## Gurth's Puzzles

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

### Gurth's Puzzles

re: Gurth's puzzles

GA12

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

... might as well keep them together, and take up fewer threads.

Composing and solving this was my morning's fun. (It's now 10.30am).

It's no pearl, but I found it quite resistant and interesting. In my 4 nets, I was tempted to go beyond SSTS by some tempting leads.
gurth

Posts: 358
Joined: 11 February 2006
Location: Cape Town, South Africa

### Emerald Challenge

GA16Emerald

Code: Select all
` *-----------* |.7.|1.6|...| |..4|..2|...| |6..|.8.|..5| |---+---+---| |..9|..1|...| |.3.|...|...| |7..|...|...| |---+---+---| |...|...|...| |...|...|...| |...|...|...| *-----------*`

Emerald Challenge :

I redefine an Emerald as a puzzle having a symmetrical SOLUTION.

In this case (GA16) the symmetry is defined thus:
Taking 180-degree rotational symmetry, each digit n in the solution must be mirrored by 10-n. (Exactly as in Tso's famous puzzle used by Carcul in the "Riddle of Sho".) E.g. if there is a 6 at c1, there must be a 4 at g9.

Defining an Emerald this way, in terms of the Solution rather than the Clues, a lot of clues become redundant. Removing these redundant clues presents a new challenge. Can it be solved without relying entirely on a computer?

GA16 has a unique EMERALD solution. Can you find it?
gurth

Posts: 358
Joined: 11 February 2006
Location: Cape Town, South Africa

Emerald Uniqueness Rectangles and Parallelograms (EURs, EUPs) solve this puzzle nicely...
udosuk

Posts: 2698
Joined: 17 July 2005

### Emerald Challenge

Maybe I should explain what I have in mind with my Emerald Challenge. My letter to David, posted on the other forum, explains it all very well, so I might just as well post a copy of it here:

David, all these disputes ought to be seen in relation to the backward culture in which we find ourselves. On this forum, we are dished up with sub-9.0 puzzles as "unsolvable", and asked to solve them with backward, retarded, obsolete methods. On the other forum, people are composing, presenting and tackling puzzles in the range 9.7-10.0. A recipe for progress; ours is a recipe for stagnation.

All restrictive practices, bans on "Nishio, Bowman's, T&E etc" are counterproductive to progress.

One reason to oppose efficient techniques is Carcul's: sudoku puzzles become too easy if you allow them. The answer to this is : NOT to restrict procedures (that is always false, arbitrary and reactionary), but to devise more difficult puzzles for these geniuses.

I think I may have hit on a way to produce such puzzles: through my Emerald Challenge. (Which probably won't interest any of the brain-washed members of this forum). But I entertain still some hopes of the other forum. There seems to be a wider range of interests there. People are interested in all sorts of properties: shapes, touchability, rating systems etc.

David, note that I have disposed of all 11 new unsolvables, plus Ruud's #1/50000 which is a sight harder than any of them, with the greatest of ease using perfectly legitimate FN technique, even going to the unnecessary extravagance of proving uniqueness in each and every case. My work is there for all to study and learn from. I don't know what more you expect from me. ("If you provided the example..." you said.) But it all falls on deaf ears.

Once you have reached this level, I suggest facing the Emerald Challenge. This is still brand new in the world: I have offered only GA16 as a forerunner, a kindergarten introduction. I can promise, for the future, puzzles increasing to over 100 times the difficulty of all current sudokus.

_________________________

udosuk, re GA16: you seem to be the only one with any ideas on the subject at all. But I think you will find that the techniques you mention are not enough to solve this puzzle. Care to prove me wrong?
gurth

Posts: 358
Joined: 11 February 2006
Location: Cape Town, South Africa

gurth wrote:Emerald Challenge : ... Can it be solved without relying entirely on a computer?

