## Benoku (Part II)

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Benoku (Part II)

Benoku is a logic-based number-placement puzzle. It is distinct from but shares some properties and rules with Str8ts and Sudoku.
Rules:
• All sudoku rules apply.
• Some rows and columns are divided into two compartments by a gray cell
• Each compartment, vertically or horizontally, must contain a straight – a set of consecutive numbers, but in any order. For example: 7, 6, 4, 5 is valid, but 1, 3, 8, 7 is not.
• Each puzzle has a UNIQUE solution

Benoku Definition:PDF

Benoku Sample (Easy):PDF

Gray cells are shown with [ ]
Code: Select all
` 0   0   3   |  9   0   0   |  0   0   5   0   0   0   |  5   0   0   | [9]  0   0   0   0   9   |  0   0  [0]  |  0   0   0  ---------------------------------------- 6   0   0   |  0   0   0   |  0   9   7   0   0   0   |  0   0   0   |  8   1   0  [0]  0   1   |  0   3   8   |  0   4   0  ---------------------------------------- 0   0   0   |  3  [0]  0   |  0   0   0   0  [0]  6   |  2   0   0   |  0   0   4   3   0   0   |  4   0   0   |  0   5  [0]`

Posts: 75
Joined: 31 July 2019

### Re: Benoku (Part II)

Your puzzle was unique and easy to solve.

Leren
Last edited by Leren on Tue Aug 20, 2019 10:57 am, edited 2 times in total.
Leren

Posts: 3916
Joined: 03 June 2012

### Re: Benoku (Part II)

Leren wrote:Your puzzle was unique and easy to solve.

I concur. That's a good (new) beginning! More challenge can be added as long as uniqueness is not compromised.

You can also remove the clue from the gray cell and the puzzle solution is unique but difficult.

I didn't notice it changed anything. I'd ask what you meant, but since you choose to ignore everything I say, never mind

Code: Select all
`  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] `

This is the fully reduced puzzle with a unique solution that is more difficult to solve.

Not much. Still just basics and compartment rules, as far as I see.

SpAce

Posts: 2519
Joined: 22 May 2017

### Re: Benoku (Part II)

Code: Select all
`  0    1    2    0    0    0    0    0    8    0    6    0    0    0    0    0    0    0    0    0    0    0    0   [0]   0    0    0    0    0    0    9    0    0    0    7    0    0    0    8    0    0    0    0    0    3   [0]   0    0    0    0    0    2    0    0    0    0    0    0   [0]   0    0    0    0    0   [0]   0    0    0    0    0    4    0    0    0    0    0    0    0    7    0   [0] `

Code: Select all
`  8    0    0    0    0    0    0    0    0    0    5    0    6    0    0    1    8    0    0    0    3    5    0    0    0    0    0    0    0    0    0    0    0    2    0    0    0    0    0    0   [5]   0    0    0    0    3    0    7    0    0    0    0    0    9    1    0    6    0    0    0    3    0    0    0    2    0    0    0    0    0    0    0    0    0    0    0    0    4    9    0    7 `

Leren
Leren

Posts: 3916
Joined: 03 June 2012

### Re: Benoku (Part II)

Looks like I wasn't quite done with your puzzles after all.

Leren wrote:
Code: Select all
`  0    1    2    0    0    0    0    0    8    0    6    0    0    0    0    0    0    0    0    0    0    0    0   [0]   0    0    0    0    0    0    9    0    0    0    7    0    0    0    8    0    0    0    0    0    3   [0]   0    0    0    0    0    2    0    0    0    0    0    0   [0]   0    0    0    0    0   [0]   0    0    0    0    0    4    0    0    0    0    0    0    0    7    0   [0] `

solution: Show
Code: Select all
`.------------.------------.------------.|  4   1   2 | 7   3   9  | 5   6   8  ||  5   6   3 | 2   4   8  | 1   9   7  ||  8   7   9 | 5   6  [1] | 4   3   2  |:------------+------------+------------:|  6   2   1 | 9   5   3  | 8   7   4  ||  7   4   8 | 1   2   6  | 9   5   3  || [9]  3   5 | 8   7   4  | 2   1   6  |:------------+------------+------------:|  2   5   4 | 3  [1]  7  | 6   8   9  ||  1  [9]  7 | 6   8   2  | 3   4   5  ||  3   8   6 | 4   9   5  | 7   2  [1] |'------------'------------'------------'`

Besides basics & compartment rules:

XY-Wing (46-65-54)r4c91,r6c3 => -4 r4c2,r6c9
W-Wing (54) => -5 r2c4
W-Wing (65) => -6 r4c5
AIC => -2 r45c6
XY-Chain => -7 r1c1,r56c2; stte

