## Making a puzzle harder by adding a number

Everything about Sudoku that doesn't fit in one of the other sections
TTHsieh wrote:
Code: Select all
`.1.....2..34.1....5.67..........238.1.......4.735..........87.2....4.61..9.....4. # 7.1/7.0/3.4.1...5.2..34.1....5.67..........238.1.......4.735..........87.2....4.61..9.....4. # 7.2/7.0/3.4`

This is a bug in the Sudoku Explainer.
With the 5 in r1c6 it gets here, when it uses a 7.2 chain:
Code: Select all
` *-----------------------------------------------------------* | 79    1     79    |#48    3     5     | 48    2     6     | | 28    3     4     |#28    1     6     | 9     57    57    | | 5     28    6     | 7    #29   #49    |#14    3     18    | |-------------------+-------------------+-------------------| | 469   456   59    |#14    67    2     | 3     8     17    | | 1     268   28    | 369   679   379   | 25    57    4     | |#24    7     3     | 5     8    #14    |#12    6     9     | |-------------------+-------------------+-------------------| | 346   456   15    | 136   56    8     | 7     9     2     | | 2378  58    28    | 39    4     379   | 6     1     358   | | 3678  9     157   | 1236  2567  137   | 58    4     358   | *-----------------------------------------------------------*`
But without the 5 it "sees" a "bidirectional Y-cycle" rated 6.9, which eliminates 2 from r2c1 (either r2c4 is 2 or r6c1) - i would do it with the 5 cell xy-chain included.

I did not look at other samples.
eleven

Posts: 1873
Joined: 10 February 2008

### Just destroy a Unique rectangle!

In my experience there is one constellation in which adding one clue may make a puzzle harder. Mostly you destroy in placing a number a Unique Rectangle. You then have to take resort to a Turbot-Fish, which is considered much harder to spot than a unique rectangle which is confined to two boxes, while Turbot-Fishes extend to 4 boxes.

Example:
Code: Select all
`9 . . . 2 . . . .. 8 . 4 . 3 . . .. . 7 . 6 . . . .. 6 . . 1 . . 3 .4 . 9 6 . 8 1 . 2. 5 . . 7 . . 9 .. . . . 8 . 7 . .. . . 5 . 6 . 8 .. . . . 4 . 2 . 9`
SE 6.6

This Sudoku can be solved with 2 Hidden Pair, 5 Pointing (Line vs. Box), a Naked Triplet, one Swordfish, a unique Rectangle Typ4 and 2 Turbotfishes.

Code: Select all
`9 . 5 . 2 . . . .. 8 . 4 . 3 . . .. . 7 . 6 . . . .. 6 . . 1 . . 3 .4 . 9 6 . 8 1 . 2. 5 . . 7 . . 9 .. . . . 8 . 7 . .. . . 5 . 6 . 8 .. . . . 4 . 2 . 9`
SE 6.6

Making the puzzle more symmetric by adding 5 in r1c3 we get a puzzle still with the same rating, but without the Swordfish and you have to use 1 Turbot-Fish and two Sky-Crapers instead. In this case the Unique Rectangle is not destroyed. I will bring an example where this happens soon.

With best wishes

Claudia
claudiarabia

Posts: 288
Joined: 14 May 2006

Both puzzles solve with (pairs, triple and) a single skyscraper in 3.

When programs rate swordfish easier than turbot fish (skyscraper, kite etc.), this only can have historical reasons. Turbot fish are easier to spot, much more common and mostly more effective than swordfish.
eleven

Posts: 1873
Joined: 10 February 2008

vidarino wrote:I'd like to add that you don't get any less information by adding that clue, though, despite that being what it looks like.

Without the extra clue, you have the information that the rectangle in question does *not* contain 4-9-9-4 or 9-4-4-9, and therefore you use that information to make a deduction.

With the added clue, however, you lose this information, replacing it with the knowledge about a single cell. Therefore you can not make the above deduction any more, and have to find alternative ways around the obstacle. The alternative way being much harder is coincidental.

I'd also like to add that the difficulty of the technique in itself is "SUBJECTIVE!!"

vidarino is absolutely right in two ways.
1. of course its coincidental because obviously that number would have had to go there anyways to keep the puzzle unique.