It sure can, quite simple actually! Basic symmetry rules gives:
Code: Select all
` *-----------* |.7.|1.6|...| |..4|..2|...| |6..|.8.|..5| |---+---+---| |..9|..1|..3| |.3.|.5.|.7.| |7..|9..|1..| |---+---+---| |5..|.2.|..4| |...|8..|6..| |...|4.9|.3.| *-----------*`

Code: Select all
` *-----------* |.75|1.6|...| |..4|5.2|...| |6..|.8.|..5| |---+---+---| |..9|..1|..3| |.3.|.5.|.7.| |7..|9..|1..| |---+---+---| |5..|.2.|..4| |...|8.5|6..| |...|4.9|53.| *-----------*`

Now we have an symetrical UR:
If r46c5=73 => r37c46=37, which would give two symmetrical solutions => r46c5<>37

Advances the puzzle a bit and we get here:
Code: Select all
` *--------------------------------------------------------------------* |-238    7      5      | 1      9      6      | 2348   248   *28     | | 1389   189    4      | 5      7      2      | 389    1689  #1689   | | 6      129    12     | 3      8      4      | 7      129    5      | |----------------------+----------------------+----------------------| | 248    24568  9      | 7      46     1      | 248    24568  3      | | 14     3      16     | 2      5      8      | 49     7      69     | | 7      24568  268    | 9      46     3      | 1      24568  268    | |----------------------+----------------------+----------------------| | 5      189    3      | 6      2      7      | 89     189    4      | |%1249   1249   127    | 8      3      5      | 6      129   #1279   | |*28     268    2678   | 4      1      9      | 5      3     -278    | *--------------------------------------------------------------------*`

Simple elimination, r1c9&r9c1=28 (if r1c9=2 => r9c1=8, and the other way around), r1c1=r9c9<>28

Then have a look at the candidates for digit 1. We know that both r2c9 and r8c1 cannot hold digit 1, but there is a strong link on 1 in r28c9 => r8c1<>1. Due to symmetry we may also eliminate 9 from r2c9.

Using one strong link more we can do the same for digit 8:
[r9c1]-8-[r9c3]=8=[r6c3]-8-[r6c9]=8=[r1c9]
=> r9c1<>8
=> r9c1=2 => r1c9=8

Now we got here:
Code: Select all
` *--------------------------------------------------------------------* | 3      7      5      | 1      9      6      | 24     24     8      | | 189    89     4      | 5      7      2      | 3      169    16     | | 6      29    *12     | 3      8      4      | 7     -19     5      | |----------------------+----------------------+----------------------| | 48     24568  9      | 7      46     1      | 248    24568  3      | | 14     3      16     | 2      5      8      | 49     7      69     | | 7      24568  268    | 9      46     3      | 1      24568  26     | |----------------------+----------------------+----------------------| | 5      19     3      | 6      2      7      | 89    #18     4      | | 49     149    7      | 8      3      5      | 6     #12    *129    | | 2      68     68     | 4      1      9      | 5      3      7      | *--------------------------------------------------------------------*`

Forcing chain:
either r78c8=1 or r8c9=1 => r1c2<>1 => r3c3=1

in either case r3c8<>1, puzzle solved.

The last move is very similar to the two earlier eliminations, using a grouped link:
[r2c1]-1-[r3c3]=1=[r3c8]-1-[r78c8]=1=[r8c9]

I liked this emerald gurth, have you got any more of them?

RW
Last edited by RW on Wed Oct 18, 2006 10:03 am, edited 1 time in total.
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

RW wrote:It sure can, quite simple actually! Basic symmetry rules gives:(...) Which easily advances to: (...)

But, that puzzle you solved has a total of 59 solutions. Or that doesn't matter for the "Emerald Challenge"? If not, then it seems ridiculous to solve such a puzzle with logical arguments beyond those referring only to simmetry.

Carcul
Carcul

Posts: 724
Joined: 04 November 2005

carcul wrote:But, that puzzle you solved has a total of 59 solutions. Or that doesn't matter for the "Emerald Challenge"?

gurth wrote:In this case (GA16) the symmetry is defined thus:
Taking 180-degree rotational symmetry, each digit n in the solution must be mirrored by 10-n. (Exactly as in Tso's famous puzzle used by Carcul in the "Riddle of Sho".) E.g. if there is a 6 at c1, there must be a 4 at g9.

...

GA16 has a unique EMERALD solution.

All my deductions where based on the given information by Gurth. The puzzle might have 59 solutions that would apply to a normal sudoku, but 58 of them will not be solutions to an emerald.

carcul wrote:If not, then it seems ridiculous to solve such a puzzle with logical arguments beyond those referring only to simmetry.

