## Tight Bottlenecks

Everything about Sudoku that doesn't fit in one of the other sections

### Tight Bottlenecks

An important feature for a hard puzzle is that it should bottleneck at some point, reach a point where there isn't many options to proceed. For a computer, a bottleneck doesn't really have any effect, if a certain technique is programmed, then it will always find the easiest path. For a human though, it might sometimes be that you don't see the obvious and do it the hard way. This thread is for puzzles that will give you a very hard time, unless you spot the single move that let's you solve the puzzle easily.

Here's a starter:

Code: Select all
` *-----------* |..2|..7|...| |..4|...|..5| |9.6|...|81.| |---+---+---| |...|...|34.| |2..|.1.|...| |.8.|..5|7..| |---+---+---| |.4.|8.1|.6.| |...|...|...| |3..|69.|...| *-----------*`

This puzzle solves with singles, locked candidates and one naked triplet. Now what if the solver cannot see the naked triplet? Here's the path Sudoku Explainer will take if the triplet is ignored:

Nishio Forcing Chains: 2
Region Forcing Chains: 3
Dynamic Region Forcing Chains: 1
Explainer rating: 8.9

Sudocue also finds it quite hard when I turn of all subsets:

XYZ-wing: 1
APE: 3
Finned swordfish: 2
Tabling: 19
Total score: 16180

If I allow sudocue to use naked and hidden pairs, then it's enough with 10 tabling steps...

What can I say, better learn your basic techniques well! What's the tightest bottleneck you can find? Hint on finding tight bottlenecks: If suexrat gives > 400 points and gsf's program < 100, then there probably is a tight bottleneck that solves with subsets. Apparently suexrat doesn't find naked triplets...

RW
RW
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Joined: 16 March 2006

This is the one i mentioned to have suexrate 390, but can be solved with a naked triple:
Code: Select all
`------------.27|...|6.....|...|...6..|..4|.78------------5..|..7|.16...|...|....6.|.3.|5.9------------..9|651|...8..|.2.|9..4.6|...|...------------`
Similarly in SE i counted at least 15 forcing chains (2 with 8.9). susser needs tabling with subsets disabled.

AFAIK suexrat only uses singles and guesses and counts the "nodes" (average for 100 solutions).
ravel

Posts: 998
Joined: 21 February 2006

Nice puzzle ravel! Here's one with 16 forcing chains, ER 9.0, or then you could of course solve it with one UR type 1:

Code: Select all
`2.59.....1..............96...3.........68.....8..54....4...8..7.5.1..82......7.41`

Next puzzle, suexrat9 rating 508, solves with hidden pairs:
Code: Select all
`...1.2.....3......9.84..2..5..8.3.1........6...17....4.49....8....3..7..8...5.4..`

It has several hidden pairs, you need at least three to avoid the forcing chains, not necessarily the same three. Only the one in column 2 must be used in all options, without it ER 8.8. Without using any subsets you get ER 9.1.

RW
RW
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This nice superior with some tight bottlenecks
Code: Select all
` . . 8 | . . 1 | . . 4   . 1 . | . 3 . | . 2 .   3 . . | 7 . . | 9 . .  -------+-------+------  . . 9 | 4 . 3 | . . 8   . 5 . | . . . | . 7 .   1 . . | . . 6 | 4 . .  -------+-------+------  . . 6 | . . 9 | . . 2   . . . | . 4 . | . 1 .   . . . | 2 . . | 5 . .   `

How much does it take to avoid ANY of the following:
Code: Select all
`1. X-wing2. Hidden Triple (Quad counterpart)3. Hidden double (Triple counterpart)`

tarek

tarek

Posts: 3366
Joined: 05 January 2006

Very nice puzzle tarek! The first pair is the hardest to avoid for SE, several forcing chains upto 8.4. SE also uses several forcing chains for the triplet, upto 7.1, and a few forcing chains for the x-wing, upto 7.1. At the point where the x-wing is to be applied there is also an APE and a turbot, but none of them helps. So in case you don't know how to use forcing chains, this puzzle has 3 bottlenecks!

RW
RW
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Posts: 1000
Joined: 16 March 2006

tarek wrote:How much does it take to avoid ANY of the following:
Code: Select all
`1. X-wing2. Hidden Triple (Quad counterpart)3. Hidden double (Triple counterpart)`

One Forcing Chain and one XY-Chain are all that's exceptional.