2. the extra techniques needed to solve the puzzle after the candidate was added may not be viewed in a subjective manner as harder ( or easier ).

consider how you got started in sudoku in the first place and you solved your first 10 puzzles. there is always a stage in the puzzle where the going gets a little rough. for some its in the beginning, for some its in the middle. the point is, no one has the same experience. and that is the key.

just because I think that finding ALS is hard (or easy) doesn't mean its hard ( or easy ) for someone else.

who says the puzzle is harder ( or easier ) if you add one number if the new puzzle needs an extra x-wing or UR to solve??

someone might find that "new" puzzle much easier ( or harder ). correct??

what gives anyone the right to tell me that a particular stage in the puzzle is harder ( or easier )than the last stage??
why is the point at which I need to use a UR suppose to be harder ( or easier ) than the point where I had to find that hidden single?
anyone??

because ultimately, the forum is driving towards making ALL techniques available for ALL to learn "OBJECTIVELY" so that no technique is viewed as harder than the other.
storm_norm

Posts: 85
Joined: 27 February 2008

### Making harder by adding a number

This time I took the way vice versa:

I created this puzzle:
Code: Select all
`. . 7 . 1 . . . .. 5 . 6 . 9 . . .8 . . . . . 4 . .. 6 . 2 . . . 5 .1 . . . 3 . . . 9. 9 . . . 7 . 4 .. . 2 . . . . . 8. . . 9 . 4 . 6 .. . . . 5 . 1 . .`

ER 7,1

Naked single
7 Pointing
X-Wing
2xy-wing
XYZ-Wing
Jellyfish
Turbot-Fish
2 forcing chains

By leaving away the 9 in r5c9 in the picture below the rating will drop to 4,7 with the following Strategies:

Code: Select all
`. . 7 . 1 . . . .. 5 . 6 . 9 . . .8 . . . . . 4 . .. 6 . 2 . . . 5 .1 . . . 3 . . . .. 9 . . . 7 . 4 .. . 2 . . . . . 8. . . 9 . 4 . 6 .. . . . 5 . 1 . .`
ER 4,7

5 hidden pairs
5x Pointing
X-Wing
XYZ-Wing (actually not really needed)
Unique Loop Type 4 (8 nodes)
claudiarabia

Posts: 288
Joined: 14 May 2006

storm_norm wrote:because ultimately, the forum is driving towards making ALL techniques available for ALL to learn "OBJECTIVELY" so that no technique is viewed as harder than the other.

There are objective ways of comparing the difficulty of two techniques. We can look at the number of cells you have to scan, the number of comparisons you have to make, the number of candidates you have to consider and other measures of the computation you have to go through in order to apply a particular tecahnique. That's why it's generally considered that finding hidden singles is easier than finding hidden pairs, triple or quads.

By this measure hidden sets, naked sets and simple fish (locked candidates, x wings, swordfish and jellyfish) should have the same difficulty ie naked pairs should be the same difficulty as x wings. But for some reason most people tend to find fish harder than hidden and naked sets. Maybe that has something to do with the way our eyes scan the grid. Also some people think naked singles are harder than some pairs.

It's true that in a particular puzzle occasionally a hard technique will jump out at you, but generally speaking it's safe to say that in order to find that technique, on average you have to make so many computations.

I think there's scope for creating a puzzle rating based on the computational steps required to solve the puzzle, although I see the difficulties posed by this too. If you start your search with rows as opposed to columns or boxes, you may reach a placement or elimination faster.
ab

Posts: 451
Joined: 06 September 2005

Very nice sample, claudiarabia.

This time i fully agree, that the second puzzle is easier, though not many solvers (humans or programs) will catch the uniqueness placement.
Code: Select all
` *--------------------------------------------------------------------* |#69     4      7      | 38     1      238    |#69     23     5      | | 23     5      13     | 6      4      9      | 2378   1378   237    | | 8      123   #69     | 357    27     35     | 4      13    #69     | |----------------------+----------------------+----------------------| | 37     6      4      | 2      9      18     | 378    5      137    | | 1      278    58     | 4      3      5*68   |#26789  78    #2679   | | 235    9      358    | 158    68     7      | 2368   4      1236   | |----------------------+----------------------+----------------------| | 4      37     2      | 137    67     136    | 5      9      8      | | 357    1378   1358   | 9      278    4      | 237    6      237    | |#69     378   #69     | 378    5      238    | 1      237    4      | *--------------------------------------------------------------------*`
eleven