Which logical argument did not refer to symmetry?

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

Forcing chains? Strong links? You must be kidding me...
Code: Select all
` *--------------------------------------------------------------------* | 2389   7      5      | 1      349    6      | 23489  2489   289    | | 1389   189    4      | 5      379    2      | 3789   1689   16789  | | 6      129    123    |*37     8     *347    | 23479  1249   5      | |----------------------+----------------------+----------------------| | 248    24568  9      | 267    467    1      | 248    24568  3      | | 1248   3      1268   | 26     5      48     | 2489   7      2689   | | 7      24568  268    | 9      346    348    | 1      24568  268    | |----------------------+----------------------+----------------------| | 5      1689   13678  |*367    2     *37     | 789    189    4      | | 12349  1249   1237   | 8      137    5      | 6      129    1279   | | 128    1268   12678  | 4      167    9      | 5      3      1278   | *--------------------------------------------------------------------*EUR r37c46 => r3c6=4, r7c4=6 *--------------------------------------------------------------------* |*238    7      5      | 1      9      6      | 2348   248   *28     | | 1389   189    4      | 5      7      2      | 389    1689   1689   | | 6      129    12     | 3      8      4      | 7      129    5      | |----------------------+----------------------+----------------------| | 248    24568  9      | 7      46     1      | 248    24568  3      | | 14     3      16     | 2      5      8      | 49     7      69     | | 7      24568  268    | 9      46     3      | 1      24568  268    | |----------------------+----------------------+----------------------| | 5      189    3      | 6      2      7      | 89     189    4      | | 1249   1249   127    | 8      3      5      | 6      129    1279   | |*28     268    2678   | 4      1      9      | 5      3     *278    | *--------------------------------------------------------------------*EUR r19c19 => r1c1=3, r9c9=7  *--------------------------------------------------------------------* | 3      7      5      | 1      9      6      | 248    248    28     | | 189    189    4      | 5      7      2      | 3      169    169    | | 6     *129    12     | 3      8      4      | 7     *19     5      | |----------------------+----------------------+----------------------| | 248    24568  9      | 7      46     1      | 248    24568  3      | | 14     3      16     | 2      5      8      | 49     7      69     | | 7      24568  268    | 9      46     3      | 1      24568  268    | |----------------------+----------------------+----------------------| | 5     *19     3      | 6      2      7      | 89    *189    4      | | 149    149    7      | 8      3      5      | 6      129    129    | | 28     268    268    | 4      1      9      | 5      3      7      | *--------------------------------------------------------------------*EUR r37c28 => r3c2=2, r7c8=8 *--------------------------------------------------* | 3    7    5    | 1    9    6    | 28   4    28   | | 89   89   4    | 5    7    2    | 3    16   16   | | 6    2    1    | 3    8    4    | 7    9    5    | |----------------+----------------+----------------| |*248  458  9    | 7    46   1    |*28   256  3    | | 1    3    6    | 2    5    8    | 4    7    9    | | 7    458 *28   | 9    46   3    | 1    256 *268  | |----------------+----------------+----------------| | 5    1    3    | 6    2    7    | 9    8    4    | | 49   49   7    | 8    3    5    | 6    12   12   | | 28   6    28   | 4    1    9    | 5    3    7    | *--------------------------------------------------*EUP r4c17+r6c39 => r4c1=4, r6c9=6Singles remain...`

As easy as breeze...
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:Forcing chains? Strong links? You must be kidding me...

Well, I think my eliminations can also be defined as Emerald seafood patterns, which aren't any more complicated than your eliminations. Without pms they are actually easier to spot.

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

RW wrote:Well, I think my eliminations can also be defined as Emerald seafood patterns, which aren't any more complicated than your eliminations. Without pms they are actually easier to spot.