Code: Select all
`    b5  -  7     Locked Candidate (1)r4c8    =  5     [r4c8]=6 => [r5c1]=6 => [r5c3]=4 => [r6c2]=8 => [r6c3]=3 => [r6]=INVALIDr6  b6  -  39    Naked  Pair (row and box)    b3  -  7     Locked Candidate (1)    b7  -  4     Locked Candidate (1)r7      -  7     Locked Candidate (2)r6c3    <> 7     XY-Chain on [r2c3] (5-cells)`
daj95376
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Here is one that SE wants a forcing chain for, after a UR that isn't needed.
The trick is a grouped x-wing on sixes solves it. (Finned x-wing, in this case.)

Code: Select all
`8..13.4..5............796.17.8...5...5..4.1.24..3....8..9..8.......9.......4.5.79`
wapati
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Posts: 527
Joined: 13 September 2006

This one needs many methods, unless you use the UR1. <sigh>

Code: Select all
`. . 3 | 9 . 6 | 7 5 .. . . | . 8 . | . . .. 6 . | 7 . . | . 9 .---------------------3 . . | . 5 . | . . .. 9 5 | 2 . . | . . 46 . . | . 1 . | . . .---------------------. 2 . | 6 . . | . 7 .. . . | . 3 . | . . .. . 4 | 1 . 8 | 5 6 .`
wapati
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I have seen "skyscraper", a pattern belonging to grouped X-wing, shatter many puzzles that popular solvers rate much higher.

SS, using colors, I assume, handles it fine.

I don't use colors or chains, or other guesses.
(The less paperwork the better! )

Code: Select all
`9 3 . | . . . | . 7 .. 4 . | . . 1 | . . .. . . | . 9 . | 3 5 4---------------------. . 8 | . . 6 | . . .7 6 . | 5 . . | 1 . .. . 4 | . . 2 | . . .---------------------. . . | . 2 . | 6 1 8. 1 . | . . 9 | . . .3 8 . | . . . | . 9 .`
wapati
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Posts: 527
Joined: 13 September 2006

wapati, I'm afraid you misunderstood the purpose of this thread. None of the puzzles have what I would call 'tight bottlenecks'. As example your first puzzle:
Code: Select all
` *-----------* |8..|13.|4..| |5..|...|...| |...|.79|6.1| |---+---+---| |7.8|...|5..| |.5.|.4.|1.2| |4..|3..|..8| |---+---+---| |..9|..8|...| |...|.9.|...| |...|4.5|.79| *-----------*`

I stopped counting in SE after it had found 15 different forcing chains, rating < 7.3, that solve the puzzle in one step... This is not tight, it doesn't narrow the solving path and require the solver to find the only simple move that solves it. With 15 simple forcing chains, all equally simple to your suggested grouped x-wing, I could pretty much start a forcing chain at any candidate and solve it by that! I could probably not find a puzzle with more possible solving paths if I tried!

RW
RW
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RW wrote:I could probably not find a puzzle with more possible solving paths if I tried!

RW

Hmm, perhaps that is why I like patterns.

Any guess on how many MORE chains there are than patterns?

There may not be a first pattern, but a few usually solve a puzzle.

If a few patterns don't solve it, I go to the next one.

Chains, you can do them all day, SE shows that.

To get back to it, this is about using a shortcut.

It is not always a chain.

These I posted can be solved with a known method, or in a longer path with other known methods.

What I posted can be done with patterns.

I never post puzzles that require chains, they are guessing.

OK, I don't understand this thread. Sorry!
wapati
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wapati wrote:Hmm, perhaps that is why I like patterns.

Any guess on how many MORE chains there are than patterns?

No guess, it's a fact that there is as many chains as there is false candidates, if your willing to go far enough with your chaining.

wapati wrote:These I posted can be solved with a known method, or in a longer path with other known methods.

That is exactly the point of this thread, except that your puzzles didn't really require longer paths using chains.

Most hard puzzles are easier to solve for a human being if you don't restrict yourself to patterns but allow yourself to use all information provided by the grid, and perhaps some t&e related techniques. This is simply because there is so many more optional paths using chains. The puzzles I look for are Sudokus that don't provide a shorter, or even remotely as short solution using chains as when using a particular pattern. Puzzles where you get to a specific stage where you have to find the simple pattern, or else you'll be forced to do several monster chains to progress the puzzle. Puzzles like these should be interesting particulary to non-chain solvers like you, as they are puzzles where the pattern solver would actually beat the chaining solver. So the ultimate goal is to find a puzzle that you could solve with your UR or turbot fish, but carcul would require a page full of NL-notation to find a alternative solution. (How much do you need to get around the already posted subset puzzles, carcul?)

RW
RW
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