Posts: 1873
Joined: 10 February 2008

Solving Caludiarabia's puzzle manually, I came up with basically the same two nice loops to solve it both times, which I guess means I took the ER 7.1 path both times. UR's are not in my arsenal yet, so of course I missed that. Fun puzzle, claudiarabia!

susume
susume

Posts: 19
Joined: 03 August 2008

i have an open question regarding this thread.

given the puzzles in here is it possible to utilize this as a solving technique. but in reverse of the question of this thread.

removing clues to make a puzzle easier from its current postion.

like removing a clue to set up reductions based on uniqueness or other wise. and then place the removed clue back into its position after following said decutions that where discoverd.

for example the opposit of the first post:

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

removing the 9 @ R3C6to utilize the ur it blocks.

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

UR (type 3) in r35c46 allows to eliminate 4 from r35c4
Some do, some teach, the rest look it up.

StrmCkr

Posts: 860
Joined: 05 September 2006

I suggest that removing a clue to make an easier technique available is a special case of adding candidate(s) in one or more cells to make an easier technique available.

Champagne asserts in the Ratings thread:
basically, you can freely add any candidate in a cell without disturbing the logic).

StrmCkr, what do you think? Could your thoughts be generalized to adding candidate(s) in any cell, whether given, solved, or unsolved?

susume
susume

Posts: 19
Joined: 03 August 2008

Another question about this thread, based on Denis Berthier's definition of confluence (see his post in the Ratings thread here). I gather confluence means, in part, that the order of steps does not matter - if you miss a placement or elimination at one point you can still make it at a later point. Denis asserts that uniqueness methods lack confluence - they are only possible at certain states in the puzzle. As he points out, the easier of Claudiarabia's pair of puzzles above relies on uniqueness, where the harder one does not.

My question is - does the harder puzzle of each pair on this thread add a clue that prevents using a uniqueness pattern? Or are there other types of patterns that are prevented by adding a clue that makes a puzzle harder?

If it is only uniqueness patterns that behave this way, one could look specifically for places where adding candidate(s) would make a uniqueness pattern available, to find opportunities to use StrmCkr's remove-a-clue idea.

susume
susume

Posts: 19
Joined: 03 August 2008

StrmCkr wrote:i have an open question regarding this thread.

given the puzzles in here is it possible to utilize this as a solving technique. but in reverse of the question of this thread.

removing clues to make a puzzle easier from its current postion.

like removing a clue to set up reductions based on uniqueness or other wise. and then place the removed clue back into its position after following said decutions that where discoverd.

for example the opposit of the first post:

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

removing the 9 @ R3C6to utilize the ur it blocks.

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

UR (type 3) in r35c46 allows to eliminate 4 from r35c4

I can see a problem with this suggestion. When you remove a clue how do you know you are left with a puzzle with a unique solution? If the puzzle no longer has a unique solution then you cannot logically apply uniqueness tests.
ab

Posts: 451
Joined: 06 September 2005

I agree, if you know you have removed a clue from r3c6 you cant then apply a uniqueness test to a UR which envolves this position.

At a pinch you might be able to expose a UR at another location by removing a clue.

It would open a can of worms if it were true

C
coloin

Posts: 1737
Joined: 05 May 2005

If you guys click back to p.1 & p.2 of this thread you'll find these issues have been discussed 2 years ago.

In a nutshell: if you remove a clue, you lose the knowledge if your current puzzle state still has a unique solution. Then blindly applying uniqueness techniques can get you into trouble, such as reaching a contradiction or into another (wrong) solution.

However, depending on your luck, you might reach a solution in which the cell where you removed a clue attains a value identical to the clue you removed. In that case you can be sure the solution you got is correct (because the original clueset, before you removed a clue, should guarantee you a unique solution).

Therefore my take on this approach is: it is definitely a practical way to obtain the solution of a puzzle but the logicality of it is as good as guessing/trial-and-error. If in a timed-competition it is certainly one option to try (but so as guessing).
udosuk

Posts: 2698
Joined: 17 July 2005

Indeed it has been mentioned.....but I think the question is whether a distant UR can be found after removing an [essential] clue.

This UR cant envolve the position of the removed clue and to be of any use it cant have been there before the clue removal.

If a UR can be found then the elimination is as valid as any uniqueness method.

I cant see it happening though !

Also, if a guess is able to confirm the removed clue, I perhaps think this doesnt confirm the guess was right.

C
coloin

Posts: 1737
Joined: 05 May 2005

PreviousNext