Code: Select all
`  8    0    0    0    0    0    0    0    0    0    5    0    6    0    0    1    8    0    0    0    3    5    0    0    0    0    0    0    0    0    0    0    0    2    0    0    0    0    0    0   [5]   0    0    0    0    3    0    7    0    0    0    0    0    9    1    0    6    0    0    0    3    0    0    0    2    0    0    0    0    0    0    0    0    0    0    0    0    4    9    0    7 `

solution: Show
Code: Select all
`.-----------.-----------.-----------.| 8   7   2 | 9   3   1 | 4   6   5 || 4   5   9 | 6   2   7 | 1   8   3 || 6   1   3 | 5   4   8 | 7   9   2 |:-----------+-----------+-----------:| 9   8   5 | 7   1   6 | 2   3   4 || 2   4   1 | 3  [5]  9 | 8   7   6 || 3   6   7 | 4   8   2 | 5   1   9 |:-----------+-----------+-----------:| 1   9   6 | 2   7   5 | 3   4   8 || 7   2   4 | 8   9   3 | 6   5   1 || 5   3   8 | 1   6   4 | 9   2   7 |'-----------'-----------'-----------'`

Besides basics & direct compartment rules:

(1|2)r9c4 => -1234 r6789c5; btte

SpAce

Posts: 2519
Joined: 22 May 2017

### Re: Benoku (Part II)

Has anyone figured out the exact conditions when uniqueness techniques can be used with these puzzles, if ever? Looks like they're very risky in most cases.

For example, here's a partially solved state from Leren's first puzzle above:

Code: Select all
`.----------------.--------------------.-------------.| 47   1      2  |  47   b3     9     |  5  b6   8  || 45   6      3  |  245   245   8     |  1   9   7  || 8   a57    a9  |  567  b56    1     |  4  b3   2  |:----------------+--------------------+-------------:| 56   2345   1  |  9     2456  23456 |  8   7   46 || 67   247    8  |  1     2467  246   |  9   5   3  || 9   e3457  e45 |  8     4567  3456  |  2   1   46 |:----------------+--------------------+-------------:| 2  ce45    e45 | d3     1     7     | d6   8  c9  || 1  ac9     a7  | d26    8     26    | d3   4  c5  || 3    8      6  |  45    9     45    |  7   2   1  |'----------------'--------------------'-------------'`

It contains several inviting-looking Type 1 Avoidable Rectangles and one Unique Rectangle. Except for the last one, they'd all produce false results:

AR a: (79)r38c23 => +5 r3c2 (false)
AR b: (36)r13c58 => +5 r3c5 (false)
AR c: (59)r78c29 => +4 r7c2 (false)
AR d: (36)r78c47 => +2 r8c4 (false)
UR e: (45)r67c23 => -45 r6c2 (true)

SpAce

Posts: 2519
Joined: 22 May 2017

### Re: Benoku (Part II)

I don't know whether koushanejad74 is aware of this, or anyone else has noticed it, but apart from r46c46, r5c5, the only values that can possibly go into a grey cell in a Benoku puzzle are 1 and 9.

In r5c5, 5 can also go there. In r46c6 4 and 6 can also go there.

The reason for this is easy to see. The grey cells are the only ones allowed in their row and column in a Benuko puzzle.

If the grey cell is on any edge cell of the puzzle, it sees an 8 cell compartment in its row or column, which must have values 1-8 or 2-9, so the grey cell can only be 1 or 9.

If the grey cell is in Row 5 or Column 5 (but not in r5c5), it splits the column or row into two 4 cell compartments and can only have the values 1, 5, or 9. However, in the row or column there is one compartment that contains >= 5 cells, which must contain a 5, which leaves only 1 and 9 for the grey cell.

Now look at off diagonal cells, for example r4c3. A little bit of thought will lead to the conclusion that the grey cell can only have the values 1, 3, 7 or 9 considering the row (which must contain one 2 cell compartment and one 6 cell compartment) or 1, 4, 6 and 9 considering the column (which must contain one 3 cell compartment and one 5 cell compartment). Since it can't be 3, 4, 6, or 7 at the same time the only allowed values are 1 and 9.

So far, everything I've said is valid for any Str8ts puzzle (which also allows blanks in grey cells, but Benoku does not).

Now look at a diagonal cell, (other than Box 5). Pick, say r3c3 for example. The allowed values are 1, 3, 7 or 9 for both the row and the column. So you might conclude that it could be 3 or 7. This is where the boxes come in. r3c3 is in Box 1 and the 2 cell compartments in the grey cell's row and column must contain different values, so the grey cell can't be 3 or 7 at the same time, so again it can only be 1 or 9. A similar argument can be made for any off-diagonal cell (I think).

In r5c5. The allowed values are 1, 9 and 5, and 5 does not cause any sort of contradiction of the sort described above. Similarly in r46c6 you can have 1, 4, 6 and 9 and this does not cause a contradiction. QED (hopefully).