RW, I think you're the undisputed world champion of solving sudoku without PMs... I'm lazy and will resort to solving with PMs... Where my EURs & EUPs are definitely much easier than your chains... Why would you say my eliminations are complicated? The EURs are as easy as any UR type 1...
udosuk

Posts: 2698
Joined: 17 July 2005

RW, btw when I tried to study your moves, it's really unclear to me...
(Sorry I'm not a fast follower of chains from smart people like you & Carcul...)
RW wrote:
Code: Select all
` *--------------------------------------------------------------------* |-238    7      5      | 1      9      6      | 2348   248   *28     | | 1389   189    4      | 5      7      2      | 389    1689  #1689   | | 6      129    12     | 3      8      4      | 7      129    5      | |----------------------+----------------------+----------------------| | 248    24568  9      | 7      46     1      | 248    24568  3      | | 14     3      16     | 2      5      8      | 49     7      69     | | 7      24568  268    | 9      46     3      | 1      24568  268    | |----------------------+----------------------+----------------------| | 5      189    3      | 6      2      7      | 89     189    4      | |%1249   1249   127    | 8      3      5      | 6      129   #1279   | |*28     268    2678   | 4      1      9      | 5      3     -278    | *--------------------------------------------------------------------*`

Simple elimination, r1c9&r9c1=28 (if r1c9=2 => r9c1=8, and the other way around), r1c1=r9c9<>28

Then have a look at the candidates for digit 1. We know that both r2c9 and r8c1 cannot hold digit 1, but there is a strong link on 1 in r28c9 => r8c1<>1. Due to symmetry we may also eliminate 9 from r2c9.

Using one strong link more we can do the same for digit 8:
[r9c1]-8-[r9c3]=8=[r6c3]-8-[r6c9]=8=[r1c9]
=> r9c1<> => r9c1=2 => r1c9=8

"r9c1<>" what? And sorry I don't see a strong link [r6c9]=8=[r1c9]... When did you eliminate the 8 from r2c9?

RW wrote:Now we got here:
Code: Select all
` *--------------------------------------------------------------------* | 3      7      5      | 1      9      6      | 24     24     8      | | 189    89     4      | 5      7      2      | 3      169    16     | | 6      29    *12     | 3      8      4      | 7     -19     5      | |----------------------+----------------------+----------------------| | 48     24568  9      | 7      46     1      | 248    24568  3      | | 14     3      16     | 2      5      8      | 49     7      69     | | 7      24568  268    | 9      46     3      | 1      24568  26     | |----------------------+----------------------+----------------------| | 5      19     3      | 6      2      7      | 89    #18     4      | | 49     149    7      | 8      3      5      | 6     #12    *129    | | 2      68     68     | 4      1      9      | 5      3      7      | *--------------------------------------------------------------------*`

Forcing chain:
either r78c8=1 or r8c9=1 => r1c2<>1 => r3c3=1

in either case r3c8<>1, puzzle solved.

The last move is very similar to the two earlier eliminations, using a grouped strong link:
[r8c9]-1-[r78c8]=1=[r3c8]-1-[r3c3]=1=[r2c1]

Same here! Why is there a strong link [r78c8]=1=[r3c8]? What about the 1 in r2c8?

Puzzled...
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:"r9c1<>" what? And sorry I don't see a strong link [r6c9]=8=[r1c9]... When did you eliminate the 8 from r2c9?

Oops, forgot to mention what I eliminated... Should of coure be r9c1<>1. The 8 from r2c9 gets eliminated by locked candidates after the earlier elimination r1c1<>28.
udosuk wrote:Same here! Why is there a strong link [r78c8]=1=[r3c8]? What about the 1 in r2c8?

Ah, mistake when writing the chain. Should be the other way around:
[r2c1]-1-[r3c3]=1=[r3c8]-1-[r78c8]=1=[r8c9]

Silly that I didn't happen to spot the shorter version [r2c1]-1-[r2c9]=1=[r8c9], same as the previous eliminations of candidate 1...

Then to define some emerald seafood:

Note: Digit 5 is very special in Emerald bay and is not affected by these rules.

Emerald Eel
Code: Select all
`...|...|...-..|...|..a...|...|...---+---+---...|...|......|...|......|...|...---+---+---...|...|...-..|...|..a...|...|...`

If there is a strong link on candidate A between to symmetrically placed cells in a row or column r(x)c(y), r(10-x)c(y), then we may eliminate candidate A from r(x)c(10-y) and r(10-x)(10-y) and it's symmetrical opposite 10-A from r(x)c(y) and r(10-x)c(y).

