seeing them as completely empty - indeed, the first thing I do when solving one of these puzzles in something like Hodoku is to remove all candidates from the black cells.

Here's the clearest way (IMO) of explaining the logic from the POV of the black cells being completely empty. Looking row-wise, the middle tier must contain in total three 6s, because each row has one. Looking at the same cells box-wise instead of row-wise, there must therefore be one in each box.

(This same explanation generalises in the way that I pointed out originally to say that is S is the set (with repeats) of digits missing from the three rows/columns in a tier/stack, then the set of digits missing from the boxes in that same tier/stack must also be exactly S. I can't easily convince myself that your POV is as powerful, although maybe it is.)
SCLT

Posts: 158
Joined: 06 August 2013

### Re: Introducing Benoku

Well, after putting SLCT's Tier/Stack Box techniques into my SudokuB solver all of my puzzles now solve with basics. I would suggest that SudokuB is more constrained than Benoku because you always know the exact value of the black cell, whereas with Benoku there is some latitude there. With that in mind I've come up with a SudokuB puzzle that my solver uses more than basics. The trick ? Use the minimum of black cells.

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`  0    0    0    6    0    0    0    0    0    0    5    0    4    0    0    7    0    0    0    0    0    0    0    9    0    3    5    7    0    2    0    0    0    5    0    0    0    0    0    0   [0]   0    0    0    0    0    0    4    0    0    0    0    8    0    6    0    0    0    0    0    0    0    0    1    0    0    0    0    0    2    0    0    0    0    0    0    0    3    4    9    0  `

Leren
Leren

Posts: 3916
Joined: 03 June 2012

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