This fact greatly reduces the scope for finding puzzles in Benoku. The best way to find a puzzle is to pick your grey cell positions, put 1 or 9 in them (or 5 in r5c5, 4 or 6 in r46c46) and test for a contradiction (not all 1/9/5/46 grey cell patterns are mutually compatible). If no contradiction is found then there are usually lots of possible puzzles, which you can generate by filling in clues until you get a unique solution. With luck you can then also remove some or all of the 1's and 9's (and 5/46) from the grey cells and still have a unique solution puzzle. At least that's what I've been doing.

I'll finish off by presenting just one more puzzle, which has three different values in grey cells, and is not trivial to solve.

Code: Select all
`  0    0    0    1    0    0    0    0    0    0    0    7    0    0    0    0   [9]   0    0    0    0    2    0    3    0    0    0    8    0    0    0    0    0    0    5    0    0    0    0    0   [5]   0    0    0    0    0    0    3    0    0    0    9    0    0    0   [1]   0    0    0    7    2    4    0    0    0    0    0    0    0    0    0    5    0    0    4    0    0    0    0    0    0`

Leren

PS I thought of the 46 in r46c6 thing at the last minute and tested it. It works.
Last edited by Leren on Thu Aug 22, 2019 8:51 am, edited 8 times in total.
Leren

Posts: 3916
Joined: 03 June 2012

### Re: Benoku (Part II)

Leren wrote:I don't know whether koushanejad74 is aware of this, or anyone else has noticed it, but apart from r5c5, the only values that can possibly go into a grey cell in a Benoku puzzle are 1 and 9. In r5c5, 5 can also go there.

If you were actually interested in what other people say, you might have noticed Wecoc's post about the same thing. You're also missing what he didn't: the corners of box 5. I don't see anything wrong with these configurations:

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

I don't make puzzles, though, so I have no idea if some other reasons prevent the (4/6) blockers in reality.

SpAce

Posts: 2519
Joined: 22 May 2017

### Re: Benoku (Part II)

Leren wrote:PS I thought of the 46 in r46c6 thing at the last minute and tested it. It works.

Lol. Funny that you "thought" of it only after I mentioned it.

You're a piece of work, Leren. I don't know anyone else here who's quite as reluctant to admit his mistakes and give credit to other people (and that's a lot to say, because it's not exactly easy for some others either). And no, I don't deserve the credit in this case. Wecoc does. (It's obvious that giving credit to me would be physically impossible for you, so I would never expect that anyway. I think it's pretty funny, actually.)

I have no problem admitting that when I first tried Benokus I thought only 1 or 9 could be blockers, so I really appreciated Wecoc's clarification on the matter. My excuse was that it seemed to be the case with the only two examples I'd seen, and it was hard to draw definite conclusions about anything anyway when the first samples were invalid to begin with. Still I can easily admit that I didn't really think it through, and hence missed those other possibilities.

For the same reasons I was wrong to say that Kousha's posted rules were "ambiguous and confusing". They weren't -- only the sample puzzles were. (Well, the "valid straight" example 7, 6, 4, 5 is still confusing because it can't really exist, but maybe it's on purpose to leave it up to the solver to figure out which straights can exist and which can't.)

See, those are some simple examples of owning one's mistakes. It's not that hard.

SpAce

Posts: 2519
Joined: 22 May 2017

### Re: Benoku (Part II)

Leren wrote:
Code: Select all
`  0    0    0    1    0    0    0    0    0    0    0    7    0    0    0    0   [9]   0    0    0    0    2    0    3    0    0    0    8    0    0    0    0    0    0    5    0    0    0    0    0   [5]   0    0    0    0    0    0    3    0    0    0    9    0    0    0   [1]   0    0    0    7    2    4    0    0    0    0    0    0    0    0    0    5    0    0    4    0    0    0    0    0    0`

solution: Show
Code: Select all
`.---------.---------.---------.| 3  9  6 | 1  7  5 | 4  8  2 || 2  5  7 | 6  8  4 | 3 [9] 1 || 4  8  1 | 2  9  3 | 5  6  7 |:---------+---------+---------:| 8  7  9 | 4  6  2 | 1  5  3 || 1  4  2 | 3 [5] 9 | 6  7  8 || 5  6  3 | 7  1  8 | 9  2  4 |:---------+---------+---------:| 9 [1] 5 | 8  3  7 | 2  4  6 || 6  2  8 | 9  4  1 | 7  3  5 || 7  3  4 | 5  2  6 | 8  1  9 |'---------'---------'---------'`

Besides basics and compartment rules:

Step 1. T&E (singles): (6789)r5c1234 -> contradiction => (1234)r5c1234
Step 2. Remote Pair(56) => -56 r9c1; stte

SpAce

Posts: 2519
Joined: 22 May 2017