Emerald Turbot fish
Code: Select all
`...|...|..a...|...|......|...|...---+---+---...|...|......|...|.....-|...|..a---+---+---...|...|......|...|...a.a|...|...`

If there is two strong links on digit A that have one end in symmetrically opposite cells r(x)c(y), r(10-x)c(10-y) Then you may eliminate A from all cells that can see both the other cells of the strong links.

I apparently didn't use the full power of the Eel in my solution (second step, r8c1<>1). Same pattern of course also eliminates 1 from r2c1 and makes my last move unnecessary. After both eliminations by the Eel, the puzzle advances to a state where the puzzle solves with an Emerald Turbot fish on digits 2 and 8.

RW
RW
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Okay, here are the formal definitions of my EURs/EUPs... (On 2nd thought they shouldn't be tagged with "uniqueness" because the logic has nothing to do with whether the solution is unique or not... The eliminations are made because otherwise we cannot achieve a valid "Emerald/symmetrical" state... But since there seems to be another technique called the "ER" I'll keep the "U" but change the word to "Unlocking" since that's pretty much what it does on the puzzles...)

EURs: Emerald Unlocking Rectangles
Code: Select all
`.*.|...|.*....|...|......|...|...---+---+---...|...|......|...|......|...|...---+---+---...|...|......|...|....*.|...|.*.`

For all rectangles with corners lying on symmetrical pair of cells (i.e. centred at r5c5):

We must not allow all 4 of them to have the 2 candidates (A,10-A) only, otherwise we'll be forced to have 2 of the same values within a row/column.

Note: unlike URs, these rectangles don't need to involve exactly 2 boxes... The 4 corners can be in 4 different boxes (b1379)...

EUPs: Emerald Unlocking Parallelograms
Code: Select all
`...|...|......|...|......|...|...---+---+---.*.|...|*.....|...|.....*|...|.*.---+---+---...|...|......|...|......|...|...`

For all parallelograms with a pair of horizontal/vertical opposite lines, the other pair of opposite lines lying entirely within 2 corresponding boxes, and opposite vertices located in symmetrical pair of cells:

We must not allow all 4 of them to have the 2 candidates (A,10-A) only, otherwise we'll be forced to have 2 of the same values within a row/column/box.

Perhaps not as common as RW's eels etc, but these don't rely on strong links... Remains to see which will appear more in Gurth's other EC (Emerald Challenge) puzzles...
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:Okay, here are the formal definitions on my EURs/EUPs...

Hmm... IMHO the eliminations made in the puzzle are actually NEPs, Naked Emerald Pairs. One cell has to have one value, the symmetrically opposite has the other value, both values may be eliminated from all cells that can see both cells in the NEP. You could also give a more general description:
Code: Select all
`If a bivalue cell C has two symmetrically opposite values A,10-A you may eliminate candidates A and 10-A from all cells that can see both C and it's symmetrical opposite.`

I also think the rectangles could be easier defined as
Code: Select all
`No digit (except digit 5) may appear in an x-wing formation of 4 symmetrically opposite cells`

example of elimination:
Code: Select all
`...|.a.|......|aaa|......|.a.|...---+---+---...|.-.|......|.-.|......|.-.|...---+---+---...|.a.|......|aaa|......|.a.|...Eliminate a from r456c5`

Btw, just realized that my uniqueness argument was wrong, there wouldn't have been 2 symmetrical solutions, but no symmetrical solutions... Actually a good example of the pattern above.

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

RW wrote:Hmm... IMHO the eliminations made in the puzzle are actually NEPs, Naked Emerald Pairs. One cell has to have one value, the symmetrically opposite has the other value, both values may be eliminated from all cells that can see both cells in the NEP...

Geez what a great observation RW! OK, so RIP to EURs and EUPs... Up come the NEPs!

Wait a minute... Perhaps there is still some value to the EUR/Ps...

For example, suppose this is r4:

. . 149 | . 46 . | . 149 .

r4c5 cannot be 4... (can you see why?)

I think this EUP is equivalent to a type 2|3|4 (whatever) UR...
udosuk

Posts: 2698
Joined: 17 July 2005

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